Arithmetic coding/As a generalized change of radix: Difference between revisions
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Revision as of 03:19, 18 June 2019
Arithmetic coding/As a generalized change of radix is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Arithmetic coding is a form of entropy encoding used in lossless data compression. Normally, a string of characters such as the words "hello there" is represented using a fixed number of bits per character, as in the ASCII code. When a string is converted to arithmetic encoding, frequently used characters will be stored with fewer bits and not-so-frequently occurring characters will be stored with more bits, resulting in fewer bits used in total. Arithmetic coding differs from other forms of entropy encoding, such as Huffman coding, in that rather than separating the input into component symbols and replacing each with a code, arithmetic coding encodes the entire message into a single number.
- Task
Create a program which implements the arithmetic coding as a generalized change of radix.
Show the results, in base 10, for all the following strings:
- "DABDDB"
- "DABDDBBDDBA"
- "ABRACADABRA"
- "TOBEORNOTTOBEORTOBEORNOT"
Verify the implementation by decoding the results back into strings and checking for equality with the given strings.
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Numerics; using System.Text;
namespace AruthmeticCoding {
using Freq = Dictionary<char, long>; using Triple = Tuple<BigInteger, int, Dictionary<char, long>>;
class Program {
static Freq CumulativeFreq(Freq freq) {
long total = 0;
Freq cf = new Freq();
for (int i = 0; i < 256; i++) {
char c = (char)i;
if (freq.ContainsKey(c)) {
long v = freq[c];
cf[c] = total;
total += v;
}
}
return cf;
}
static Triple ArithmeticCoding(string str, long radix) {
// The frequency of characters
Freq freq = new Freq();
foreach (char c in str) {
if (freq.ContainsKey(c)) {
freq[c] += 1;
} else {
freq[c] = 1;
}
}
// The cumulative frequency
Freq cf = CumulativeFreq(freq);
// Base
BigInteger @base = str.Length;
// Lower bound
BigInteger lower = 0;
// Product of all frequencies
BigInteger pf = 1;
// Each term is multiplied by the product of the
// frequencies of all previously occuring symbols
foreach (char c in str) {
BigInteger x = cf[c];
lower = lower * @base + x * pf;
pf = pf * freq[c];
}
// Upper bound
BigInteger upper = lower + pf;
int powr = 0;
BigInteger bigRadix = radix;
while (true) {
pf = pf / bigRadix;
if (pf == 0) break;
powr++;
}
BigInteger diff = (upper - 1) / (BigInteger.Pow(bigRadix, powr));
return new Triple(diff, powr, freq);
}
static string ArithmeticDecoding(BigInteger num, long radix, int pwr, Freq freq) {
BigInteger powr = radix;
BigInteger enc = num * BigInteger.Pow(powr, pwr);
long @base = freq.Values.Sum();
// Create the cumulative frequency table
Freq cf = CumulativeFreq(freq);
// Create the dictionary
Dictionary<long, char> dict = new Dictionary<long, char>();
foreach (char key in cf.Keys) {
long value = cf[key];
dict[value] = key;
}
// Fill the gaps in the dictionary
long lchar = -1;
for (long i = 0; i < @base; i++) {
if (dict.ContainsKey(i)) {
lchar = dict[i];
