Compare sorting algorithms' performance

Measure a relative performance of sorting algorithms implementations.

Plot execution time vs. input sequence length dependencies for various implementation of sorting algorithm and different input sequence types (example figures).

Consider three type of input sequences:

ones: sequence of all 1's. Example: {1, 1, 1, 1, 1}
range: ascending sequence, i.e. already sorted. Example: {1, 2, 3, 10, 15}
shuffled range: sequence with elements randomly distributed. Example: {5, 3, 9, 6, 8}

Consider at least two different sorting functions (different algorithms or/and different implementation of the same algorithm).

For example, consider Bubble Sort, Insertion sort, Quicksort or/and implementations of Quicksort with different pivot selection mechanisms. Where possible, use existing implementations.

Preliminary subtask:

General steps:

Define sorting routines to be considered.
Define appropriate sequence generators and write timings.
Plot timings.
What conclusions about relative performance of the sorting routines could be made based on the plots?

AutoHotkey

This example is under development. It was marked thus on 16/1/2010. Please help complete the example.

<lang ahk>; BUGGY - FIX

Persistent
SingleInstance OFF

SetBatchLines, -1 SortMethods := "Bogo,Bubble,Cocktail,Counting,Gnome,Insertion,Merge,Permutation,Quick,Selection,Shell,BuiltIn" Gui, Add, Edit, vInput, numbers,separated,by,commas,without,spaces,afterwards Loop, PARSE, SortMethods, `, Gui, Add, CheckBox, v%A_LoopField%, %A_LoopField% Sort Gui, Add, Button, gTest, Test! Gui, Show,, SortTest! Return Test: SplashTextOn,,, Test Commencing Sleep 2500 SplashTextOff Gui, +OwnDialogs Gui, Submit, NoHide Loop, PARSE, SortMethods, `, { If (%A_LoopField%) { DllCall("QueryPerformanceCounter", "Int64 *", %A_LoopField%Begin) %A_LoopField%Out := %A_LoopField%Sort(Input) DllCall("QueryPerformanceCounter", "Int64 *", %A_LoopField%Time) %A_LoopField%End := %A_LoopField%Begin + %A_LoopField%Time %A_LoopField%Time -= %A_LoopField%Begin } } Time := "" Loop, PARSE, SortMethods, `, If (%A_LoopField%) Time .= A_LoopField . " Sort: " . %A_LoopField%Time . "`t`t" . %A_LoopField%Out . "`r`n" MsgBox,, Results!, %Time% Return

Sorting funtions (Bogo, Bubble, Cocktail, Counting, Gnome, Insertion, Merge, Permutation, Quick, Selection, Shell, BuiltIn)

BogoSort(var) { sorted := 1 Loop, Parse, var { current := A_LoopField rest := SubStr(var, A_Index) Loop, Parse, rest { If (current > A_LoopField) sorted := 0 } } While !sorted { sorted := 1 Loop, Parse, var, `, { current := A_LoopField rest := SubStr(var, A_Index) Loop, Parse, rest, `, { If (current > A_LoopField) sorted := 0 } }

Sort, var, D`, Random } Return var }

BubbleSort(var) { StringSplit, array, var, `, hasChanged = 1 size := array0 While hasChanged { hasChanged = 0 Loop, % (size - 1) { i := array%A_Index% aj := A_Index + 1 j := array%aj% If (j < i) { temp := array%A_Index% array%A_Index% := array%aj% array%aj% := temp hasChanged = 1 } } } Loop, % size sorted .= "," . array%A_Index% Return substr(sorted,2) }

CocktailSort(var) { StringSplit array, var, `, i0 := 1, i1 := array0 Loop { Changed = Loop % i1-- -i0 { j := i0+A_Index, i := j-1 If (array%j% < array%i%) t := array%i%, array%i% := array%j%, array%j% := t ,Changed = 1 } IfEqual Changed,, Break Loop % i1-i0++ { i := i1-A_Index, j := i+1 If (array%j% < array%i%) t := array%i%, array%i% := array%j%, array%j% := t ,Changed = 1 } IfEqual Changed,, Break } Loop % array0 sorted .= "," . array%A_Index% Return SubStr(sorted,2) }

CountingSort(var) { max := min := substr(var, 1, instr(var, ",")) Loop, parse, var, `, { If (A_LoopField > max) max := A_LoopField

Else If (A_LoopField < min) min := A_LoopField } Loop % max-min+1 i := A_Index-1, a%i% := 0 Loop, Parse, var, `, i := A_LoopField-min, a%i%++ Loop % max-min+1 { i := A_Index-1, v := i+min Loop % a%i% t .= "," v } Return SubStr(t,2) }

GnomeSort(var) { StringSplit, a, var, `, i := 2, j := 3 While i <= a0 { u := i-1 If (a%u% < a%i%) i := j, j := j+1 Else { t := a%u%, a%u% := a%i%, a%i% := t If (--i = 1) i := j, j++ } } Loop % a0 sorted .= "," . a%A_Index% Return SubStr(sorted,2) }

InsertionSort(var) { StringSplit, a, var, `, Loop % a0-1 { i := A_Index+1, v := a%i%, j := i-1 While j>0 and a%j%>v u := j+1, a%u% := a%j%, j-- u := j+1, a%u% := v } Loop % a0 sorted .= "," . a%A_Index% Return SubStr(sorted,2) }

MergeSort(var) { StringReplace, t, var, `,,, UseErrorLevel L := ((t = "") ? 0 : ErrorLevel+1) If (2 > L) Return var StringGetPos, p, var, `,, % "L" L//2 list0 := MergeSort(SubStr(var,1,p)) list1 := MergeSort(SubStr(var,p+2)) If (list0 = "") Return list1 Else If (list1 = "") Return list0 list := list0 i0 := (p0 := InStr(list,",",0,i:=p0+1)) ? SubStr(list,i,p0-i) : SubStr(list,i) list := list1 i1 := (p1 := InStr(list,",",0,i:=p1+1)) ? SubStr(list,i,p1-i) : SubStr(list,i) Loop { i := i0>i1 list .= "," i%i% If (p%i%) { list := list%i% i%i% := (p%i% := InStr(list,",",0,i:=p%i%+1)) ? SubStr(list,i,p%i%-i) : SubStr(list,i) } Else { i ^= 1 rtv := SubStr(list "," i%i% (p%i% ? "," SubStr(list%i%,p%i%+1) : ""), 2) } } Return rtv }

PermutationSort(var) { static a:="a",v:="v" StringSplit, a, var, `, v0 := a0 Loop %v0% v%A_Index% := A_Index unsorted := 0 Loop % %a%0-1 { i := %v%%A_Index%, j := A_Index+1, j := %v%%j% If (%a%%i% > %a%%j%) unSorted := 1 } While unSorted { i := %v%0, i1 := i-1 While %v%%i1% >= %v%%i% { --i, --i1 IfLess i1,1, Return 1 } j := %v%0 While %v%%j% <= %v%%i1% --j t := %v%%i1%, %v%%i1% := %v%%j%, %v%%j% := t, j := %v%0 While i < j t := %v%%i%, %v%%i% := %v%%j%, %v%%j% := t, ++i, --j unsorted := 0 Loop % %a%0-1 { i := %v%%A_Index%, j := A_Index+1, j := %v%%j% If (%a%%i% > %a%%j%) unSorted := 1 } } Loop % a0 i := v%A_Index%, sorted .= "," . a%i% Return SubStr(sorted,2) }

QuickSort(var) { StringSplit, list, var, `, If (list0 <= 1) Return list pivot := list1 Loop, Parse, var, `, { If (A_LoopField < pivot) less .= "," . A_LoopField Else If (A_LoopField > pivot) more .= "," . A_LoopField Else pivotlist .= "," . A_LoopField } less := QuickSort(substr(less,2)) more := QuickSort(substr(more,2)) Return substr(less,2) . pivotList . more }

SelectionSort(var) { StringSplit, a, var, `, Loop % a0-1 { i := A_Index, mn := a%i%, j := m := i Loop % a0-i { j++ If (a%j% < mn) mn := a%j%, m := j } t := a%i%, a%i% := a%m%, a%m% := t } Loop % a0 sorted .= "," . a%A_Index% Return SubStr(sorted,2) }

ShellSort(var) { StringSplit, a, var, `, inc := a0 While inc:=round(inc/2.2) Loop % a0-inc { i := A_Index+inc, t := a%i%, j := i, k := j-inc While j > inc && a%k% > t a%j% := a%k%, j := k, k -= inc a%j% := t } Loop % a0 s .= "," . a%A_Index% Return SubStr(s,2) }

BuiltInSort(var) { Sort, var, N D`, Return var }</lang>

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> HIMEM = PAGE + 2000000

     INSTALL @lib$+"SORTLIB"
     INSTALL @lib$+"TIMERLIB"
     Sort% = FN_sortinit(0,0)
     Timer% = FN_ontimer(1000, PROCtimer, 1)
     
     PRINT "Array size:", 1000, 10000, 100000
     @% = &2020A
     
     FOR patt% = 1 TO 4
       CASE patt% OF
         WHEN 1: PRINT '"Data set to all ones:"
         WHEN 2: PRINT '"Data ascending sequence:"
         WHEN 3: PRINT '"Data randomly shuffled:"
         WHEN 4: PRINT '"Data descending sequence:"
       ENDCASE
       
       FOR type% = 1 TO 6
         CASE type% OF
           WHEN 1: PRINT "Internal (lib)";
           WHEN 2: PRINT "Quicksort   ";
           WHEN 3: PRINT "Radix sort  ";
           WHEN 4: PRINT "Shellsort   ";
           WHEN 5: PRINT "Bubblesort  ";
           WHEN 6: PRINT "Insertion sort";
         ENDCASE
         
         FOR power% = 3 TO 5
           PROCsorttest(patt%, type%, 10^power%)
         NEXT
         PRINT
         
       NEXT type%
     NEXT patt%
     END
     
     DEF PROCsorttest(patt%, type%, size%)
     LOCAL a%(), C%, I%
     DIM a%(size%-1)
     
     CASE patt% OF
       WHEN 1: a%() = 1 : a%() = 1
       WHEN 2: FOR I% = 0 TO size%-1 : a%(I%) = I% : NEXT
       WHEN 3: FOR I% = 0 TO size%-1 : a%(I%) = I% : NEXT
         C% = RND(-123456) : REM Seed
         FOR I% = size% TO 2 STEP -1 : SWAP a%(I%-1),a%(RND(I%)-1) : NEXT
       WHEN 4: FOR I% = 0 TO size%-1 : a%(I%) = size%-1-I% : NEXT
     ENDCASE
     
     Start% = TIME
     ON ERROR LOCAL PRINT , "   >100.00" ; : ENDPROC
     CASE type% OF
       WHEN 1: C% = size% : CALL Sort%, a%(0)
       WHEN 2: PROCquicksort(a%(), 0, size%)
       WHEN 3: PROCradixsort(a%(), size%, 10)
       WHEN 4: PROCshellsort(a%(), size%)
       WHEN 5: PROCbubblesort(a%(), size%)
       WHEN 6: PROCinsertionsort(a%(), size%)
     ENDCASE
     PRINT , (TIME - Start%)/100;
     
     FOR I% = 0 TO size%-2
       IF a%(I%) > a%(I%+1) ERROR 100, "Sort failed!"
     NEXT
     ENDPROC
     
     DEF PROCtimer
     Start% += 0
     IF (TIME - Start%) > 10000 ERROR 111, ""
     ENDPROC
     
     DEF PROCbubblesort(a%(), n%)
     LOCAL i%, l%
     REPEAT
       l% = 0
       FOR i% = 1 TO n%-1
         IF a%(i%-1) > a%(i%) THEN
           SWAP a%(i%-1),a%(i%)
           l% = i%
         ENDIF
       NEXT
       n% = l%
     UNTIL l% = 0
     ENDPROC
     
     DEF PROCinsertionsort(a%(), n%)
     LOCAL i%, j%, t%
     FOR i% = 1 TO n%-1
       t% = a%(i%)
       j% = i%
       WHILE j%>0 AND t%<a%(ABS(j%-1))
         a%(j%) = a%(j%-1)
         j% -= 1
       ENDWHILE
       a%(j%) = t%
     NEXT
     ENDPROC
     
     DEF PROCquicksort(a%(), s%, n%)
     LOCAL l%, p%, r%, t%
     IF n% < 2 THEN ENDPROC
     t% = s% + n% - 1
     l% = s%
     r% = t%
     p% = a%((l% + r%) DIV 2)
     REPEAT
       WHILE a%(l%) < p% l% += 1 : ENDWHILE
       WHILE a%(r%) > p% r% -= 1 : ENDWHILE
       IF l% <= r% THEN
         SWAP a%(l%), a%(r%)
         l% += 1
         r% -= 1
       ENDIF
     UNTIL l% > r%
     IF s% < r% PROCquicksort(a%(), s%, r% - s% + 1)
     IF l% < t% PROCquicksort(a%(), l%, t% - l% + 1 )
     ENDPROC
     
     DEF PROCshellsort(a%(), n%)
     LOCAL h%, i%, j%, k%
     h% = n%
     WHILE h%
       IF h% = 2 h% = 1 ELSE h% DIV= 2.2
       FOR i% = h% TO n% - 1
         k% = a%(i%)
         j% = i%
         WHILE j% >= h% AND k% < a%(ABS(j% - h%))
           a%(j%) = a%(j% - h%)
           j% -= h%
         ENDWHILE
         a%(j%) = k%
       NEXT
     ENDWHILE
     ENDPROC
     
     DEF PROCradixsort(a%(), n%, r%)
     LOCAL d%, e%, i%, l%, m%, b%(), bucket%()
     DIM b%(DIM(a%(),1)), bucket%(r%-1)
     FOR i% = 0 TO n%-1
       IF a%(i%) < l% l% = a%(i%)
       IF a%(i%) > m% m% = a%(i%)
     NEXT
     a%() -= l%
     m% -= l%
     e% = 1
     WHILE m% DIV e%
       bucket%() = 0
       FOR i% = 0 TO n%-1
         bucket%(a%(i%) DIV e% MOD r%) += 1
       NEXT
       FOR i% = 1 TO r%-1
         bucket%(i%) += bucket%(i% - 1)
       NEXT
       FOR i% = n%-1 TO 0 STEP -1
         d% = a%(i%) DIV e% MOD r%
         bucket%(d%) -= 1
         b%(bucket%(d%)) = a%(i%)
       NEXT
       a%() = b%()
       e% *= r%
     ENDWHILE
     a%() += l%
     ENDPROC</lang>

Output:

Array size:               1000     10000    100000

Data set to all ones:
Internal (lib)            0.00      0.01      0.03
Quicksort                 0.02      0.27      3.18
Radix sort                0.01      0.05      0.53
Shellsort                 0.03      0.36      4.44
Bubblesort                0.00      0.01      0.09
Insertion sort            0.00      0.02      0.26

Data ascending sequence:
Internal (lib)            0.00      0.00      0.02
Quicksort                 0.02      0.15      1.82
Radix sort                0.02      0.18      2.10
Shellsort                 0.03      0.37      4.44
Bubblesort                0.00      0.01      0.09
Insertion sort            0.01      0.03      0.27

Data randomly shuffled:
Internal (lib)            0.00      0.02      0.44
Quicksort                 0.02      0.26      3.17
Radix sort                0.02      0.17      2.08
Shellsort                 0.04      0.73     11.57
Bubblesort                0.69     69.70   >100.00
Insertion sort            0.55     55.54   >100.00

Data descending sequence:
Internal (lib)            0.00      0.01      0.10
Quicksort                 0.01      0.15      1.90
Radix sort                0.02      0.17      2.06
Shellsort                 0.03      0.50      6.39
Bubblesort                0.95     94.77   >100.00
Insertion sort            1.11   >100.00   >100.00

C

(The reference example is Python)

Examples of sorting routines

We can use the codes in the category Sorting Algorithms; since these codes deal with integer arrays, we should change them a little. To accomplish this task I've also renamed them more consistently algorithm_sort; so we have e.g. bubble_sort, quick_sort and so on.

Sequence generators

csequence.h <lang c>#ifndef _CSEQUENCE_H

define _CSEQUENCE_H
include <stdlib.h>

void setfillconst(double c); void fillwithconst(double *v, int n); void fillwithrrange(double *v, int n); void shuffledrange(double *v, int n);

endif</lang>

csequence.c <lang c>#include "csequence.h"

static double fill_constant;

void setfillconst(double c) {

 fill_constant = c;

}

void fillwithconst(double *v, int n) {

 while( --n >= 0 ) v[n] = fill_constant;

}

void fillwithrrange(double *v, int n) {

 int on = n;
 while( --on >= 0 ) v[on] = n - on;

}

void shuffledrange(double *v, int n) {

 int on = n;
 fillwithrrange(v, n);
 while( --n >= 0 ) {
   int r = rand() % on;
   double t = v[n];
   v[n] = v[r];
   v[r] = t;
 }

}</lang>

Write timings

We shall use the code from Query Performance. Since the action is a generic function with a single argument, we need wrappers which encapsule each sorting algorithms we want to test.

writetimings.h <lang c>#ifndef _WRITETIMINGS_H

define _WRITETIMINGS_H
include "sorts.h"
include "csequence.h"
include "timeit.h"

/* we repeat the same MEANREPEAT times, and get the mean; this *should*

  give "better" results ... */

define MEANREPEAT 10.0
define BUFLEN 128
define MAKEACTION(ALGO) \

 int action_ ## ALGO (int size) {				\
   ALGO ## _sort(tobesorted, size);				\
   return 0; }

define MAKEPIECE(N) { #N , action_ ## N }

int action_bubble(int size); int action_shell(int size); int action_quick(int size); int action_insertion(int size); int action_merge(int size); int doublecompare( const void *a, const void *b ); int action_qsort(int size); int get_the_longest(int *a);

struct testpiece {

 const char *name;
 int (*action)(int);

}; typedef struct testpiece testpiece_t;

struct seqdef {

 const char *name;
 void (*seqcreator)(double *, int);

}; typedef struct seqdef seqdef_t;

endif</lang>

writetimings.c <lang c>#include <stdio.h>

include <stdlib.h>

include "writetimings.h"

double *tobesorted = NULL; const char *bname = "data_"; const char *filetempl = "%s%s_%s.dat"; int datlengths[] = {100, 200, 300, 500, 1000, 5000, 10000, 50000, 100000};

testpiece_t testpieces[] = { // MAKEPIECE(bubble),

 MAKEPIECE(shell),
 MAKEPIECE(merge),
 MAKEPIECE(insertion),
 MAKEPIECE(quick),
 MAKEPIECE(qsort),
 { NULL, NULL }

};

seqdef_t seqdefs[] = {

 { "c1", fillwithconst },
 { "rr", fillwithrrange },
 { "sr", shuffledrange },
 { NULL, NULL }

};

MAKEACTION(bubble) MAKEACTION(insertion) MAKEACTION(quick) MAKEACTION(shell)

int action_merge(int size) {

 double *res = merge_sort(tobesorted, size);
 free(res); /* unluckly this affects performance */
 return 0;

}

int doublecompare( const void *a, const void *b ) {

 if ( *(const double *)a < *(const double *)b ) return -1;
 else return *(const double *)a > *(const double *)b;

} int action_qsort(int size) {

 qsort(tobesorted, size, sizeof(double), doublecompare);
 return 0;

}

int get_the_longest(int *a) {

 int r = *a;
 while( *a > 0 ) {
   if ( *a > r ) r = *a;
   a++;
 }
 return r;

}

int main() {

 int i, j, k, z, lenmax;
 char buf[BUFLEN];
 FILE *out;
 double thetime;

 lenmax = get_the_longest(datlengths);
 printf("Bigger data set has %d elements\n", lenmax);
 tobesorted = malloc(sizeof(double)*lenmax);
 if ( tobesorted == NULL ) return 1;

 setfillconst(1.0);

 for(i=0; testpieces[i].name != NULL; i++) {
   for(j=0; seqdefs[j].name != NULL; j++) {
     snprintf(buf, BUFLEN, filetempl, bname, testpieces[i].name,

seqdefs[j].name);

     out = fopen(buf, "w");
     if ( out == NULL ) goto severe;
     printf("Producing data for sort '%s', created data type '%s'\n",

testpieces[i].name, seqdefs[j].name);

     for(k=0; datlengths[k] > 0; k++) {

printf("\tNumber of elements: %d\n", datlengths[k]); thetime = 0.0; seqdefs[j].seqcreator(tobesorted, datlengths[k]); fprintf(out, "%d ", datlengths[k]); for(z=0; z < MEANREPEAT; z++) { thetime += time_it(testpieces[i].action, datlengths[k]); } thetime /= MEANREPEAT; fprintf(out, "%.8lf\n", thetime);

     }
     fclose(out);
   }
 }

severe:

 free(tobesorted);
 return 0;

}</lang> This code produce several files with the following naming convention:

data_algorithm_sequence.dat

Where algorithm is one of the following: insertion, merge, shell, quick, qsort (the quicksort in the libc library) (bubble sort became too slow for longest sequences). Sequence is c1 (constant value 1), rr (reverse range), sr (shufled range). These data can be easily plotted by Gnuplot, which can also do fitting. Instead we do our fitting using Polynomial Fitting. <lang c>#include <stdio.h>

include <stdlib.h>
include <math.h>

include "polifitgsl.h"

define MAXNUMOFDATA 100

double x[MAXNUMOFDATA], y[MAXNUMOFDATA]; double cf[2];

int main() {

 int i, nod;
 int r;

 for(i=0; i < MAXNUMOFDATA; i++)
 {
   r = scanf("%lf %lf\n", &x[i], &y[i]);
   if ( (r == EOF) || (r < 2) ) break;
   x[i] = log10(x[i]);
   y[i] = log10(y[i]);
 }
 nod = i;

 polynomialfit(nod, 2, x, y, cf);
 printf("C0 = %lf\nC1 = %lf\n", cf[0], cf[1]);

 return 0;

}</lang> Here we search for a fit with C₀+C₁x "in the log scale", since we supposed the data, once plotted on a logscale graph, can be fitted by a line. We can use e.g. a shell one-liner to produce the parameters for the line for each data file previously output. In particular I've used the following

for el in *.dat ; do fitdata <$el >${el%.dat}.fd ; done

Plot timings and Figures

Once we have all the ".dat" files and associated ".fd", we can use Gnuplot to draw our data and think about conclusions (we could also use the idea in Plot x, y arrays, but it needs too much enhancements to be usable for this analysis). Here an example of such a draw for a single file (using Gnuplot)

gnuplot> f(x) = C0 + C1*x
gnuplot> set logscale xy
gnuplot> load 'data_quick_sr_u.fd'
gnuplot> set xrange [100:100000]
gnuplot> set key left
gnuplot> plot 10**f(log10(x)), 'data_quick_sr_u.dat'

(The _u.dat are produced by a modified version of the code in order to write timings in microseconds instead of seconds) We can easily write another shell script/one-liner to produce a single file driver for Gnuplot in order to produce all the graph we can be interested in. These graphs show that the linear (in log scale) fit do not always fit the data... I haven't repeated the tests; the problems are when the sequence length becomes huge; for some algorithm that uses extra memory (like implementation of the merge sort), this could depend on the allocation of the needed memory. Another extraneous factor could be system load (the CLOCK_MONOTONIC used by the timing function is system wide rather than per process, so counting time spent in other processes too?). The "most stable" algorithms seem to be quick sort (but not qsort, which indeed is just the libc quick sort, here not plotted!) and shell sort (except for reversed range).

Conclusion: we should repeat the tests...

D

<lang d>import std.stdio, std.algorithm, std.container, std.datetime,

      std.random, std.typetuple;

immutable int[] allOnes, sortedData, randomData;

static this() { // Initialize global Arrays.

   immutable size_t arraySize = 10_000;

   allOnes = new int[arraySize];
   //allOnes[] = 1;
   foreach (ref d; allOnes)
       d = 1;

   sortedData = new int[arraySize];
   foreach (immutable i, ref d; sortedData)
       d = i;

   randomData = new int[arraySize];
   foreach (ref d; randomData)
       d = uniform(0, int.max);

}

// BubbleSort:

void bubbleSort(T)(T[] list) {

   for (int i = list.length - 1; i > 0; i--)
       for (int j = i -1; j >= 0; j--)
           if (list[i] < list[j])
               swap(list[i], list[j]);

}

void allOnesBubble() {

   auto data = allOnes.dup;
   data.bubbleSort;
   assert(data.isSorted);

}

void sortedBubble() {

   auto data = sortedData.dup;
   data.bubbleSort;
   assert(data.isSorted);

}

void randomBubble() {

   auto data = randomData.dup;
   data.bubbleSort;
   assert(data.isSorted);

}

// InsertionSort:

void insertionSort(T)(T[] list) {

   foreach (immutable i, currElem; list) {
       size_t j = i;
       for (; j > 0 && currElem < list[j - 1]; j--)
           list[j] = list[j - 1];
       list[j] = currElem;
   }

}

void allOnesInsertion() {

   auto data = allOnes.dup;
   data.insertionSort;
   assert(data.isSorted);

}

void sortedInsertion() {

   auto data = sortedData.dup;
   data.insertionSort;
   assert(data.isSorted);

}

void randomInsertion() {

   auto data = randomData.dup;
   data.insertionSort;
   assert(data.isSorted);

}

// HeapSort:

void heapSort(T)(T[] data) {

   auto h = data.heapify;
   while (!h.empty)
       h.removeFront;

}

void allOnesHeap() {

   auto data = allOnes.dup;
   data.heapSort;
   assert(data.isSorted);

}

void sortedHeap() {

   auto data = sortedData.dup;
   data.heapSort;
   assert(data.isSorted);

}

void randomHeap() {

   auto data = randomData.dup;
   data.heapSort;
   assert(data.isSorted);

}

// Built-in sort:

void allOnesBuiltIn() {

   auto data = allOnes.dup;
   data.sort!q{a < b};
   assert(data.isSorted);

}

void sortedBuiltIn() {

   auto data = sortedData.dup;
   data.sort!q{a < b};
   assert(data.isSorted);

}

void randomBuiltIn() {

   auto data = randomData.dup;
   data.sort!q{a < b};
   assert(data.isSorted);

}

static void show(in TickDuration[4u] r) {

   alias args = TypeTuple!("usecs", int);
   writefln("    Bubble Sort:    %10d", r[0].to!args);
   writefln("    Insertion Sort: %10d", r[1].to!args);
   writefln("    Heap Sort:      %10d", r[3].to!args);
   writefln("    Built-in Sort:  %10d", r[2].to!args);

}

void main() {

   enum nRuns = 100;
   writeln("Timings in microseconds:");

   writeln("  Testing against all ones:");
   nRuns.benchmark!(allOnesBubble, allOnesInsertion,
                    allOnesHeap, allOnesBuiltIn).show;

   writeln("\n  Testing against sorted data.");
   nRuns.benchmark!(sortedBubble, sortedInsertion,
                    sortedHeap, sortedBuiltIn).show;

   writeln("\n  Testing against random data.");
   nRuns.benchmark!(randomBubble, randomInsertion,
                    randomHeap, randomBuiltIn).show;

}</lang>

Output:

Timings in microseconds:
  Testing against all ones:
    Bubble Sort:       7377065
    Insertion Sort:       5868
    Heap Sort:           25173
    Built-in Sort:       34538

  Testing against sorted data.
    Bubble Sort:       7370520
    Insertion Sort:       6006
    Heap Sort:           18127
    Built-in Sort:      176235

  Testing against random data.
    Bubble Sort:      27293705
    Insertion Sort:    3762374
    Heap Sort:           85053
    Built-in Sort:      218268

(With 10,000 elements in each array. A naive HeapSort seems faster than the built-in sort in all three cases.)

Erlang

The sort routines are borrowed from bubble sort, insertion sort and quick sort. Plots by eplot. Bubble sort does ones and ranges best. Insertion sort does reversed ranges best. Quick sort handles shuffled numbers best. <lang Erlang> -module( compare_sorting_algorithms ).