} else if (lchar != -1) {
dict[i] = (char)lchar;
}
}
// Decode the input number
StringBuilder decoded = new StringBuilder((int)@base);
BigInteger bigBase = @base;
for (long i = @base - 1; i >= 0; --i) {
BigInteger pow = BigInteger.Pow(bigBase, (int)i);
BigInteger div = enc / pow;
char c = dict[(long)div];
BigInteger fv = freq[c];
BigInteger cv = cf[c];
BigInteger diff = enc - pow * cv;
enc = diff / fv;
decoded.Append(c);
}
// Return the decoded output
return decoded.ToString();
}
static void Main(string[] args) {
long radix = 10;
string[] strings = { "DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT" };
foreach (string str in strings) {
Triple encoded = ArithmeticCoding(str, radix);
string dec = ArithmeticDecoding(encoded.Item1, radix, encoded.Item2, encoded.Item3);
Console.WriteLine("{0,-25}=> {1,19} * {2}^{3}", str, encoded.Item1, radix, encoded.Item2);
if (str != dec) {
throw new Exception("\tHowever that is incorrect!");
}
}
}
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
D
<lang D>import std.array; import std.bigint; import std.stdio; import std.typecons;
BigInt bigPow(BigInt b, BigInt e) {
if (e == 0) {
return BigInt(1);
}
BigInt result = 1;
while (e > 1) {
if (e % 2 == 0) {
b *= b;
e /= 2;
} else {
result *= b;
b *= b;
e = (e - 1) / 2;
}
}
return b * result;
}
long[byte] cumulative_freq(long[byte] freq) {
long[byte] cf;
long total;
foreach (i; 0..256) {
byte b = cast(byte) i;
if (b in freq) {
cf[b] = total;
total += freq[b];
}
}
return cf;
}
Tuple!(BigInt, BigInt, long[byte]) arithmethic_coding(string str, long radix) {
// Convert the string into a slice of bytes auto chars = cast(byte[]) str;
// The frequency characters
long[byte] freq;
foreach (c; chars) {
freq[c]++;
}
// The cumulative frequency auto cf = cumulative_freq(freq);
// Base BigInt base = chars.length;
// Lower bound BigInt lower = 0;
// Product of all frequencies BigInt pf = 1;
// Each term is multiplied by the product of the
// frequencies of all previously occurring symbols
foreach (c; chars) {
BigInt x = cf[c];
lower = lower*base + x*pf;
pf = pf*freq[c];
}
// Upper bound auto upper = lower + pf;
BigInt tmp = pf;
auto powr = BigInt("0");
while (true) {
tmp = tmp / radix;
if (tmp == 0) {
break;
}
powr++;
}
auto diff = (upper-1) / bigPow(BigInt(radix), powr);
return tuple(diff, powr, freq);
}
string arithmethic_decoding(BigInt num, long radix, BigInt pow, long[byte] freq) {
BigInt powr = radix;
BigInt enc = num * bigPow(powr, pow);
BigInt base = 0;
foreach (v; freq) {
base += v;
}
// Create the cumulative frequency table auto cf = cumulative_freq(freq);
// Create the dictionary
byte[long] dict;
foreach (k,v; cf) {
dict[v] = k;
}
// Fill the gaps in the dictionary
long lchar = -1;
for (long i=0; i<base; i++) {
if (i in dict) {
lchar = dict[i];
} else if (lchar != -1) {
dict[i] = cast(byte) lchar;
}
}
// Decode the input number
auto decoded = appender!string;
for (BigInt i=base-1; i>=0; i--) {
pow = bigPow(base, i);
auto div = enc / pow;
auto c = dict[div.toLong];
auto fv = freq[c];
auto cv = cf[c];
auto prod = pow * cv;
auto diff = enc - prod;
enc = diff / fv;
decoded.put(c); }
// Return the decoded output return decoded.data;
}
void main() {
long radix = 10;
foreach (str; ["DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"]) {
auto output = arithmethic_coding(str, radix);
auto dec = arithmethic_decoding(output[0], radix, output[1], output[2]);
writefln("%-25s=> %19s * %s^%s", str, output[0], radix, output[1]);
if (str != dec) {
throw new Exception("\tHowever that is incorrect!");
}
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Go
<lang go>package main
import (
"fmt" "math/big"
)
func cumulative_freq(freq map[byte]int64) map[byte]int64 {
total := int64(0)
cf := make(map[byte]int64)
for i := 0; i < 256; i++ {
b := byte(i)
if v, ok := freq[b]; ok {
cf[b] = total
total += v
}
}
return cf
}
func arithmethic_coding(str string, radix int64) (*big.Int,
*big.Int, map[byte]int64) {
// Convert the string into a slice of bytes chars := []byte(str)
// The frequency characters
freq := make(map[byte]int64)
for _, c := range chars {
freq[c] += 1
}
// The cumulative frequency cf := cumulative_freq(freq)
// Base base := len(chars)
// Lower bound L := big.NewInt(0)
// Product of all frequencies pf := big.NewInt(1)
// Each term is multiplied by the product of the // frequencies of all previously occurring symbols bigBase := big.NewInt(int64(base))
for _, c := range chars {
x := big.NewInt(cf[c])
L.Mul(L, bigBase)
L.Add(L, x.Mul(x, pf))
pf.Mul(pf, big.NewInt(freq[c]))
}
// Upper bound U := big.NewInt(0) U.Set(L) U.Add(U, pf)
bigOne := big.NewInt(1) bigZero := big.NewInt(0) bigRadix := big.NewInt(radix)
tmp := big.NewInt(0).Set(pf) powr := big.NewInt(0)
for {
tmp.Div(tmp, bigRadix)
if tmp.Cmp(bigZero) == 0 {
break
}
powr.Add(powr, bigOne)
}
diff := big.NewInt(0) diff.Sub(U, bigOne) diff.Div(diff, big.NewInt(0).Exp(bigRadix, powr, nil))
return diff, powr, freq
}
func arithmethic_decoding(num *big.Int, radix int64,
pow *big.Int, freq map[byte]int64) string {
powr := big.NewInt(radix)
enc := big.NewInt(0).Set(num) enc.Mul(enc, powr.Exp(powr, pow, nil))
base := int64(0)
for _, v := range freq {
base += v
}
// Create the cumulative frequency table cf := cumulative_freq(freq)
// Create the dictionary
dict := make(map[int64]byte)
for k, v := range cf {
dict[v] = k
}
// Fill the gaps in the dictionary
lchar := -1
for i := int64(0); i < base; i++ {
if v, ok := dict[i]; ok {
lchar = int(v)
} else if lchar != -1 {
dict[i] = byte(lchar)
}
}
// Decode the input number decoded := make([]byte, base) bigBase := big.NewInt(base)
for i := base - 1; i >= 0; i-- {
pow := big.NewInt(0)
pow.Exp(bigBase, big.NewInt(i), nil)
div := big.NewInt(0)
div.Div(enc, pow)
c := dict[div.Int64()]
fv := freq[c]
cv := cf[c]
prod := big.NewInt(0).Mul(pow, big.NewInt(cv))
diff := big.NewInt(0).Sub(enc, prod)
enc.Div(diff, big.NewInt(fv))
decoded[base-i-1] = c }
// Return the decoded output return string(decoded)
}
func main() {
var radix = int64(10)
strSlice := []string{
`DABDDB`,
`DABDDBBDDBA`,
`ABRACADABRA`,
`TOBEORNOTTOBEORTOBEORNOT`,
}
for _, str := range strSlice {
enc, pow, freq := arithmethic_coding(str, radix)
dec := arithmethic_decoding(enc, radix, pow, freq)
fmt.Printf("%-25s=> %19s * %d^%s\n", str, enc, radix, pow)
if str != dec {
panic("\tHowever that is incorrect!")