-export( [task/0] ).

task() -> Ns = [100, 1000, 10000], Lists = [{"ones", fun list_of_ones/1, Ns}, {"ranges", fun list_of_ranges/1, Ns}, {"reversed_ranges", fun list_of_reversed_ranges/1, Ns}, {"shuffleds", fun list_of_shuffleds/1, Ns}], Sorts = [{bubble_sort, fun bubble_sort:list/1}, {insertion_sort, fun sort:insertion/1}, {iquick_sort, fun quicksort:qsort/1}], Results = [time_list(X, Sorts) || X <- Lists], [file:write_file(X++".png", egd_chart:graph(Y, [{x_label, "log N"}, {y_label, "log ms"}])) || {X, Y} <- Results].

list_of_ones( N ) -> [1 || _X <- lists:seq(1, N)]. list_of_ranges( N ) -> [X || X <- lists:seq(1, N)]. list_of_reversed_ranges( N ) -> lists:reverse( list_of_ranges(N) ). list_of_shuffleds( N ) -> [random:uniform(N) || _X <- lists:seq(1, N)].

time_list( {List, List_fun, Values}, Sorts ) -> Results = [{Sort, time_sort(Sort_fun, List_fun, Values)} || {Sort, Sort_fun} <- Sorts], {List, Results}.

time_sort( Sort_fun, List_fun, Values ) -> [time(Sort_fun, List_fun, X) || X <- Values].

time( Fun, List_fun, N ) -> {Time, _Result} = timer:tc( fun() -> Fun( List_fun(N) ) end ), {math:log10(N), math:log10(Time)}. </lang>

Go

Library: gonum/plot

<lang go>package main

import (

   "log"
   "math/rand"
   "testing"
   "time"

   "github.com/gonum/plot"
   "github.com/gonum/plot/plotter"
   "github.com/gonum/plot/plotutil"
   "github.com/gonum/plot/vg"

)

// Step 1, sort routines. // These functions are copied without changes from the RC tasks Bubble Sort, // Insertion sort, and Quicksort.

func bubblesort(a []int) {

   for itemCount := len(a) - 1; ; itemCount-- {
       hasChanged := false
       for index := 0; index < itemCount; index++ {
           if a[index] > a[index+1] {
               a[index], a[index+1] = a[index+1], a[index]
               hasChanged = true
           }
       }
       if hasChanged == false {
           break
       }
   }

}

func insertionsort(a []int) {

   for i := 1; i < len(a); i++ {
       value := a[i]
       j := i - 1
       for j >= 0 && a[j] > value {
           a[j+1] = a[j]
           j = j - 1
       }
       a[j+1] = value
   }

}

func quicksort(a []int) {

   var pex func(int, int)
   pex = func(lower, upper int) {
       for {
           switch upper - lower {
           case -1, 0:
               return
           case 1:
               if a[upper] < a[lower] {
                   a[upper], a[lower] = a[lower], a[upper]
               }
               return
           }
           bx := (upper + lower) / 2
           b := a[bx]
           lp := lower
           up := upper
       outer:
           for {
               for lp < upper && !(b < a[lp]) {
                   lp++
               }
               for {
                   if lp > up {
                       break outer
                   }
                   if a[up] < b {
                       break
                   }
                   up--
               }
               a[lp], a[up] = a[up], a[lp]
               lp++
               up--
           }
           if bx < lp {
               if bx < lp-1 {
                   a[bx], a[lp-1] = a[lp-1], b
               }
               up = lp - 2
           } else {
               if bx > lp {
                   a[bx], a[lp] = a[lp], b
               }
               up = lp - 1
               lp++
           }
           if up-lower < upper-lp {
               pex(lower, up)
               lower = lp
           } else {
               pex(lp, upper)
               upper = up
           }
       }
   }
   pex(0, len(a)-1)

}

// Step 2.0 sequence routines. 2.0 is the easy part. 2.5, timings, follows.

func ones(n int) []int {

   s := make([]int, n)
   for i := range s {
       s[i] = 1
   }
   return s

}

func ascending(n int) []int {

   s := make([]int, n)
   v := 1
   for i := 0; i < n; {
       if rand.Intn(3) == 0 {
           s[i] = v
           i++
       }
       v++
   }
   return s

}

func shuffled(n int) []int {

   return rand.Perm(n)

}

// Steps 2.5 write timings, and 3 plot timings are coded together. // If write means format and output human readable numbers, step 2.5 // is satisfied with the log output as the program runs. The timings // are plotted immediately however for step 3, not read and parsed from // any formated output. const (

   nPts = 7    // number of points per test
   inc  = 1000 // data set size increment per point

)

var (

   p        *plot.Plot
   sortName = []string{"Bubble sort", "Insertion sort", "Quicksort"}
   sortFunc = []func([]int){bubblesort, insertionsort, quicksort}
   dataName = []string{"Ones", "Ascending", "Shuffled"}
   dataFunc = []func(int) []int{ones, ascending, shuffled}

)

func main() {

   rand.Seed(time.Now().Unix())
   var err error
   p, err = plot.New()
   if err != nil {
       log.Fatal(err)
   }
   p.X.Label.Text = "Data size"
   p.Y.Label.Text = "microseconds"
   p.Y.Scale = plot.LogScale{}
   p.Y.Tick.Marker = plot.LogTicks{}
   p.Y.Min = .5 // hard coded to make enough room for legend

   for dx, name := range dataName {
       s, err := plotter.NewScatter(plotter.XYs{})
       if err != nil {
           log.Fatal(err)
       }
       s.Shape = plotutil.DefaultGlyphShapes[dx]
       p.Legend.Add(name, s)
   }
   for sx, name := range sortName {
       l, err := plotter.NewLine(plotter.XYs{})
       if err != nil {
           log.Fatal(err)
       }
       l.Color = plotutil.DarkColors[sx]
       p.Legend.Add(name, l)
   }
   for sx := range sortFunc {
       bench(sx, 0, 1) // for ones, a single timing is sufficient.
       bench(sx, 1, 5) // ascending and shuffled have some randomness though,
       bench(sx, 2, 5) // so average timings on 5 different random sets.
   }

   if err := p.Save(5*vg.Inch, 5*vg.Inch, "comp.png"); err != nil {
       log.Fatal(err)
   }

}

func bench(sx, dx, rep int) {

   log.Println("bench", sortName[sx], dataName[dx], "x", rep)
   pts := make(plotter.XYs, nPts)
   sf := sortFunc[sx]
   for i := range pts {
       x := (i + 1) * inc
       // to avoid timing sequence creation, create sequence before timing
       // then just copy the data inside the timing loop.  copy time should
       // be the same regardless of sequence data.
       s0 := dataFunc[dx](x) // reference sequence
       s := make([]int, x)   // working copy
       var tSort int64
       for j := 0; j < rep; j++ {
           tSort += testing.Benchmark(func(b *testing.B) {
               for i := 0; i < b.N; i++ {
                   copy(s, s0)
                   sf(s)
               }
           }).NsPerOp()
       }
       tSort /= int64(rep)
       log.Println(x, "items", tSort, "ns") // step 2.5, write timings
       pts[i] = struct{ X, Y float64 }{float64(x), float64(tSort) * .001}
   }
   pl, ps, err := plotter.NewLinePoints(pts) // step 3, plot timings
   if err != nil {
       log.Fatal(err)
   }
   pl.Color = plotutil.DarkColors[sx]
   ps.Color = plotutil.DarkColors[sx]
   ps.Shape = plotutil.DefaultGlyphShapes[dx]
   p.Add(pl, ps)

}</lang>

Output:

Step 4, conclusions about relative performance of the sorting routines made based on the plots.

The plots show differences in best and worse case performance for the various data sets. Bubble and insertion sorts show very good best case performance with all one and ascending sequences, beating quicksort. Quicksort shows best case performance with the ascending sequence but worst case performance with the all one sequence.

On random data (triangles) insertion and bubble sort show worse performance than quicksort.

J

<lang j> NB. extracts from other rosetta code projects ts=: 6!:2, 7!:2@] radix =: 3 : 0 256 radix y

a=. #{. z =. x #.^:_1 y e=. (-a) {."0 b =. i.x x#.1{::(<:@[;([: ; (b, {"1) <@}./. e,]))&>/^:a [ z;~a-1 NB. , ([: ; (b, {:"1) <@(}:"1@:}.)/. e,])^:(#{.z) y,.z ) bubble=: (([ (<. , >.) {.@]) , }.@])/^:_ insertion=:((>: # ]) , [ , < #])/ sel=: 1 : 'x # [' quick=: 3 : 0

if.  1 >: #y do.  y
else.
 e=. y{~?#y
 (quick y <sel e),(y =sel e),quick y >sel e
end.

) gaps =: [: }: 1 (1+3*])^:(> {:)^:a:~ # insert =: (I.~ {. ]) , [ , ] }.~ I.~ gapinss =: #@] {. ,@|:@(] insert//.~ #@] $ i.@[) shell =: [: ; gapinss &.>/@(< ,~ ]&.>@gaps) builtin =: /:~

NB. characterization of the sorting algorithms.

sorts =: bubble`insertion`shell`quick`radix`builtin generators =: #&1`(i.@-)`(?.~) NB. data generators

round =: [: <. 1r2&+

ll =: (<_1 0)&{ NB. verb to extract lower left which holds ln data length lc =: (<_1 1)&{ NB. verb to fetch lower center which holds most recent time

NB. maximum_time characterize ln_start_size NB. characterize returns a rank 4 matrix with successive indexes for NB. algorithm, input arrangement, max number of tests in group, length time space characterize =: 4 : 0

 max_time =. x
 start =. 1 3{.<:y
 for_sort. sorts do.
   for_generator. generators do.                                           NB. limit time  and  paging prevention
     t =: }. (, (, [: ts 'sort@.0 (generator@.0)' , ":@round@^)@>:@ll) ^: ((lc < max_time"_) *. ll < 17"_) ^:_ start
     if. generator -: {.generators do.
       g =. ,:t
     else.
       g =. g,t
     end.
   end.
   if. sort -: {.sorts do.
     s =. ,:g
   else.
     s =. s,g
   end.
 end.

)

NB. character cell graphics

NB. From j phrases 10E. Approximation d3=: 1&,.@[ %.~ ] NB. a and b such that y is approx. a + b*x

NB. domain and range 0 to 14. D=:14

plot =: 1 : '(=/ round@(u&.(*&(D%<:y))))i.y' NB. function plot size points =: 4 : '1(<"1|:|.round y*D%~<:x)}0$~2#x' NB. size points x,:y

show =: [: |. [: '0'&~:@{:} ' ' ,: ":

plt =: 3 : 0 30 plt y NB. default size 30

n =. >:i.-# experiments =. <@(#~"1 (0&<)@{.)"2 y pts =. n +./ .*x&points@>experiments coef =. d3/@>experiments (_*pts) + n +./ .*1 0 2|:coef&(p."1) plot x ) </lang>

   a =: 1 characterize 5
   $a  NB. a has rank 4
6 3 13 3

   'l t s' =: |:a   NB. transpose moves length time space to leading dimension
   l =: |: <: l     NB. transpose restores order
   t =:	|: 12 +^. t NB. choose arbitrary time units so that ^. time is positive
   s =:	|: ^. s     NB. ln space

   NB. 6 groups of sort methods   with 3 arrangements of data    ---> exponentially increasing data lengths,
   6j2":t   NB. ln time         negative infinity indicates avoided experiment
  3.83  4.80  5.89  7.15  8.65 10.18 11.90 13.75    __    __    __    __    __
  8.77 10.90 13.13    __    __    __    __    __    __    __    __    __    __
  8.70 10.87 13.11    __    __    __    __    __    __    __    __    __    __

  3.91  5.43  7.13  8.90 10.76 12.73    __    __    __    __    __    __    __
  3.99  5.63  7.42  9.21 11.13 13.06    __    __    __    __    __    __    __
  4.14  5.72  7.58  9.37 11.30 13.26    __    __    __    __    __    __    __

  5.56  6.75  7.90  9.05 10.27 11.60 13.13    __    __    __    __    __    __
  5.61  6.88  8.05  9.40 10.84 12.48    __    __    __    __    __    __    __
  5.67  6.93  8.09  9.43 10.89 12.48    __    __    __    __    __    __    __

  1.95  1.99  2.19  2.46  2.98  3.66  4.63  5.54  6.60  7.61  8.70  9.81 10.99
  6.06  7.13  8.09  9.09 10.10 11.10 12.12    __    __    __    __    __    __
  6.09  7.17  8.10  9.11 10.11 11.12 12.14    __    __    __    __    __    __

  3.25  3.33  3.55  4.06  4.77  5.60  6.72  7.75  8.75 10.11 11.21 12.22    __
  3.37  3.87  4.19  4.72  5.53  6.49  7.84  9.29 10.71 11.78 12.88    __    __
  3.42  4.00  4.43  5.07  5.93  7.07  8.06  9.76 10.96 12.10    __    __    __

  0.38  0.38  0.58  1.02  1.68  2.68  3.45  4.42  5.42  6.51  7.64  8.86  9.87
  0.49  0.58  0.96  1.55  2.34  3.11  4.31  5.82  7.43  8.70  9.75 10.75 11.75
  1.40  2.01  2.88  3.87  4.92  5.92  7.05  8.18  9.45 10.91 12.01    __    __

This display is no less than a bar chart and, frankly, suffices but for the curve fit requirement.

   NB. algorithms: bubble 6,  insertion 5,  shell 4,  quick 3,  radix 2,  builtin 1
   NB. rows:  log time
   NB. cols:  log size

   NB. data is all 1
   show 30 plt l ,: & (0&{)"2 t
                        5   4 _                       2 2  
                          4                         2      
                      5 _   6                     2        
                    _   4 6                     2          
                      4 _                     _           3
                  5 _   6                   2           3  
                      6                   _       _   3   1
                _ 4                     2         3 3   1  
                _   _                 _         3     1    
              1   6                 2         _   _ 1      
            _                     2         3   1          
            _   _               2 _       _   _            
          4                   2         3   1              
        _ 5   6             2 _     3 _   _                
      4 _   _             2       3     1                  
    _                   _       3 _   _                    
      5   6           2       3     1                      
  4     _           _       3 _   1                        
4   _             2       3     1 _                        
    _ 6       2 _     3 3   1 1                            
  5 6       2       3   _ 1   _                            
          2 _     3 _   1                                  
5 6 _   _       3     1 _                                  
6     2       3 _   1                                      
    2   _   _     1 _                                      
  2 _   3 3     1                                          
2     3       1 _                                          
    3       _                                              
  3 _   _ 1                                                
3     1 1                                                  

   NB. data is  reversed integer list
   show 30 plt l ,: & (1&{)"2 t
          6               4     3                         1
                      5 4     3               2         1  
        _           _ 4     3             _ 2         1    
                    _     3               2         1      
      6             4   _               2         _        
                  1   3               _                    
    _           _   _               2           1          
    6           4 3               _           _            
              1 _               2           1              
  6         _ 3               2           _                
            _               2 _         1                  
6         1               2           _                    
        _ 5             2           1                      
      3               2 _         1                        
    _ 4 _           2             _                        
  3 _                           1                          
3     5           2 _         1                            
  4 _           2           1 _                            
4   5         2 _         1                                
            _           1                                  
  5     _ 2           1 _                                  
5   _   2           1                                      
      2           1 _                                      
    2                                                      
  2             _                                          
2             1                                            
            _                                              
        _ 1                                                
    _   1                                                  
      1                                                    

   NB. data is  random
   show 30 plt l ,: & (2&{)"2 t
          6           5   4     3             2   1        
                    _   4     3             2   1          
        _           5 4     3             2   1            
                    _     3             2   1              
      6           5 4   _             _   _                
                  4   3             2     1                
    _           _   _             _   _ 1                  
    6           4 3                   1                    
              1 _               2   1                      
  6         _ 3               _   _                        
            _               2   1                          
6         1               2   1                            
        _ 5             _   1 _                            
      3 _             2   1                                
    _ 4 5           2   1                                  
  3 _ 5           2 _ 1 _                                  
3 4             2                                          
    _         2 _   _                                      
4           _     1                                        
  5       2     _                                          
        _     1                                            
5   _ 2     _                                              
    2     1                                                
  2     _                                                  
2   _ 1                                                    
    1                                                      
  1                                                        
1

The first version of the code set only a time limit. The builtin sort violated this only when the data overflowed RAM into virtual space, causing a large jump in time affecting also the next data set as the OS restored itself. The timing might be interesting for some other exercise. Here, a maximum data size test was inserted. Arbitrary time is the reasonable choice without details of the J interpreter nor of specific hardware. The radix sort involves putting data directly into the right spot. It is quick!