}
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
J
Implementation:
<lang J>NB. generate a frequency dictionary from a reference string aekDict=:verb define
d=. ~.y NB. dictionary lists unique characters
o=. /:d NB. in canonical order
f=. (#/.~%&x:#)y NB. and their relative frequencies
(o{d);o{f
)
NB. encode a string against a reference dict aekEnc=:verb define
NB. use string to generate a dict if none provided (aekDict y) aekEnc y
'u F'=.x NB. unpack dictionary
b=. x:#y NB. numeric base
f=. b*F NB. absolute frequencies
i=. u i.y NB. character indices
c=. +/\0,}:f NB. cumulative frequencies
L=. b #. (i{c)**/\1,}:i{f NB. lower bound
p=. */i{f NB. product of character frequencies
e=. x:<.10^.p NB. number of decimal positions to drop
e,~<.(L+p)%10^e
)
aekDec=:adverb define
'u F'=. x NB. unpack dictionary
f=. m*F NB. frequencies of characters
c=.+/\0,}:f NB. cumulative frequencies
C=.<:}.c,m NB. id lookup table
N=. (* 10&^)/y NB. remainder being decoded
r=. NB. result of decode
for_d. m^x:i.-m do. NB. positional values
id=. <.N%d NB. character id
i=.C I.id NB. character index
N=.<.(N -(i{c)*d)%i{f NB. corrected remainder
r=.r,u{~i NB. accumulated result
end.
)
NB. task demo utility: aek=:verb define
dict=. aekDict y
echo 'Dictionary:'
echo ' ',.(0{::dict),.' ',.":,.1{::dict
echo 'Length:'
echo ' ',":#y
echo 'Encoded:'
echo ' ',":dict aekEnc y
echo 'Decoded:'
echo ' ',":dict (#y) aekDec aekEnc y
)</lang>
Example use:
<lang J> aek 'DABDDB' Dictionary:
A 1r6 B 1r3 D 1r2
Length:
6
Encoded:
251 2
Decoded:
DABDDB
aek 'DABDDBBDDBA'
Dictionary:
A 2r11 B 4r11 D 5r11
Length:
11
Encoded:
167351 6
Decoded:
DABDDBBDDBA
aek 'ABRACADABRA'
Dictionary:
A 5r11 B 2r11 C 1r11 D 1r11 R 2r11
Length:
11
Encoded:
7954170 4
Decoded:
ABRACADABRA
aek 'TOBEORNOTTOBEORTOBEORNOT'
Dictionary:
B 1r8 E 1r8 N 1r12 O 1r3 R 1r8 T 5r24
Length:
24
Encoded:
1150764267498783364 15
Decoded:
TOBEORNOTTOBEORTOBEORNOT</lang>
Note that for this task we use our plaintext to generate our dictionary for decoding. Also note that we use rational numbers, rather than floating point, for our dictionary, because floating point tends to be inexact.
Java
<lang Java>import java.math.BigInteger; import java.util.HashMap; import java.util.Map; import java.util.Objects;
public class ArithmeticCoding {
private static class Triple<A, B, C> {
A a;
B b;
C c;
Triple(A a, B b, C c) {
this.a = a;
this.b = b;
this.c = c;
}
}
private static class Freq extends HashMap<Character, Long> {
//"type alias"
}
private static Freq cumulativeFreq(Freq freq) {
long total = 0;
Freq cf = new Freq();
for (int i = 0; i < 256; ++i) {
char c = (char) i;
Long v = freq.get(c);
if (v != null) {
cf.put(c, total);
total += v;
}
}
return cf;
}
private static Triple<BigInteger, Integer, Freq> arithmeticCoding(String str, Long radix) {
// Convert the string into a char array
char[] chars = str.toCharArray();
// The frequency characters
Freq freq = new Freq();
for (char c : chars) {
if (!freq.containsKey(c))
freq.put(c, 1L);
else
freq.put(c, freq.get(c) + 1);
}
// The cumulative frequency
Freq cf = cumulativeFreq(freq);
// Base
BigInteger base = BigInteger.valueOf(chars.length);
// LowerBound
BigInteger lower = BigInteger.ZERO;
// Product of all frequencies
BigInteger pf = BigInteger.ONE;
// Each term is multiplied by the product of the
// frequencies of all previously occurring symbols
for (char c : chars) {
BigInteger x = BigInteger.valueOf(cf.get(c));
lower = lower.multiply(base).add(x.multiply(pf));
pf = pf.multiply(BigInteger.valueOf(freq.get(c)));
}
// Upper bound
BigInteger upper = lower.add(pf);
int powr = 0;
BigInteger bigRadix = BigInteger.valueOf(radix);
while (true) {
pf = pf.divide(bigRadix);
if (pf.equals(BigInteger.ZERO)) break;
powr++;
}
BigInteger diff = upper.subtract(BigInteger.ONE).divide(bigRadix.pow(powr));
return new Triple<>(diff, powr, freq);
}
private static String arithmeticDecoding(BigInteger num, long radix, int pwr, Freq freq) {
BigInteger powr = BigInteger.valueOf(radix);
BigInteger enc = num.multiply(powr.pow(pwr));
long base = 0;
for (Long v : freq.values()) base += v;
// Create the cumulative frequency table
Freq cf = cumulativeFreq(freq);
// Create the dictionary
Map<Long, Character> dict = new HashMap<>();
for (Map.Entry<Character, Long> entry : cf.entrySet()) dict.put(entry.getValue(), entry.getKey());
// Fill the gaps in the dictionary
long lchar = -1;
for (long i = 0; i < base; ++i) {
Character v = dict.get(i);
if (v != null) {
lchar = v;
} else if (lchar != -1) {
dict.put(i, (char) lchar);
}
}
// Decode the input number
StringBuilder decoded = new StringBuilder((int) base);
BigInteger bigBase = BigInteger.valueOf(base);
for (long i = base - 1; i >= 0; --i) {
BigInteger pow = bigBase.pow((int) i);
BigInteger div = enc.divide(pow);
Character c = dict.get(div.longValue());
BigInteger fv = BigInteger.valueOf(freq.get(c));
BigInteger cv = BigInteger.valueOf(cf.get(c));
BigInteger diff = enc.subtract(pow.multiply(cv));
enc = diff.divide(fv);
decoded.append(c);
}
// Return the decoded output
return decoded.toString();
}
public static void main(String[] args) {
long radix = 10;
String[] strings = {"DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"};
String fmt = "%-25s=> %19s * %d^%s\n";
for (String str : strings) {
Triple<BigInteger, Integer, Freq> encoded = arithmeticCoding(str, radix);
String dec = arithmeticDecoding(encoded.a, radix, encoded.b, encoded.c);
System.out.printf(fmt, str, encoded.a, radix, encoded.b);
if (!Objects.equals(str, dec)) throw new RuntimeException("\tHowever that is incorrect!");
}
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Julia
Taken from the wikipedia example. <lang julia>function charfreq(s)
d = Dict()
for c in s
if haskey(d, c)
d[c] += 1
else
d[c] = 1
end
end
d
end
function charcumfreq(dfreq)
lastval = 0
d = Dict()
for c in sort!(collect(keys(dfreq)))
d[c] = lastval
lastval += dfreq[c]
end
d
end
function L(s, dfreq, dcumfreq)
nbase = BigInt(length(s))
lsum, cumprod = BigInt(0), 1
for (i, c) in enumerate(s)
lsum += nbase^(nbase - i) * dcumfreq[c] * cumprod
cumprod *= dfreq[c]
end
lsum
end
U(l, s, dfreq) = l + prod(c -> dfreq[c], s)
function mostzeros(low, high)
z = Int(floor(log10(high - low)))
if z == 0
return string(low), 0
end
if low <= parse(BigInt, string(high)[1:end- z - 1] * "0"^(z + 1)) <= high
z += 1
end
return string(high)[1:end-z], z
end
function msgnum(s)
dfreq = charfreq(s) dcumfreq = charcumfreq(dfreq) low = L(s, dfreq, dcumfreq) high = U(low, s, dfreq) return mostzeros(low, high), dfreq
end
function decode(encoded, fdict)
cdict, bas = charcumfreq(fdict), sum(values(fdict))
kys = sort!(collect(keys(cdict)))
revdict = Dict([(cdict[k], k) for k in kys])