The data fit curves of the character cell graph were combined with GCD +. function. This explains "1"s or other strange values where these curves intersect. Finally the scatter plots were multiplied by infinity and added to the best fit curves. The points didn't show up well using the same values as the curves.

Julia

Julia comes with the InsertionSort, MergeSort, and QuickSort routines built into the Base.Sort module. Here is a comparison using those system algorithms and with integers. <lang julia> function comparesorts(tosort)

   a = shuffle(["i", "m", "q"])
   iavg = mavg = qavg = 0.0
   for c in a
       if c == "i"
           iavg = sum(i -> @elapsed(sort(tosort, alg=InsertionSort)), 1:100) / 100.0
       elseif c == "m"
           mavg = sum(i -> @elapsed(sort(tosort, alg=MergeSort)), 1:100) / 100.0
       elseif c == "q"
           qavg = sum(i -> @elapsed(sort(tosort, alg=QuickSort)), 1:100) / 100.0
       end
   end
   iavg, mavg, qavg

end

allones = fill(1, 40000) sequential = collect(1:40000) randomized = collect(shuffle(1:40000))

comparesorts(allones) comparesorts(allones) iavg, mavg, qavg = comparesorts(allones) println("Average sort times for 40000 ones:") println("\tinsertion sort:\t$iavg\n\tmerge sort:\t$mavg\n\tquick sort\t$qavg")

comparesorts(sequential) comparesorts(sequential) iavg, mavg, qavg = comparesorts(sequential) println("Average sort times for 40000 presorted:") println("\tinsertion sort:\t$iavg\n\tmerge sort:\t$mavg\n\tquick sort\t$qavg")

comparesorts(randomized) comparesorts(randomized) iavg, mavg, qavg = comparesorts(randomized) println("Average sort times for 40000 randomized:") println("\tinsertion sort:\t$iavg\n\tmerge sort:\t$mavg\n\tquick sort\t$qavg") </lang>

Average sort times for 40000 ones:
        insertion sort: 0.00036058316000000005
        merge sort:     0.00040099004999999996
        quick sort      0.0003586394400000001
Average sort times for 40000 presorted:
        insertion sort: 0.0003141142199999999
        merge sort:     0.0007967360000000003
        quick sort      0.0005601127399999998
Average sort times for 40000 randomized:
        insertion sort: 0.2190664327599999
        merge sort:     0.0028818907399999986
        quick sort      0.0023325933899999997

Kotlin

This mostly reuses the code from the sorting sub-tasks except that:

1. All sorting functions have been adjusted where necessary so that they sort an IntArray 'in place'. This ensures that the timings are not affected by time spent copying arrays.

2. The bubble sort function, which is very slow when sorting 100,000 random numbers, has been optimized somewhat to try and reduce overall execution time, though the program is still taking about 5 minutes to run on my machine.

Unfortunately the code used to measure CPU time in the 'Time a function' sub-task no longer works properly on my present Windows 10 machine (many results are inexplicably zero). I've therefore had to use the Kotlin library function, measureNanoTime(), instead which measures system time elapsed. Consequently, the results are a bit erratic even when averaged over 10 runs.

Although it would be easy enough to plot the results graphically using an external library such as JFreePlot, there doesn't seem much point when we can no longer upload images to RC. I've therefore presented the results in tabular form on the terminal which is good enough to detect significant trends. <lang scala>// Version 1.2.31

import java.util.Random import kotlin.system.measureNanoTime

typealias Sorter = (IntArray) -> Unit

val rand = Random()

fun onesSeq(n: Int) = IntArray(n) { 1 }

fun ascendingSeq(n: Int) = shuffledSeq(n).sorted().toIntArray()

fun shuffledSeq(n: Int) = IntArray(n) { 1 + rand.nextInt(10 * n) }

fun bubbleSort(a: IntArray) {

   var n = a.size
   do {
       var n2 = 0
       for (i in 1 until n) {
           if (a[i - 1] > a[i]) {
               val tmp = a[i]
               a[i] = a[i - 1]
               a[i - 1] = tmp
               n2 = i
           }
       }
       n = n2
   } while (n != 0)

}

fun insertionSort(a: IntArray) {

   for (index in 1 until a.size) {
       val value = a[index]
       var subIndex = index - 1
       while (subIndex >= 0 && a[subIndex] > value) {
           a[subIndex + 1] = a[subIndex]
           subIndex--
       }
       a[subIndex + 1] = value
   }

}

fun quickSort(a: IntArray) {

   fun sorter(first: Int, last: Int) {
       if (last - first < 1) return
       val pivot = a[first + (last - first) / 2]
       var left = first
       var right = last
       while (left <= right) {
           while (a[left] < pivot) left++
           while (a[right] > pivot) right--
           if (left <= right) {
               val tmp = a[left]
               a[left] = a[right]
               a[right] = tmp
               left++
               right--
           }
       }
       if (first < right) sorter(first, right)
       if (left < last) sorter(left, last)
   }
   sorter(0, a.lastIndex)

}

fun radixSort(a: IntArray) {

   val tmp = IntArray(a.size)
   for (shift in 31 downTo 0) {
       tmp.fill(0)
       var j = 0
       for (i in 0 until a.size) {
           val move = (a[i] shl shift) >= 0
           val toBeMoved = if (shift == 0) !move else move
           if (toBeMoved)
               tmp[j++] = a[i]
           else {
               a[i - j] = a[i]
           }
       }
       for (i in j until tmp.size) tmp[i] = a[i - j]
       for (i in 0 until a.size) a[i] = tmp[i]
   }

}

val gaps = listOf(701, 301, 132, 57, 23, 10, 4, 1) // Marcin Ciura's gap sequence

fun shellSort(a: IntArray) {

   for (gap in gaps) {
       for (i in gap until a.size) {
           val temp = a[i]
           var j = i
           while (j >= gap && a[j - gap] > temp) {
               a[j] = a[j - gap]
               j -= gap
           }
           a[j] = temp
       }
   }

}

fun main(args: Array<String>) {

   val runs = 10
   val lengths = listOf(1, 10, 100, 1_000, 10_000, 100_000)
   val sorts = listOf<Sorter>(
       ::bubbleSort, ::insertionSort, ::quickSort, ::radixSort, ::shellSort
   )

   /* allow JVM to compile sort functions before timings start */
   for (sort in sorts) sort(intArrayOf(1))

   val sortTitles = listOf("Bubble", "Insert", "Quick ", "Radix ", "Shell ")
   val seqTitles = listOf("All Ones", "Ascending", "Shuffled")
   val totals = List(seqTitles.size) { List(sorts.size) { LongArray(lengths.size) } }
   for ((k, n) in lengths.withIndex()) {
       val seqs = listOf(onesSeq(n), ascendingSeq(n), shuffledSeq(n))
       repeat(runs) {
           for (i in 0 until seqs.size) {           
               for (j in 0 until sorts.size) {
                   val seq = seqs[i].copyOf()
                   totals[i][j][k] += measureNanoTime { sorts[j](seq) }
               }
           }
       }
   }
   println("All timings in micro-seconds\n")
   print("Sequence length")
   for (len in lengths) print("%8d   ".format(len))
   println("\n")
   for (i in 0 until seqTitles.size) {
       println("  ${seqTitles[i]}:")
       for (j in 0 until sorts.size) {
           print("    ${sortTitles[j]}     ")
           for (k in 0 until lengths.size) {
               val time = totals[i][j][k] / runs / 1_000
               print("%8d   ".format(time))
           }
           println()
       }
       println("\n")
   }

}</lang>

Output:

All timings in micro-seconds

Sequence length       1         10        100       1000      10000     100000

  All Ones:
    Bubble            1          2          6         24         26        264
    Insert            1         16         10         14         48        518
    Quick             2          7         18         46        397       5181
    Radix            38         79        501       3720        864       9096
    Shell            11         15         43        189        407       4105


  Ascending:
    Bubble            1          2          6          8         24        270
    Insert            0          2          9         14         47        496
    Quick             1          6         19         33        282       3347
    Radix            38         71        264        415       1869      21403
    Shell             7         10         42        171        399       4052


  Shuffled:
    Bubble            1          5        436       3292     275224   27730705
    Insert            0          3        176        754      24759    2546180
    Quick             1          7         24        106       1281      14982
    Radix            28         73        622        317       1891      21617
    Shell            11         19         88        408       1946      36980

Conclusions

As expected quick sort is faster than the other methods when applied to random data of a reasonable size though radix and shell sort are also respectable performers for large amounts of random data. In contrast, bubble and insertion sorts are orders of magnitude slower, particularly the former.

On the other hand, bubble and insertion sorts are much quicker than the other methods for constant data and for data which is already sorted in an ascending direction, bubble sort being the faster of the two.

Mathematica/Wolfram Language

Comparing bubble and shell sort: <lang Mathematica>ClearAll[BubbleSort,ShellSort] BubbleSort[in_List]:=Module[{x=in,l=Length[in],swapped},swapped=True;

While[swapped,swapped=False;
Do[If[xi>xi+1,x[[{i,i+1}]]//=Reverse;
swapped=True;],{i,l-1}];];

x] ShellSort[lst_]:=Module[{list=lst,incr,temp,i,j},incr=Round[Length[list]/2];

While[incr>0,For[i=incr+1,i<=Length[list],i++,temp=listi;j=i;
While[(j>=(incr+1))&&(listj-incr>temp),listj=listj-incr;j=j-incr;];
listj=temp;];
If[incr==2,incr=1,incr=Round[incr/2.2]]];list

]

times=Table[

arr=ConstantArray[1,n];
t1={{n,AbsoluteTiming[BubbleSort[arr];]1},{n,AbsoluteTiming[ShellSort[arr];]1}};
arr=Sort[RandomInteger[{10^6},n]];
t2={{n,AbsoluteTiming[BubbleSort[arr];]1},{n,AbsoluteTiming[ShellSort[arr];]1}};
arr=RandomInteger[{10^6},n];
t3={{n,AbsoluteTiming[BubbleSort[arr];]1},{n,AbsoluteTiming[ShellSort[arr];]1}};
{t1,t2,t3}

,

{n,2^Range[13]}

];

ListLogLogPlot[Transpose@timesAll,1,PlotLegends->{"Bubble","Shell"},PlotLabel->"Ones"] ListLogLogPlot[Transpose@timesAll,2,PlotLegends->{"Bubble","Shell"},PlotLabel->"Ascending integers"] ListLogLogPlot[Transpose@timesAll,3,PlotLegends->{"Bubble","Shell"},PlotLabel->"Shuffled"]</lang>

Output:

Outputs three graphs on a log-log scales showing the size of the array and the time taken, for each of the three different arrays.

Nim

Translation of: Kotlin

This is a direct translation of the Kotlin program. We have added the sorting algorithm provided by Nim standard library which is a merge sort. For this reason, we have been constrained to annotate the sorting functions with the pragma {.locks: "unknown".} to make their type compatible with that of the standard sort function.

We have also added the array as first parameter of the internal function “sorter” as Nim compiler doesn’t allow direct access to this mutable array in function “quicksort” (memory safety violation).

<lang Nim>import algorithm import random import sequtils import times

Data.

proc oneSeq(n: int): seq[int] = repeat(1, n)

---------------------------------------------------------------------------------------------------

proc shuffledSeq(n: int): seq[int] =

 result.setLen(n)
 for item in result.mitems: item = rand(1..(10 * n))

---------------------------------------------------------------------------------------------------

proc ascendingSeq(n: int): seq[int] = sorted(shuffledSeq(n))

Algorithms.

func bubbleSort(a: var openArray[int]) {.locks: "unknown".} =

 var n = a.len
 while true:
   var n2 = 0
   for i in 1..<n:
     if a[i - 1] > a[i]:
       swap a[i], a[i - 1]
       n2 = i
   n = n2
   if n == 0: break

---------------------------------------------------------------------------------------------------

func insertionSort(a: var openArray[int]) {.locks: "unknown".} =

 for index in 1..a.high:
   let value = a[index]
   var subIndex = index - 1
   while subIndex >= 0 and a[subIndex] > value:
     a[subIndex + 1] = a[subIndex]
     dec subIndex
   a[subIndex + 1] = value

---------------------------------------------------------------------------------------------------

func quickSort(a: var openArray[int]) {.locks: "unknown".} =

 func sorter(a: var openArray[int]; first, last: int) =
   if last - first < 1: return
   let pivot = a[first + (last - first) div 2]
   var left = first
   var right = last
   while left <= right:
     while a[left] < pivot: inc left
     while a[right] > pivot: dec right
     if left <= right:
       swap a[left], a[right]
       inc left
       dec right
   if first < right: a.sorter(first, right)
   if left < last: a.sorter(left, last)

 a.sorter(0, a.high)

---------------------------------------------------------------------------------------------------

func radixSort(a: var openArray[int]) {.locks: "unknown".} =

 var tmp = newSeq[int](a.len)

 for shift in countdown(63, 0):
   for item in tmp.mitems: item = 0
   var j = 0
   for i in 0..a.high:
     let move = a[i] shl shift >= 0
     let toBeMoved = if shift == 0: not move else: move
     if toBeMoved:
       tmp[j] = a[i]
       inc j
     else:
       a[i - j] = a[i]
   for i in j..tmp.high: tmp[i] = a[i - j]
   for i in 0..a.high: a[i] = tmp[i]

---------------------------------------------------------------------------------------------------

func shellSort(a: var openArray[int]) {.locks: "unknown".} =

 const Gaps = [701, 301, 132, 57, 23, 10, 4, 1]

 for gap in Gaps:
   for i in gap..a.high:
     let temp = a[i]
     var j = i
     while j >= gap and a[j - gap] > temp:
       a[j] = a[j - gap]
       dec j, gap
     a[j] = temp

---------------------------------------------------------------------------------------------------

func standardSort(a: var openArray[int]) =

 a.sort()

Main code.

import strformat

const

 Runs = 10
 Lengths = [1, 10, 100, 1_000, 10_000, 100_000]

 Sorts = [bubbleSort, insertionSort, quickSort, radixSort, shellSort, standardSort]

const

 SortTitles = ["Bubble", "Insert", "Quick ", "Radix ", "Shell ", "Standard"]
 SeqTitles = ["All Ones", "Ascending", "Shuffled"]

var totals: array[SeqTitles.len, array[Sorts.len, array[Lengths.len, Duration]]]

randomize()

for k, n in Lengths:

 let seqs = [oneSeq(n), ascendingSeq(n), shuffledSeq(n)]
 for _ in 1..Runs:
   for i, s in seqs:
     for j, sort in Sorts:
       var s = s
       let t0 = getTime()
       s.sort()
       totals[i][j][k] += getTime() - t0

echo "All timings in microseconds\n" stdout.write "Sequence length " for length in Lengths:

 stdout.write &"{length:6d}     "

echo '\n' for i in 0..SeqTitles.high:

 echo &"  {SeqTitles[i]}:"
 for j in 0..Sorts.high:
   stdout.write &"  {SortTitles[j]:8s}     "
   for k in 0..Lengths.high:
     let time = totals[i][j][k].inMicroseconds div Runs
     stdout.write &"{time:8d}   "
   echo ""
 echo '\n'</lang>

Output:

All timings in microseconds

Sequence length       1         10        100       1000      10000     100000     

  All Ones:
  Bubble              0          0          0          0          6         64   
  Insert              0          0          0          1          9         90   
  Quick               0          0          3          9        105       1201   
  Radix               1          4         34        103        848       8354   
  Shell               0          0          2         10         97        946   
  Standard            0          2          2          6         45        380   


  Ascending:
  Bubble              0          0          0          0          6         61   
  Insert              0          0          0          1          9         94   
  Quick               0          0          3         11         88        919   
  Radix               1          5         47        154       1435      15519   
  Shell               0          0          2         10         95        954   
  Standard            0          0          2          7         47        463   


  Shuffled:
  Bubble              0          0         38       1026     133729   16181412   
  Insert              0          0          8        152      10010    1133210   
  Quick               0          0          9         63        607       7199   
  Radix               1          5         46        157       1405      15557   
  Shell               0          0          8         69        708      10236   
  Standard            0          0         18         96        992      12394

Conclusions

Compared to the results obtained by the Kotlin program, the radix sort seems less efficient and the shell sort more efficient. Maybe some optimizations could improve the radix sort, but it seems also that the shell sort is well optimized by the Nim compiler and the C compiler.

The standard sort behaves well if the list is already sorted. For random list, it is less efficient than the quick sort or the shell sort, but is still a good performer.

Phix

Library: Phix/pGUI

-- demo\rosetta\Compare_sorting_algorithms.exw
constant XQS = 01  -- (set to 1 to exclude quick_sort and shell_sort from ones)
 
include pGUI.e
 
Ihandle dlg, tabs, plot
Ihandles plots
 
function quick_sort2(sequence x)
    integer n = length(x)
    if n<2 then
        return x    -- already sorted (trivial case)
    end if
    integer mid = floor((n+1)/2),
            midn = 1
    object  midval = x[mid]
    sequence left = {}, right = {}
    x[mid] = x[1]
    for i=2 to n do
        object xi = x[i]
        integer c = compare(xi,midval)
        if c<0 then
            left = append(left,xi)
        elsif c>0 then
            right = append(right,xi)
        else
            midn += 1
        end if
    end for
 
    return quick_sort2(left) & repeat(midval,midn) & quick_sort2(right)
end function
 
function quick_sort(sequence s)
    sequence qstack = repeat(0,floor((length(s)/5)+10))   -- create a stack
    integer first = 1,
            last = length(s),
            stackptr = 0
    while true do
        while first<last do
            object pivot = s[floor(last+first)/2],
                   si, sj
            integer I = first,
                    J = last
            while true do
                while true do
                    si = s[I]
                    if si>=pivot then exit end if
                    I += 1
                end while
                while true do
                    sj = s[J]
                    if sj<=pivot then exit end if
                    J -= 1
                end while
                if I>J then exit end if
                if I<J then
                    if si=sj then
                        {I,J} = {J+1,I-1}
                        exit
                    end if
                    s[I] = sj
                    s[J] = si
                end if
                I += 1
                J -= 1
                if I>J then exit end if
            end while
            if I<last then
                qstack[stackptr+1] = I
                qstack[stackptr+2] = last
                stackptr += 2
            end if
            last = J
        end while
        if stackptr=0 then exit end if
        stackptr -= 2
        first = qstack[stackptr+1]
        last = qstack[stackptr+2]
    end while
    return s
end function
 
function radixSortn(sequence s, integer n)
    sequence buckets = repeat({},10)
    sequence res = {}
    for i=1 to length(s) do
        integer digit = remainder(floor(s[i]/power(10,n-1)),10)+1
        buckets[digit] = append(buckets[digit],s[i])
    end for
    for i=1 to length(buckets) do
        integer len = length(buckets[i])
        if len!=0 then
            if len=1 or n=1 then
                res &= buckets[i]
            else
                res &= radixSortn(buckets[i],n-1)
            end if
        end if
    end for
    return res
end function
 
function split_by_sign(sequence s)
    sequence buckets = {{},{}}
    for i=1 to length(s) do
        integer si = s[i]
        if si<0 then
            buckets[1] = append(buckets[1],-si)
        else
            buckets[2] = append(buckets[2],si)
        end if
    end for
    return buckets
end function
 
function radix_sort(sequence s)
-- NB this is an integer-only sort
    integer mins = min(s),
            passes = floor(log10(max(max(s),abs(mins))))+1
    if mins<0 then
        sequence buckets = split_by_sign(s)
        buckets[1] = reverse(sq_uminus(radixSortn(buckets[1],passes)))
        buckets[2] = radixSortn(buckets[2],passes)
        s = buckets[1]&buckets[2]
    else
        s = radixSortn(s,passes)
    end if
    return s
end function
 
function shell_sort(sequence s)
    integer gap = floor(length(s)/2)
    while gap>0 do
        for i=gap to length(s) do
            object temp = s[i]
            integer j = i-gap
            while j>=1 and temp<=s[j] do
                s[j+gap] = s[j]
                j -= gap
            end while
            s[j+gap] = temp
        end for
        gap = floor(gap/2)
    end while
    return s
end function
 
function shell_sort2(sequence x)
    integer last = length(x),
            gap = floor(last/10)+1
    while TRUE do
        integer first = gap+1
        for i=first to last do
            object xi = x[i]
            integer j = i-gap
            while TRUE do
                object xj = x[j]
                if xi>=xj then
                    j += gap
                    exit
                end if
                x[j+gap] = xj
                if j<=gap then
                    exit
                end if
                j -= gap
            end while
            x[j] = xi
        end for
        if gap=1 then
            return x
        else
            gap = floor(gap/3.5)+1
        end if
    end while
end function
 
function siftDown(sequence arr, integer s, integer last)
    integer root = s
    while root*2<=last do
        integer child = root*2
        if child<last and arr[child]<arr[child+1] then
            child += 1
        end if
        if arr[root]>=arr[child] then exit end if
        object tmp = arr[root]
        arr[root] = arr[child]
        arr[child] = tmp
        root = child
    end while
    return arr
end function
 
function heapify(sequence arr, integer count)
    integer s = floor(count/2)
    while s>0 do
        arr = siftDown(arr,s,count)
        s -= 1
    end while
    return arr
end function
 
function heap_sort(sequence arr)
    integer last = length(arr)
    arr = heapify(arr,last)
    while last>1 do
        object tmp = arr[1]
        arr[1] = arr[last]
        arr[last] = tmp
        last -= 1
        arr = siftDown(arr,1,last)
    end while
    return arr
end function
 
include builtins/sort.e
 
enum ONES = 1, SORTED = 2, RANDOM = 3, REVERSE = 4
 
constant tabtitles = {"ones","sorted","random","reverse"}
integer tabidx = 3
 
integer STEP
 
function tr(sequence name, integer rid=routine_id(name))
    return {name,rid}
end function
 
constant tests = {tr("quick_sort"),
                  tr("quick_sort2"),
                  tr("radix_sort"),
                  tr("shell_sort"),
                  tr("shell_sort2"),
                  tr("heap_sort"),
                  tr("sort"),           -- builtin
                 }
 
sequence results = repeat(repeat({}, length(tests)),length(tabtitles))
 
sequence dsdx = repeat(repeat(0,length(tests)),length(tabtitles))
 
integer ds_index
 
function idle_action_cb()
    atom best = -1, -- fastest last
         besti = 0, -- 1..length(tests) 
         bestt = 0, -- 1..length(tabtitles)
         len
    --
    -- Search for something to do, active/visible tab first.
    -- Any result set of length 0 -> just do one.
    -- Of all result sets<8, pick the lowest [$].
    --
    sequence todo = {tabidx}
    for t=1 to length(tabtitles) do
        if t!=tabidx then todo &= t end if
    end for
 
    for t=1 to length(tabtitles) do
        integer ti = todo[t]
        for i=1 to length(results[ti]) do
            len = length(results[ti][i])
            if len=0 then
                best = 0
                besti = i
                bestt = ti
                exit
            elsif len<8 then
                if (best=-1) or (best>results[ti][i][$]) then
                    best = results[ti][i][$]
                    besti = i
                    bestt = ti
                end if
            end if
        end for
        if (t=1) and (besti!=0) then exit end if
    end for
    if best>10 then
        -- cop out if it is getting too slow
        besti = 0
    end if
    if besti!=0 then
        STEP = iff(not XQS and bestt=ONES?1000:100000)
        len = (length(results[bestt][besti])+1)*STEP
        sequence test = iff(bestt=ONES?repeat(1,len):
                        iff(bestt=SORTED?tagset(len):
                        iff(bestt=RANDOM?shuffle(tagset(len)):
                        iff(bestt=REVERSE?reverse(tagset(len)):9/0))))
        ds_index = dsdx[bestt][besti]
        atom t0 = time()
        sequence check = call_func(tests[besti][2],{test})
        t0 = time()-t0
--      if check!=sort(test) then ?9/0 end if
        plot = plots[bestt]
        IupPlotInsert(plot, ds_index, -1, len, t0)
        results[bestt][besti] = append(results[bestt][besti],t0)
        IupSetAttribute(plot,"REDRAW",NULL)
        sequence progress = {bestt}
        for i=1 to length(results[bestt]) do
            progress &= length(results[bestt][i])
        end for
        IupSetStrAttribute(dlg,"TITLE","Compare sorting algorithms %s",{sprint(progress)})
        return IUP_CONTINUE
    end if
    IupSetAttribute(dlg,"TITLE","Compare sorting algorithms (all done, idle)")
    return IUP_IGNORE   -- all done, remove callback
end function
constant cb_idle_action = Icallback("idle_action_cb")
 
function tabchange_cb(Ihandle /*self*/, Ihandle /*new_tab*/)
    tabidx = IupGetInt(tabs,"VALUEPOS")+1
    plot = plots[tabidx]
    return IUP_DEFAULT;
end function
 
procedure main()
    IupOpen()
 
    plots = {}
    for i=1 to length(tabtitles) do
        if XQS then
--          results[ONES][1] = repeat(0,8)
            results[ONES][4] = repeat(0,8)
        end if
        plot = IupPlot()
        IupSetAttribute(plot,"MENUITEMPROPERTIES","YES")
        IupSetAttribute(plot,"TABTITLE",tabtitles[i])
        IupSetAttribute(plot,"GRID","YES")
        IupSetAttribute(plot,"MARGINLEFT","50")
        IupSetAttribute(plot,"MARGINBOTTOM","40")
        IupSetAttribute(plot,"LEGEND","YES")
        IupSetAttribute(plot,"LEGENDPOS","TOPLEFT")
--      IupSetAttribute(plot,"AXS_YSCALE","LOG10")
--      IupSetAttribute(plot,"AXS_XSCALE","LOG10")
        for j=1 to length(tests) do
            IupPlotBegin(plot)
            dsdx[i][j] = IupPlotEnd(plot)
            IupSetAttribute(plot,"DS_NAME",tests[j][1])
        end for
        plots = append(plots,plot)
    end for
    tabs = IupTabs(plots)
    IupSetCallback(tabs, "TABCHANGE_CB", Icallback("tabchange_cb"))
    dlg = IupDialog(tabs, "RASTERSIZE=800x480")
    IupSetAttribute(dlg, "TITLE", "Compare sorting algorithms")
    IupShow(dlg)
    IupSetInt(tabs, "VALUEPOS", tabidx-1)
    IupSetGlobalFunction("IDLE_ACTION", cb_idle_action)
    if platform()!=JS then
        IupMainLoop()
        IupClose()
    end if
end procedure
 
main()

Conclusions

I knew bubblesort and insertion sort would be bad, but not so bad that you cannot meaningfully plot them against better sorts. (logarithmic scale helps, but is still not enough) I had no idea that (these particular implementations of) quicksort and shellsort would be so bad on a sequence of all 1s. (so bad in fact that I had to cap that test length to 8,000 instead of 800,000 as used for the other tests) The builtin sort and shell_sort2 were the clear winners, until I found a non-recursive quicksort that seems quite good. IupPlot is brilliant! It is actually quite fun to watch the graphs grow as you get more results in! There is a point where you realise you are currently wasting your life fretting over 0.015 seconds...

The ultimate conclusion is, of course, that there are some differences, but as long as you weed out the really bad algorithms, and at least in the majority of cases, you will probably never notice whether sorting 800,000 items takes 0.25s or 0.1s - more significant gains are likely to be found elsewhere.