lastkey = revdict[0]
for i in 0:bas
if !haskey(revdict, i)
revdict[i] = lastkey
else
lastkey = revdict[i]
end
end
rem = parse(BigInt, encoded)
s = ""
for i in 1:bas
basepow = BigInt(bas)^(bas -i)
c = revdict[div(rem, basepow)]
s *= c
rem = div(rem - basepow * cdict[c], fdict[c])
end
s
end
for s in ["DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"]
(encoded, z), dfreq = msgnum(s)
println(lpad(s, 30), " ", rpad(encoded, 19), " * 10^", rpad(z, 4), " ",
decode(encoded * "0"^z, dfreq))
end
</lang>
- Output:
DABDDB 251 * 10^2 DABDDB
DABDDBBDDBA 16735 * 10^7 DABDDBBDDBA
ABRACADABRA 795417 * 10^5 ABRACADABRA
TOBEORNOTTOBEORTOBEORNOT 1150764267498783364 * 10^15 TOBEORNOTTOBEORTOBEORNOT
Kotlin
<lang scala>// version 1.2.10
import java.math.BigInteger
typealias Freq = Map<Char, Long>
val bigZero = BigInteger.ZERO val bigOne = BigInteger.ONE
fun cumulativeFreq(freq: Freq): Freq {
var total = 0L
val cf = mutableMapOf<Char, Long>()
for (i in 0..255) {
val c = i.toChar()
val v = freq[c]
if (v != null) {
cf[c] = total
total += v
}
}
return cf
}
fun arithmeticCoding(str: String, radix: Long): Triple<BigInteger, Int, Freq> {
// Convert the string into a char array val chars = str.toCharArray()
// The frequency characters
val freq = mutableMapOf<Char, Long>()
for (c in chars) {
if (c !in freq)
freq[c] = 1L
else
freq[c] = freq[c]!! + 1
}
// The cumulative frequency val cf = cumulativeFreq(freq)
// Base val base = chars.size.toBigInteger()
// LowerBound var lower = bigZero
// Product of all frequencies var pf = BigInteger.ONE
// Each term is multiplied by the product of the
// frequencies of all previously occurring symbols
for (c in chars) {
val x = cf[c]!!.toBigInteger()
lower = lower * base + x * pf
pf *= freq[c]!!.toBigInteger()
}
// Upper bound val upper = lower + pf
var powr = 0 val bigRadix = radix.toBigInteger()
while (true) {
pf /= bigRadix
if (pf == bigZero) break
powr++
}
val diff = (upper - bigOne) / bigRadix.pow(powr) return Triple(diff, powr, freq)
}
fun arithmeticDecoding(num: BigInteger, radix: Long, pwr: Int, freq: Freq): String {
val powr = radix.toBigInteger() var enc = num * powr.pow(pwr) var base = 0L for ((_, v) in freq) base += v
// Create the cumulative frequency table val cf = cumulativeFreq(freq)
// Create the dictionary val dict = mutableMapOf<Long, Char>() for ((k, v) in cf) dict[v] = k
// Fill the gaps in the dictionary
var lchar = -1
for (i in 0L until base) {
val v = dict[i]
if (v != null) {
lchar = v.toInt()
}
else if(lchar != -1) {
dict[i] = lchar.toChar()
}
}
// Decode the input number
val decoded = StringBuilder(base.toInt())
val bigBase = base.toBigInteger()
for (i in base - 1L downTo 0L) {
val pow = bigBase.pow(i.toInt())
val div = enc / pow
val c = dict[div.toLong()]
val fv = freq[c]!!.toBigInteger()
val cv = cf[c]!!.toBigInteger()
val diff = enc - pow * cv
enc = diff / fv
decoded.append(c)
}
// Return the decoded output
return decoded.toString()
}
fun main(args: Array<String>) {
val radix = 10L
val strings = listOf(
"DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"
)
val fmt = "%-25s=> %19s * %d^%s"
for (str in strings) {
val (enc, pow, freq) = arithmeticCoding(str, radix)
val dec = arithmeticDecoding(enc, radix, pow, freq)
println(fmt.format(str, enc, radix, pow))
if (str != dec) throw Exception("\tHowever that is incorrect!")