Python

Works with: Python version 2.5

Examples of sorting routines

<lang python>def builtinsort(x):

   x.sort()

def partition(seq, pivot):

  low, middle, up = [], [], []
  for x in seq:
      if x < pivot:
          low.append(x)
      elif x == pivot:
          middle.append(x)
      else:
          up.append(x)
  return low, middle, up

import random def qsortranpart(seq):

  size = len(seq)
  if size < 2: return seq
  low, middle, up = partition(seq, random.choice(seq))
  return qsortranpart(low) + middle + qsortranpart(up)</lang>

Sequence generators

<lang python>def ones(n):

   return [1]*n

def reversedrange(n):

   return reversed(range(n))

def shuffledrange(n):

   x = range(n)
   random.shuffle(x)
   return x</lang>

Write timings

<lang python>def write_timings(npoints=10, maxN=10**4, sort_functions=(builtinsort,insertion_sort, qsort),

                 sequence_creators = (ones, range, shuffledrange)):    
  Ns = range(2, maxN, maxN//npoints)
  for sort in sort_functions:
      for make_seq in sequence_creators:
          Ts = [usec(sort, (make_seq(n),)) for n in Ns]
          writedat('%s-%s-%d-%d.xy' % (sort.__name__,  make_seq.__name__, len(Ns), max(Ns)), Ns, Ts)</lang>

Where writedat() is defined in the Write float arrays to a text file, usec() - Query Performance, insertion_sort() - Insertion sort, qsort - Quicksort subtasks, correspondingly.

Plot timings

Library: matplotlib

Library: NumPy

<lang python>import operator import numpy, pylab def plotdd(dictplotdict):

  """See ``plot_timings()`` below."""
  symbols = ('o', '^', 'v', '<', '>', 's', '+', 'x', 'D', 'd',
             '1', '2', '3', '4', 'h', 'H', 'p', '|', '_')
  colors = list('bgrcmyk') # split string on distinct characters
  for npoints, plotdict in dictplotdict.iteritems():
      for ttle, lst in plotdict.iteritems():            
          pylab.hold(False)                                
          for i, (label, polynom, x, y) in enumerate(sorted(lst,key=operator.itemgetter(0))):
              pylab.plot(x, y, colors[i % len(colors)] + symbols[i % len(symbols)], label='%s %s' % (polynom, label))
              pylab.hold(True)
              y = numpy.polyval(polynom, x)
              pylab.plot(x, y, colors[i % len(colors)], label= '_nolegend_')                
          pylab.legend(loc='upper left')
          pylab.xlabel(polynom.variable)
          pylab.ylabel('log2( time in microseconds )')                
          pylab.title(ttle, verticalalignment='bottom')
          figname = '_%(npoints)03d%(ttle)s' % vars()
          pylab.savefig(figname+'.png')
          pylab.savefig(figname+'.pdf')
          print figname</lang>

See Plot x, y arrays and Polynomial Fitting subtasks for a basic usage of pylab.plot() and numpy.polyfit().

<lang python>import collections, itertools, glob, re import numpy def plot_timings():

  makedict = lambda: collections.defaultdict(lambda: collections.defaultdict(list))
  df = makedict()
  ds = makedict()
  # populate plot dictionaries
  for filename in glob.glob('*.xy'):
      m = re.match(r'([^-]+)-([^-]+)-(\d+)-(\d+)\.xy', filename)
      print filename
      assert m, filename
      funcname, seqname, npoints, maxN = m.groups()
      npoints, maxN = int(npoints), int(maxN)        
      a = numpy.fromiter(itertools.imap(float, open(filename).read().split()), dtype='f')
      Ns = a[::2]  # sequences lengths
      Ts = a[1::2] # corresponding times 
      assert len(Ns) == len(Ts) == npoints
      assert max(Ns) <= maxN
      #
      logsafe = numpy.logical_and(Ns>0, Ts>0)
      Ts = numpy.log2(Ts[logsafe])
      Ns = numpy.log2(Ns[logsafe])
      coeffs = numpy.polyfit(Ns, Ts, deg=1)
      poly = numpy.poly1d(coeffs, variable='log2(N)')
      #
      df[npoints][funcname].append((seqname, poly, Ns, Ts))
      ds[npoints][seqname].append((funcname, poly, Ns, Ts))
  # actual plotting
  plotdd(df)
  plotdd(ds) # see ``plotdd()`` above</lang>

Figures: log2( time in microseconds ) vs. log2( sequence length )

log(Time) vs. log(N): Relative performance on [1]*N as an input

log(Time) vs. log(N): Relative performance on range(N) as an input

log(Time) vs. log(N): Relative performance on random permutation of range(N) as an input

<lang python>sort_functions = [

   builtinsort,         # see implementation above
   insertion_sort,      # see Insertion sort
   insertion_sort_lowb, # insertion_sort, where sequential search is replaced
                        #     by lower_bound() function
   qsort,               # see Quicksort
   qsortranlc,          # qsort with randomly choosen pivot
                        #     and the filtering via list comprehension
   qsortranpart,        # qsortranlc with filtering via partition function
   qsortranpartis,      # qsortranpart, where for a small input sequence lengths
   ]                    #     insertion_sort is called

if __name__=="__main__":

  import sys
  sys.setrecursionlimit(10000)
  write_timings(npoints=100, maxN=1024, # 1 <= N <= 2**10 an input sequence length
                sort_functions=sort_functions,
                sequence_creators = (ones, range, shuffledrange))
  plot_timings()</lang>

Executing above script we get belowed figures.

ones

ones.png (143KiB)

builtinsort     - O(N)
insertion_sort  - O(N)
qsort           - O(N**2)
qsortranpart    - O(N)

range

range.png (145KiB)

builtinsort     - O(N)
insertion_sort  - O(N)
qsort           - O(N**2)
qsortranpart    - O(N*log(N))

shuffled range

shuffledrange.png (152KiB)

builtinsort     - O(N)  
insertion_sort  - O(N**4) ???
qsort           - O(N*log(N))
qsortranpart    - O(N) ???

REXX

<lang rexx>/*REXX pgm compares various sorts for 3 types of input sequences: ones/ascending/random.*/ parse arg ranges seed . /*obtain optional arguments from the CL*/ if ranges== | ranges=="," then ranges= 5 /*Not Specified" Then use the default.*/ if seed== | seed=="," then seed= 1946 /*use a repeatable seed for RANDOM BIF*/ if datatype(seed, 'W') then call random ,,seed /*Specified? Then use as a RANDOM seed*/ kinds= 3; #= 250; hdr= /*hardcoded/fixed number of datum kinds*/

  hdr= hdr || center( commas(#) "numbers", 25)'│'  /*(top) header for the output title.*/
    do ki=1  for kinds
    call gen@@ #, ki
    call set@;  call time 'R';  call bubble     #;     bubble.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call cocktail   #;   cocktail.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call cocktailSB #; cocktailSB.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call comb       #;       comb.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call gnome      #;      gnome.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call heap       #;       heap.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call insertion  #;  insertion.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call merge      #;      merge.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call pancake    #;    pancake.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call quick      #;      quick.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call radix      #;      radix.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call selection  #;  selection.ra.ki= format(time("E"),,2)
    call set@;  call time 'R';  call shell      #;      shell.ra.ki= format(time("E"),,2)
    end   /*ki*/
  #= # + #                                                         /*double # elements.*/
  end     /*ra*/

say; say; say /*say blank sep line*/ say center(' ', 11 ) "│"left(hdr, length(hdr)-1)"│" /*replace last char.*/

                           reps= ' allONES  ascend  random │'      /*build a title bar.*/

xreps= copies( center(reps, length(reps)), ranges) /*replicate ranges. */ creps= left(xreps, length(xreps)-1)"│" /*replace last char.*/ say center('sort type', 11 ) "│"creps; Lr= length(reps)

                                       xcreps= copies( left(, Lr-1, '─')"┼", ranges)

say center( call show 'bubble' call show 'cocktail' call show 'cocktailSB' call show 'comb' call show 'gnome' call show 'heap' call show 'insertion' call show 'merge' call show 'pancake' call show 'quick' call show 'radix' call show 'selection' call show 'shell' say translate(center( exit 0 /*─────── commas: parse arg ?; inOrder: parse arg n; do j=1 set@: @.=; /*─────── gen@@: procedure , 12, '─')"┼"left(xcreps, length(xcreps)-1)"┤" /* ◄──── show results for bubble sort.*/ /* ◄──── " " " cocktail " */ /*+Shifting Bounds*/ /* ◄──── " " " cocktailSB " */ /* ◄──── " " " comb " */ /* ◄──── " " " gnome " */ /* ◄──── " " " heap " */ /* ◄──── " " " insertion " */ /* ◄──── " " " merge " */ /* ◄──── " " " pancake " */ /* ◄──── " " " quick " */ /* ◄──── " " " radix " */ /* ◄──── " " " shell " */ /* ◄──── " " " shell " */ , 12, '─')"┴"left(xcreps, length(xcreps)-1)"┘", '┴', "┼") /*stick a fork in it, we're all done. */ ───────────────────────────────────────────────────────────────────────────────*/ do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? for n-1; k= j+1; if @.j>@.k then return 0; end; return 1 do a=1 for #; @.a= @@.a; end; return ───────────────────────────────────────────────────────────────────────────────*/ expose @@.; parse arg n,kind; nn= min(n, 100000) /*1e5≡REXX's max.*/

             do j=1 for nn;      select
                                 when kind==1  then  @@.j= 1               /*all ones. */
                                 when kind==2  then  @@.j= j               /*ascending.*/
                                 when kind==3  then  @@.j= random(, nn)    /*random.   */
                                 end   /*select*/
             end   /*j*/;                                              return

/*──────────────────────────────────────────────────────────────────────────────────────*/ show: parse arg aa; _= left(aa, 11) "│"

                               do   ra=1  for ranges
                                 do ki=1  for kinds
                                 _= _  right( value(aa || . || ra || . || ki),  7, ' ')
                                 end   /*k*/
                               _= _  "│"
                               end     /*r*/;       say _;             return

/*──────────────────────────────────────────────────────────────────────────────────────*/ bubble: procedure expose @.; parse arg n /*N: is the number of @ elements. */

           do m=n-1  by -1  until ok;         ok=1 /*keep sorting  @  array until done.*/
             do j=1  for m;  k=j+1;  if @.j<=@.k  then iterate    /*elements in order? */
             _=@.j;  @.j=@.k;  @.k=_;         ok=0 /*swap 2 elements; flag as not done.*/
             end   /*j*/
           end     /*m*/;                                              return

/*──────────────────────────────────────────────────────────────────────────────────────*/ cocktail: procedure expose @.; parse arg N; nn= N-1 /*N: is number of items. */

           do until done;   done= 1
               do j=1   for nn;                jp= j+1
               if @.j>@.jp  then do;  done=0;  _=@.j;  @.j=@.jp;  @.jp=_;  end
               end   /*j*/
           if done  then leave                              /*No swaps done?  Finished.*/
               do k=nn  for nn  by -1;         kp= k+1
               if @.k>@.kp  then do;  done=0;  _=@.k;  @.k=@.kp;  @.kp=_;  end
               end   /*k*/
           end       /*until*/;                                        return

/*──────────────────────────────────────────────────────────────────────────────────────*/ cocktailsb: procedure expose @.; parse arg N /*N: is number of items. */

                            end$= N - 1;     beg$= 1
           do while beg$ <= end$
           beg$$= end$;               end$$= beg$
               do j=beg$ to end$;                   jp= j + 1
               if @.j>@.jp  then do;  _=@.j;  @.j=@.jp;  @.jp=_;  end$$=j;  end
               end   /*j*/
           end$= end$$ - 1
               do k=end$  to beg$  by -1;           kp= k + 1
               if @.k>@.kp  then do;  _=@.k;  @.k=@.kp;  @.kp=_;  beg$$=k;  end
               end   /*k*/
           beg$= beg$$ + 1
           end       /*while*/;                                        return

/*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure expose @.; parse arg n /*N: is the number of @ elements. */

      g= n-1                                    /*G:  is the gap between the sort COMBs*/
            do  until g<=1 & done;    done= 1   /*assume sort is done  (so far).       */
            g= g * 0.8  % 1                     /*equivalent to:   g= trunc( g / 1.25) */
            if g==0  then g= 1                  /*handle case of the gap is too small. */
               do j=1  until $>=n;    $= j + g  /*$:     a temporary index  (pointer). */
               if @.j>@.$  then do;   _= @.j;     @.j= @.$;    @.$= _;    done= 0;    end
               end   /*j*/                      /* [↑]  swap two elements in the array.*/
            end      /*until*/;       return

/*──────────────────────────────────────────────────────────────────────────────────────*/ gnome: procedure expose @.; parse arg n; k= 2 /*N: is number items. */

         do j=3  while k<=n;                  p= k - 1           /*P: is previous item.*/
         if @.p<<=@.k  then do;      k= j;    iterate;   end     /*order is OK so far. */
         _= @.p;       @.p= @.k;     @.k= _                      /*swap two @ entries. */
         k= k - 1;     if k==1  then k= j;    else j= j-1        /*test for 1st index. */
         end    /*j*/;                                return

/*──────────────────────────────────────────────────────────────────────────────────────*/ heap: procedure expose @.; arg n; do j=n%2 by -1 to 1; call heapS j,n; end /*j*/

            do n=n  by -1  to 2;    _= @.1;    @.1= @.n;    @.n= _;   call heapS 1,n-1
            end   /*n*/;           return       /* [↑]  swap two elements; and shuffle.*/

heapS: procedure expose @.; parse arg i,n; $= @.i /*obtain parent.*/

            do  while i+i<=n;   j= i+i;   k= j+1;    if k<=n  then  if @.k>@.j  then j= k
            if $>=@.j  then leave;      @.i= @.j;    i= j
            end   /*while*/;            @.i= $;      return            /*define lowest.*/

/*──────────────────────────────────────────────────────────────────────────────────────*/ insertion: procedure expose @.; parse arg n

                  do i=2  to n;   $= @.i;       do j=i-1  by -1  to 1  while @.j>$
                                                _= j + 1;        @._= @.j
                                                end   /*j*/
                  _= j + 1;       @._= $
                  end   /*i*/;                                         return

/*──────────────────────────────────────────────────────────────────────────────────────*/ merge: procedure expose @. !.; parse arg n, L; if L== then do; !.=; L= 1; end

         if n==1  then return;     h= L + 1
         if n==2  then do; if @.L>@.h  then do; _=@.h; @.h=@.L; @.L=_; end; return;  end
         m= n % 2                                     /* [↑]  handle case of two items.*/
         call merge  n-m, L+m                         /*divide items  to the left   ···*/
         call merger m,   L,   1                      /*   "     "     "  "  right  ···*/
         i= 1;                     j= L + m
                    do k=L  while k<j                 /*whilst items on right exist ···*/
                    if j==L+n  |  !.i<=@.j  then do;     @.k= !.i;     i= i + 1;      end
                                            else do;     @.k= @.j;     j= j + 1;      end
                    end   /*k*/;                         return

merger: procedure expose @. !.; parse arg n,L,T

          if n==1  then do;  !.T= @.L;                                       return;  end
          if n==2  then do;  h= L + 1;   q= T + 1;  !.q= @.L;    !.T= @.h;   return;  end
          m= n % 2                                    /* [↑]  handle case of two items.*/
          call merge  m,   L                          /*divide items  to the left   ···*/
          call merger n-m, L+m, m+T                   /*   "     "     "  "  right  ···*/
          i= L;                     j= m + T
                    do k=T  while k<j                 /*whilst items on left exist  ···*/
                    if j==T+n  |  @.i<=!.j  then do;     !.k= @.i;     i= i + 1;      end
                                            else do;     !.k= !.j;     j= j + 1;      end
                    end   /*k*/;                         return