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Perl
<lang perl>use Math::BigInt (try => 'GMP');
sub cumulative_freq {
my ($freq) = @_;
my %cf;
my $total = Math::BigInt->new(0);
foreach my $c (sort keys %$freq) {
$cf{$c} = $total;
$total += $freq->{$c};
}
return %cf;
}
sub arithmethic_coding {
my ($str, $radix) = @_; my @chars = split(//, $str);
# The frequency characters
my %freq;
$freq{$_}++ for @chars;
# The cumulative frequency table my %cf = cumulative_freq(\%freq);
# Base my $base = Math::BigInt->new(scalar @chars);
# Lower bound my $L = Math::BigInt->new(0);
# Product of all frequencies my $pf = Math::BigInt->new(1);
# Each term is multiplied by the product of the
# frequencies of all previously occurring symbols
foreach my $c (@chars) {
$L->bmuladd($base, $cf{$c} * $pf);
$pf->bmul($freq{$c});
}
# Upper bound my $U = $L + $pf;
my $pow = Math::BigInt->new($pf)->blog($radix); my $enc = ($U - 1)->bdiv(Math::BigInt->new($radix)->bpow($pow));
return ($enc, $pow, \%freq);
}
sub arithmethic_decoding {
my ($enc, $radix, $pow, $freq) = @_;
# Multiply enc by radix^pow $enc *= $radix**$pow;
# Base
my $base = Math::BigInt->new(0);
$base += $_ for values %{$freq};
# Create the cumulative frequency table my %cf = cumulative_freq($freq);
# Create the dictionary
my %dict;
while (my ($k, $v) = each %cf) {
$dict{$v} = $k;
}
# Fill the gaps in the dictionary
my $lchar;
foreach my $i (0 .. $base - 1) {
if (exists $dict{$i}) {
$lchar = $dict{$i};
}
elsif (defined $lchar) {
$dict{$i} = $lchar;
}
}
# Decode the input number
my $decoded = ;
for (my $pow = $base**($base - 1) ; $pow > 0 ; $pow /= $base) {
my $div = $enc / $pow;
my $c = $dict{$div};
my $fv = $freq->{$c};
my $cv = $cf{$c};
$enc = ($enc - $pow * $cv) / $fv;
$decoded .= $c;
}
# Return the decoded output return $decoded;
}
my $radix = 10; # can be any integer greater or equal with 2
foreach my $str (qw(DABDDB DABDDBBDDBA ABRACADABRA TOBEORNOTTOBEORTOBEORNOT)) {
my ($enc, $pow, $freq) = arithmethic_coding($str, $radix); my $dec = arithmethic_decoding($enc, $radix, $pow, $freq);
printf("%-25s=> %19s * %d^%s\n", $str, $enc, $radix, $pow);
if ($str ne $dec) {
die "\tHowever that is incorrect!";
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Perl 6
<lang perl6>sub cumulative_freq(%freq) {
my %cf;
my $total = 0;
for %freq.keys.sort -> $c {
%cf{$c} = $total;
$total += %freq{$c};
}
return %cf;
}
sub arithmethic_coding($str, $radix) {
my @chars = $str.comb;
# The frequency characters
my %freq;
%freq{$_}++ for @chars;
# The cumulative frequency table my %cf = cumulative_freq(%freq);
# Base my $base = @chars.elems;
# Lower bound my $L = 0;
# Product of all frequencies my $pf = 1;
# Each term is multiplied by the product of the
# frequencies of all previously occurring symbols
for @chars -> $c {
$L = $L*$base + %cf{$c}*$pf;
$pf *= %freq{$c};
}
# Upper bound my $U = $L + $pf;
my $pow = 0;
loop {
$pf div= $radix;
last if $pf == 0;
++$pow;
}