/*──────────────────────────────────────────────────────────────────────────────────────*/ pancake: procedure expose @.; parse arg n .; if inOrder(n) then return

            do n=n  by -1  for n-1
            != @.1;   ?= 1;                do j=2  to n;    if @.j<=!  then iterate
                                           != @.j;          ?= j
                                           end   /*j*/
            call panFlip ?;   call panFlip n
            end   /*n*/;                                               return

panFlip: parse arg y; do i=1 for (y+1)%2; yi=y-i+1; _=@.i; @.i=@.yi; @.yi=_; end; return /*──────────────────────────────────────────────────────────────────────────────────────*/ quick: procedure expose @.; a.1=1; parse arg b.1; $= 1 /*access @.; get #; define pivot.*/

      if inOrder(b.1)  then return
            do  while  $\==0;     L= a.$;    t= b.$;     $= $-1;   if t<2  then iterate
            H= L+t-1;             ?= L+t%2
            if @.H<@.L  then if @.?<@.H  then do;  p=@.H;  @.H=@.L;  end
                                         else if @.?>@.L  then p=@.L
                                                          else do;  p=@.?;  @.?=@.L;  end
                        else if @.?<@.L  then p=@.L
                                         else if @.?>@.H  then do;  p=@.H;  @.H=@.L;  end
                                                          else do;  p=@.?;  @.?=@.L;  end
            j= L+1;                           k=h
                   do forever
                       do j=j         while j<k & @.j<=p;  end     /*a tinie─tiny loop.*/
                       do k=k  by -1  while j<k & @.k>=p;  end     /*another   "    "  */
                   if j>=k  then leave                             /*segment finished? */
                   _= @.j;       @.j= @.k;             @.k= _      /*swap J&K elements.*/
                   end   /*forever*/
            $= $+1;            k= j-1;   @.L= @.k;     @.k= p
            if j<=?  then do;  a.$= j;   b.$= H-j+1;   $= $+1;  a.$= L;  b.$= k-L;    end
                     else do;  a.$= L;   b.$= k-L;     $= $+1;  a.$= j;  b.$= H-j+1;  end
            end          /*while $¬==0*/;              return

/*──────────────────────────────────────────────────────────────────────────────────────*/ radix: procedure expose @.; parse arg size,w; mote= c2d(' '); #= 1; !.#._n= size !.#._b= 1; if w== then w= 8 !.#._i= 1; do i=1 for size; y=@.i; @.i= right(abs(y), w, 0); if y<0 then @.i= '-'@.i

           end  /*i*/                                            /* [↑]  negative case.*/

    do  while #\==0;  ctr.= 0;  L= 'ffff'x;   low= !.#._b;   n= !.#._n;   $= !.#._i;   H=
    #= #-1                                                      /* [↑]   is the radix. */
          do j=low  for n;      parse var  @.j  =($)  _  +1;    ctr._= ctr._ + 1
          if ctr._==1 & _\==  then do;  if _<<L  then L=_;    if _>>H  then H=_
                                     end  /*  ↑↑                                       */
          end   /*j*/                     /*  └┴─────◄───  <<   is a strict comparison.*/
    _=                                    /*      ┌──◄───  >>    " "    "        "     */
    if L>>H  then iterate                 /*◄─────┘                                    */
    if L==H & ctr._==0  then do; #= #+1;  !.#._b= low;  !.#._n= n;  !.#._i= $+1;  iterate
                             end
    L= c2d(L);   H= c2d(H);      ?= ctr._ + low;        top._= ?;          ts= mote
    max= L
                 do k=L  to H;   _= d2c(k, 1);   c= ctr._  /* [↓]  swap 2 item radices.*/
                 if c>ts  then parse value  c k  with  ts max;     ?= ?+c;       top._= ?
                 end   /*k*/
    piv= low                                    /*set PIVot to the low part of the sort*/
            do  while piv<low+n
            it= @.piv
                       do forever;     parse var it  =($)  _  +1;         c= top._ -1
                       if piv>=c  then leave;   top._= c;    ?= @.c;    @.c= it;    it= ?
                       end   /*forever*/
            top._= piv;                         @.piv= it;        piv= piv + ctr._
            end   /*while piv<low+n */
    i= max
         do  until i==max;  _= d2c(i, 1);     i= i+1;     if i>H  then i= L;     d= ctr._
         if d<=mote  then do;         if d<2  then iterate;          b= top._
                            do k=b+1  for d-1;                       q= @.k
                              do j=k-1  by -1  to b  while q<<@.j;  jp= j+1;   @.jp= @.j
                              end   /*j*/
                                                                    jp= j+1;   @.jp= q
                            end     /*k*/
                          iterate
                          end
         #= #+1;       !.#._b= top._;       !.#._n= d;        !.#._i= $ + 1
         end   /*until i==max*/
    end        /*while #\==0 */

= 0 /* [↓↓↓] handle neg. and pos. arrays. */

       do i=size  by -1  for size;    if @.i>=0  then iterate;     #= #+1;      @@.#= @.i
       end   /*i*/
       do j=1  for size;   if @.j>=0  then do;  #= #+1;   @@.#= @.j;  end;    @.j= @@.j+0
       end   /*j*/                              /* [↑↑↑]  combine 2 lists into 1 list. */

return /*──────────────────────────────────────────────────────────────────────────────────────*/ selection: procedure expose @.; parse arg n

             do j=1  for n-1;         _= @.j;         p= j
                 do k=j+1  to n;      if @.k>=_  then iterate
                 _= @.k;              p= k      /*this item is out─of─order, swap later*/
                 end   /*k*/
             if p==j  then iterate              /*if the same, the order of items is OK*/
             _= @.j;     @.j= @.p;    @.p=      /*swap 2 items that're out─of─sequence.*/
             end       /*j*/;         return

/*──────────────────────────────────────────────────────────────────────────────────────*/ shell: procedure expose @.; parse arg N /*obtain the N from the argument list*/

      i= N % 2                                 /*%   is integer division in REXX.     */
                 do  while i\==0
                        do j=i+1  to N;    k= j;      p= k-i        /*P: previous item*/
                        _= @.j
                               do  while k>=i+1 & @.p>_;

                                   @.k= @.p;
                                   k= k-i;
                                   p= k-i
                               end   /*while k≥i+1*/
                        @.k= _
                        end         /*j*/
                 if i==2  then i= 1
                          else i= i * 5 % 11
                 end                  /*while i¬==0*/;                return</lang>

output when using the default inputs:

(Shown at 90% size.)

<pre>
            │       250 numbers       │       500 numbers       │      1,000 numbers      │      2,000 numbers      │      4,000 numbers      │
 sort type  │ allONES  ascend  random │ allONES  ascend  random │ allONES  ascend  random │ allONES  ascend  random │ allONES  ascend  random │
────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┤
bubble      │    0.00    0.00    0.06 │    0.02    0.00    0.27 │    0.00    0.00    1.05 │    0.00    0.00    4.30 │    0.00    0.02   17.78 │
cocktail    │    0.00    0.00    0.08 │    0.00    0.02    0.27 │    0.00    0.00    1.05 │    0.02    0.00    4.27 │    0.02    0.00   17.86 │
cocktailSB  │    0.00    0.00    0.05 │    0.00    0.00    0.22 │    0.00    0.00    0.88 │    0.00    0.02    3.47 │    0.00    0.02   14.17 │
comb        │    0.00    0.02    0.02 │    0.00    0.00    0.02 │    0.03    0.03    0.03 │    0.06    0.06    0.09 │    0.16    0.16    0.20 │
gnome       │    0.00    0.05    0.05 │    0.00    0.11    0.22 │    0.00    0.16    0.86 │    0.02    3.41    3.47 │    0.02    8.70   14.34 │
heap        │    0.02    0.02    0.02 │    0.00    0.03    0.03 │    0.03    0.06    0.05 │    0.03    0.11    0.13 │    0.09    0.25    0.23 │
insertion   │    0.00    0.00    0.03 │    0.00    0.00    0.11 │    0.00    0.00    0.47 │    0.00    0.00    1.83 │    0.02    0.02    7.52 │
merge       │    0.00    0.00    0.00 │    0.02    0.02    0.02 │    0.03    0.03    0.03 │    0.06    0.05    0.09 │    0.11    0.11    0.19 │
pancake     │    0.00    0.00    0.08 │    0.00    0.00    0.30 │    0.00    0.00    1.17 │    0.00    0.00    4.69 │    0.00    0.02   19.22 │
quick       │    0.00    0.00    0.00 │    0.00    0.00    0.00 │    0.00    0.00    0.02 │    0.00    0.02    0.05 │    0.02    0.00    0.09 │
radix       │    0.02    0.02    0.02 │    0.02    0.02    0.02 │    0.03    0.03    0.03 │    0.05    0.06    0.08 │    0.09    0.13    0.14 │
selection   │    0.02    0.02    0.02 │    0.08    0.08    0.09 │    0.28    0.31    0.34 │    1.25    1.30    1.47 │    4.95    4.77    5.34 │
shell       │    0.00    0.00    0.00 │    0.00    0.00    0.02 │    0.03    0.02    0.05 │    0.06    0.05    0.08 │    0.13    0.11    0.20 │
────────────┴─────────────────────────┴─────────────────────────┴─────────────────────────┴─────────────────────────┴─────────────────────────┘

Ruby

<lang ruby>class Array

 def radix_sort(base=10)       # negative value is inapplicable.
   ary = dup
   rounds = (Math.log(ary.max)/Math.log(base)).ceil
   rounds.times do |i|
     buckets = Array.new(base){[]}
     base_i = base**i
     ary.each do |n|
       digit = (n/base_i) % base
       buckets[digit] << n
     end
     ary = buckets.flatten
   end
   ary
 end
 
 def quick_sort
   return self  if size <= 1
   pivot = sample
   g = group_by{|x| x<=>pivot}
   g.default = []
   g[-1].quick_sort + g[0] + g[1].quick_sort
 end
 
 def shell_sort
   inc = size / 2
   while inc > 0
     (inc...size).each do |i|
       value = self[i]
       while i >= inc and self[i - inc] > value
         self[i] = self[i - inc]
         i -= inc
       end
       self[i] = value
     end
     inc = (inc == 2 ? 1 : (inc * 5.0 / 11).to_i)
   end
   self
 end
 
 def insertion_sort
   (1...size).each do |i|
     value = self[i]
     j = i - 1
     while j >= 0 and self[j] > value
       self[j+1] = self[j]
       j -= 1
     end
     self[j+1] = value
   end
   self
 end
 
 def bubble_sort
   (1...size).each do |i|
     (0...size-i).each do |j|
       self[j], self[j+1] = self[j+1], self[j]  if self[j] > self[j+1]
     end
   end
   self
 end

end

data_size = [1000, 10000, 100000, 1000000] data = [] data_size.each do |size|

 ary = *1..size
 data << [ [1]*size, ary, ary.shuffle, ary.reverse ]

end data = data.transpose

data_type = ["set to all ones", "ascending sequence", "randomly shuffled", "descending sequence"] print "Array size: " puts data_size.map{|size| "%9d" % size}.join

data.each_with_index do |arys,i|

 puts "\nData #{data_type[i]}:"
 [:sort, :radix_sort, :quick_sort, :shell_sort, :insertion_sort, :bubble_sort].each do |m|
   printf "%20s ", m
   flag = true
   arys.each do |ary|
     if flag
       t0 = Time.now
       ary.dup.send(m)
       printf "  %7.3f", (t1 = Time.now - t0)
       flag = false  if t1 > 2
     else
       print "   --.---"
     end
   end
   puts
 end

end</lang> Array#sort is a built-in method.

Output:

Array size:               1000    10000   100000  1000000

Data set to all ones:
                sort     0.000    0.001    0.005    0.043
          radix_sort     0.000    0.002    0.012    0.084
          quick_sort     0.000    0.002    0.020    0.197
          shell_sort     0.002    0.018    0.234    2.897
      insertion_sort     0.000    0.002    0.020    0.198
         bubble_sort     0.064    6.328   --.---   --.---

Data ascending sequence:
                sort     0.000    0.000    0.002    0.020
          radix_sort     0.001    0.010    0.128    1.546
          quick_sort     0.004    0.058    0.521    5.996
          shell_sort     0.001    0.019    0.234    2.882
      insertion_sort     0.000    0.002    0.021    0.195
         bubble_sort     0.065    6.453   --.---   --.---

Data randomly shuffled:
                sort     0.000    0.002    0.024    0.263
          radix_sort     0.001    0.011    0.126    1.529
          quick_sort     0.004    0.081    0.522    6.192
          shell_sort     0.003    0.033    0.498    5.380
      insertion_sort     0.027    2.627   --.---   --.---
         bubble_sort     0.122   11.779   --.---   --.---

Data descending sequence:
                sort     0.000    0.001    0.001    0.021
          radix_sort     0.001    0.012    0.125    1.560
          quick_sort     0.004    0.061    0.522    5.873
          shell_sort     0.003    0.028    0.316    3.829
      insertion_sort     0.053    5.298   --.---   --.---
         bubble_sort     0.206   17.232   --.---   --.---

Tcl

Background

The lsort command is Tcl's built-in sorting engine. It is implemented in C as a mergesort, so while it is theoretically slower than quicksort, it is a stable sorting algorithm too, which produces results that tend to be less surprising in practice. This task will be matching it against multiple manually-implemented sorting procedures.

Observations

Obviously, the built-in compiled sort command will be much faster than any Tcl-coded implementation. The Tcl-coded mergesort is up to 3 orders of magnitude slower.

The shellsort implementation suffers, relative to other algorithms, in the case where the list is already sorted.

Code

Library: Tk

Library: Tcllib (Package: struct::list)

<lang tcl>###############################################################################

measure and plot times

package require Tk package require struct::list namespace path ::tcl::mathfunc

proc create_log10_plot {title xlabel ylabel xs ys labels shapes colours} {

   set w [toplevel .[clock clicks]]
   wm title $w $title
   pack [canvas $w.c -background white]
   pack [canvas $w.legend -background white]
   update
   plot_log10 $w.c $w.legend $title $xlabel $ylabel $xs $ys $labels $shapes $colours
   $w.c config -scrollregion [$w.c bbox all]
   update

}

proc plot_log10 {canvas legend title xlabel ylabel xs ys labels shapes colours} {

   global xfac yfac
   set log10_xs [map {_ {log10 $_}} $xs]
   foreach series $ys {
       lappend log10_ys [map {_ {log10 $_}} $series]
   }
   set maxx [max {*}$log10_xs]
   set yvalues [lsort -real [struct::list flatten $log10_ys]]
   set firstInf [lsearch $yvalues Inf]
   set maxy [lindex $yvalues [expr {$firstInf == -1 ? [llength $yvalues] - 1 : $firstInf - 1}]]

   set xfac [expr {[winfo width $canvas] * 0.8/$maxx}]
   set yfac [expr {[winfo height $canvas] * 0.8/$maxy}]

   scale $canvas x 0 $maxx $xfac "log10($xlabel)"
   scale $canvas y 0 $maxy $yfac "log10($ylabel)" $maxx $xfac

   $legend create text 30 0 -text $title -anchor nw
   set count 1
   foreach series $log10_ys shape $shapes colour $colours label $labels {
       plotxy $canvas $log10_xs $series $shape $colour
       legenditem $legend [incr count] $shape $colour $label
   }

}

proc map {lambda list} {

   set res [list]
   foreach item $list {lappend res [apply $lambda $item]}
   return $res

}

proc legenditem {legend pos shape colour label} {

   set y [expr {$pos * 15}]
   $shape $legend 20 $y -fill $colour
   $legend create text 30 $y -text $label -anchor w

}

The actual plotting engine

proc plotxy {canvas _xs _ys shape colour} {

   global xfac yfac
   foreach x $_xs y $_ys {
       if {$y < Inf} {
           lappend xs $x
           lappend ys $y
       }
   }
   set coords [list]
   foreach x $xs y $ys {
       set coord_x [expr {$x*$xfac}]
       set coord_y [expr {-$y*$yfac}]
       $shape $canvas $coord_x $coord_y -fill $colour
       lappend coords $coord_x $coord_y
   }
   $canvas create line $coords -smooth true

}

Rescales the contents of the given canvas

proc scale {canvas direction from to fac label {other_to 0} {other_fac 0}} {

   set f [expr {$from*$fac}]
   set t [expr {$to*$fac}]
   switch -- $direction {
       x {
           set f [expr {$from * $fac}]
           set t [expr {$to * $fac}]
           # create x-axis
           $canvas create line $f 0 $t 0
           $canvas create text $f 0 -anchor nw -text $from
           $canvas create text $t 0 -anchor n -text [format "%.1f" $to]
           $canvas create text [expr {($f+$t)/2}] 0 -anchor n -text $label

       }
       y {
           set f [expr {$from * -$fac}]
           set t [expr {$to * -$fac}]
           # create y-axis
           $canvas create line 0 $f 0 $t
           $canvas create text 0 $f -anchor se -text $from
           $canvas create text 0 $t -anchor e -text [format "%.1f" $to]
           $canvas create text 0 [expr {($f+$t)/2}] -anchor e -text $label
           # create gridlines
           set xmax [expr {$other_to * $other_fac}]
           for {set i 1} {$i < $to} {incr i} {
               set y [expr {$i * -$fac}]
               $canvas create line 0 $y $xmax $y -dash .
           }
       }
   }

}

Helper to make points, which are otherwise not a native item type

proc dot {canvas x y args} {

   set id [$canvas create oval [expr {$x-3}] [expr {$y-3}] \
               [expr {$x+3}] [expr {$y+3}]]
   $canvas itemconfigure $id {*}$args

} proc square {canvas x y args} {

   set id [$canvas create rectangle [expr {$x-3}] [expr {$y-3}] \
               [expr {$x+3}] [expr {$y+3}]]
   $canvas itemconfigure $id {*}$args

} proc cross {canvas x y args} {

   set l1 [$canvas create line [expr {$x-3}] $y [expr {$x+3}] $y]
   set l2 [$canvas create line $x [expr {$y-3}] $x [expr {$y+3}]]
   $canvas itemconfigure $l1 {*}$args
   $canvas itemconfigure $l2 {*}$args

} proc x {canvas x y args} {

   set l1 [$canvas create line [expr {$x-3}] [expr {$y-3}] [expr {$x+3}] [expr {$y+3}]]
   set l2 [$canvas create line [expr {$x+3}] [expr {$y-3}] [expr {$x-3}] [expr {$y+3}]]
   $canvas itemconfigure $l1 {*}$args
   $canvas itemconfigure $l2 {*}$args

} proc triangleup {canvas x y args} {

   set id [$canvas create polygon $x [expr {$y-4}] \
               [expr {$x+4}] [expr {$y+4}] \
               [expr {$x-4}] [expr {$y+4}]]
   $canvas itemconfigure $id {*}$args

} proc triangledown {canvas x y args} {

   set id [$canvas create polygon $x [expr {$y+4}] \
               [expr {$x+4}] [expr {$y-4}] \
               [expr {$x-4}] [expr {$y-4}]]
   $canvas itemconfigure $id {*}$args

}

wm withdraw .

list creation procedures

proc ones n {

   lrepeat $n 1

} proc reversed n {

   while {[incr n -1] >= 0} {
       lappend result $n
   }
   return $result

} proc random n {

   for {set i 0} {$i < $n} {incr i} {
       lappend result [expr {int($n * rand())}]
   }
   return $result

}

set algorithms {lsort quicksort shellsort insertionsort bubblesort mergesort} set sizes {1 10 100 1000 10000 100000} set types {ones reversed random} set shapes {dot square cross triangleup triangledown x} set colours {red blue black brown yellow black}

create some lists to be used by all sorting algorithms

array set lists {} foreach size $sizes {

   foreach type $types {
       set lists($type,$size) [$type $size]
   }

}

set runs 10

header

fconfigure stdout -buffering none puts -nonewline [format "%-16s" "list length:"] foreach size $sizes {

   puts -nonewline [format " %10d" $size]

} puts ""

perform the sort timings and output results

foreach type $types {

   puts "\nlist type: $type"
   set times [list]
   foreach algo $algorithms {
       set errs [list]
       set thesetimes [list]
       $algo {} ;# call it once to ensure it's compiled
       
       puts -nonewline [format "   %-13s" $algo]
       foreach size $sizes {
           # some implementations are just too slow
           if {$type ne "ones" && (
               ($algo eq "insertionsort" && $size > 10000) ||
               ($algo eq "bubblesort" && $size > 1000))
           } {
               set time Inf
           } else {
               # OK, do it
               if {[catch {time [list $algo $lists($type,$size)] $runs} result] != 0} {
                   set time Inf
                   lappend errs $result
               } else {
                   set time [lindex [split $result] 0]
               }
           }
           lappend thesetimes $time
           puts -nonewline [format " %10s" $time]
       }
       puts ""
       if {[llength $errs] > 0} {
           puts [format "      %s" [join $errs "\n      "]]
       }
       lappend times $thesetimes
   }
   create_log10_plot "Sorting a '$type' list" size time $sizes $times $algorithms $shapes $colours

} puts "\ntimes in microseconds, average of $runs runs"</lang>

Output

list length:              1         10        100       1000      10000     100000

list type: ones
   lsort                0.8        1.2        7.2       71.9     1042.7    11428.9
   quicksort            1.1        9.0       40.6      369.5     3696.4    37478.4
   shellsort            1.4       26.0      249.1     4003.4    56278.7   717790.6
   insertionsort        1.1        6.4       59.0      528.1     5338.9    54913.0
   bubblesort           1.9        5.1       31.9      308.8     3259.1    31991.2
   mergesort            1.3       61.1      704.2     9275.4   224784.4 14599414.6

list type: reversed
   lsort                1.0        1.6        9.9      112.1     1434.9    20181.0
   quicksort            1.5       55.3      495.6     6705.9    79984.0   963975.0
   shellsort            1.5       25.9      457.0     7118.6    92497.5  1210143.9
   insertionsort        1.2       21.0     1645.0   159262.2 15859610.8        Inf
   bubblesort           1.9      445.0    46526.6  4665550.4        Inf        Inf
   mergesort            1.4       61.7      842.8     9572.1   215536.6 16938651.0

list type: random
   lsort                1.0        1.7       15.7      300.9     3275.0    58779.5
   quicksort            1.2       28.0      429.1     5609.5    71743.3   923630.4
   shellsort            1.6       26.7      571.0     9031.1   140526.9  2244152.7
   insertionsort        1.3       15.4      832.6    79018.0  7893722.6        Inf
   bubblesort           1.8      256.2    23753.1  2422926.0        Inf        Inf
   mergesort            1.9       60.2      883.5    12505.6   399672.6 49225509.8

times in microseconds, average of 10 runs

Wren

Translation of: Kotlin

Library: Wren-sort

Library: Wren-fmt

The quick, insertion and shell sorts all use the 'in place' implementations in the Wren-sort module.

The radix sort is lifted from the task of that name and, although more complicated, appears to be much faster than the Kotlin version.

For the bubble sort, I have used the optimized Kotlin implementation.

I've limited the size of the arrays to 50,000 though even then the program takes the best part of half an hour to run, due to the extreme slowness of the bubble and insertion sorts for large amounts of shuffled data.

Results presented in tabular form as Wren doesn't have a plotting library available at the present time. <lang ecmascript>import "random" for Random import "/sort" for Sort import "/fmt" for Fmt

var rand = Random.new()

var onesSeq = Fn.new { |n| List.filled(n, 1) }

var shuffledSeq = Fn.new { |n|

   var seq = List.filled(n, 0)
   for (i in 0...n) seq[i] = 1 + rand.int(10 * n)
   return seq

}

var ascendingSeq = Fn.new { |n|

   var seq = shuffledSeq.call(n)
   seq.sort()
   return seq

}

var bubbleSort = Fn.new { |a|

   var n = a.count
   while (true) {
       var n2 = 0
       for (i in 1...n) {
           if (a[i - 1] > a[i]) {
               a.swap(i, i - 1)
               n2 = i
           }
       }
       n = n2
       if (n == 0) break
   }

}

// counting sort of 'a' according to the digit represented by 'exp' var countSort = Fn.new { |a, exp|

   var n = a.count
   var output = [0] * n
   var count  = [0] * 10
   for (i in 0...n) {
       var t = (a[i]/exp).truncate % 10
       count[t] = count[t] + 1
   }
   for (i in 1..9) count[i] = count[i] + count[i-1]
   for (i in n-1..0) {
       var t = (a[i]/exp).truncate % 10
       output[count[t] - 1] = a[i]
       count[t] = count[t] - 1
   }
   for (i in 0...n) a[i] = output[i]

}

// sorts 'a' in place var radixSort = Fn.new { |a|

   // check for negative elements
   var min = a.reduce { |m, i| (i < m) ? i : m }
   // if there are any, increase all elements by -min
   if (min < 0) (0...a.count).each { |i| a[i] = a[i] - min }
   // now get the maximum to know number of digits
   var max = a.reduce { |m, i| (i > m) ? i : m }
   // do counting sort for each digit
   var exp = 1
   while ((max/exp).truncate > 0) {
       countSort.call(a, exp)
       exp = exp * 10
   }
   // if there were negative elements, reduce all elements by -min
   if (min < 0) (0...a.count).each { |i| a[i] = a[i] + min }

}

var measureTime = Fn.new { |sort, seq|

   var start = System.clock
   sort.call(seq)
   return ((System.clock - start) * 1e6).round // microseconds

}

var runs = 10 var lengths = [1, 10, 100, 1000, 10000, 50000] var sorts = [

   bubbleSort,
   Fn.new { |a| Sort.insertion(a) },
   Fn.new { |a| Sort.quick(a) },
   radixSort,
   Fn.new { |a| Sort.shell(a) }

]

var sortTitles = ["Bubble", "Insert", "Quick ", "Radix ", "Shell "] var seqTitles = ["All Ones", "Ascending", "Shuffled"] var totals = List.filled(seqTitles.count, null) for (i in 0...totals.count) {

   totals[i] = List.filled(sorts.count, null)
   for (j in 0...sorts.count) totals[i][j] = List.filled(lengths.count, 0)

} var k = 0 for (n in lengths) {

   var seqs = [onesSeq.call(n), ascendingSeq.call(n), shuffledSeq.call(n)]
   for (r in 0...runs) {
       for (i in 0...seqs.count) {
           for (j in 0...sorts.count) {
               var seq = seqs[i].toList
               totals[i][j][k] = totals[i][j][k] + measureTime.call(sorts[j], seq)
           }
       }
   }
   k = k + 1

} System.print("All timings in microseconds\n") System.write("Sequence length") for (len in lengths) Fmt.write("$8d ", len) System.print("\n") for (i in 0...seqTitles.count) {

   System.print("  %(seqTitles[i]):")
   for (j in 0...sorts.count) {
       System.write("    %(sortTitles[j])     ")
       for (k in 0...lengths.count) {
           var time = (totals[i][j][k] / runs).round
           Fmt.write("$8d   ", time)
       }
       System.print()
   }
   System.print("\n")

}</lang>

Output:

All timings in microseconds

Sequence length       1         10        100       1000      10000      50000   

  All Ones:
    Bubble            1          2          9         61        643       3225   
    Insert            1          3         16        118       1256       6338   
    Quick             1          8         92        983      14746      87660   
    Radix             6         13         62        460       4823      24379   
    Shell             1          5         59        770       9542      48873   


  Ascending:
    Bubble            1          2          8         61        637       3221   
    Insert            1          3         16        118       1251       6371   
    Quick             1          7         65        643       9149      54199   
    Radix             7         23        169       1648      21428     130609   
    Shell             1          5         60        779       9537      49041   


  Shuffled:
    Bubble            1          7        451      37271    4025966   99834073   
    Insert            0          7        295      24040    2597162   64875212   
    Quick             1          8        101       1149      16256      95590   
    Radix             5         24        163       1688      22443     136228   
    Shell             1          8        111       1514      25180     230897

The results are much the same as one might have expected beforehand.

As far as the shuffled data is concerned, quick sort is the fastest though radix and shell sorts are also reasonable performers. Bubble and insertion sorts are very slow indeed for large amounts of data.

Conversely, if the data is already sorted into ascending order, bubble and insertion sorts are much faster than the others and radix sort is the slowest.

If all data is the same, a similar pattern emerges except that radix sort performs better than both shell and quick sorts, the latter being the slowest.