my $enc = ($U - 1) div ($radix ** $pow); ($enc, $pow, %freq);
}
sub arithmethic_decoding($encoding, $radix, $pow, %freq) {
# Multiply encoding by radix^pow my $enc = $encoding * $radix**$pow;
# Base my $base = [+] %freq.values;
# Create the cumulative frequency table my %cf = cumulative_freq(%freq);
# Create the dictionary
my %dict;
for %cf.kv -> $k,$v {
%dict{$v} = $k;
}
# Fill the gaps in the dictionary
my $lchar;
for ^$base -> $i {
if (%dict{$i}:exists) {
$lchar = %dict{$i};
}
elsif (defined $lchar) {
%dict{$i} = $lchar;
}
}
# Decode the input number
my $decoded = ;
for reverse(^$base) -> $i {
my $pow = $base**$i;
my $div = $enc div $pow;
my $c = %dict{$div};
my $fv = %freq{$c};
my $cv = %cf{$c};
my $rem = ($enc - $pow*$cv) div $fv;
$enc = $rem;
$decoded ~= $c;
}
# Return the decoded output return $decoded;
}
my $radix = 10; # can be any integer greater or equal with 2
for <DABDDB DABDDBBDDBA ABRACADABRA TOBEORNOTTOBEORTOBEORNOT> -> $str {
my ($enc, $pow, %freq) = arithmethic_coding($str, $radix); my $dec = arithmethic_decoding($enc, $radix, $pow, %freq);
printf("%-25s=> %19s * %d^%s\n", $str, $enc, $radix, $pow);
if ($str ne $dec) {
die "\tHowever that is incorrect!";
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Phix
<lang Phix>include bigatom.e
function cumulative_freq(sequence freq)
integer total = 0,
l = length(freq)
sequence cf = repeat(-1,l)
for c=1 to l do
integer v = freq[c]
if v!=0 then
cf[c] = total
total += v
end if
end for
return cf
end function
function arithmethic_coding(string str, integer radix)
sequence freq = repeat(0,256)
for i=1 to length(str) do
freq[str[i]+1] += 1
end for
sequence cf = cumulative_freq(freq)
integer base = length(str)
bigatom lo = BA_ZERO, -- lower bound
pf = BA_ONE -- product of all frequencies
-- Each term is multiplied by the product of the
-- frequencies of all previously occurring symbols
for i=1 to length(str) do
integer c = str[i]+1
integer x = cf[c]
lo = ba_add(ba_mul(lo,base),ba_mul(x,pf))
pf = ba_mul(pf,freq[c])
end for
bigatom hi = ba_add(lo,pf) -- upper bound
integer powr = 0
while true do
pf = ba_idiv(pf,radix)
if pf=BA_ZERO then exit end if
powr += 1
end while
bigatom diff = ba_idiv(ba_sub(hi,1),ba_power(radix,powr))
return {diff, powr, freq}
end function
function arithmethic_decoding(bigatom num, integer radix, pwr, sequence freq)
bigatom enc = ba_mul(num,ba_power(radix,pwr))
integer base = sum(freq)
sequence cf = cumulative_freq(freq)
sequence dict = repeat(0,base+1)
for i=1 to length(cf) do
integer v = cf[i]
if v!=-1 then dict[v+1] = i-1 end if
end for
-- Fill the gaps in the dictionary
integer lchar = -1
for i=0 to base do
integer v = dict[i+1]
if v!=0 then
lchar = v
elsif lchar!=-1 then
dict[i+1] = lchar
end if
end for
-- Decode the input number
string decoded = ""
for i=base-1 to 0 by -1 do
bigatom pow = ba_power(base,i)
integer div = bigatom_to_atom(ba_idiv(enc,pow)),
c = dict[div+1],
fv = freq[c+1],
cv = cf[c+1]
bigatom diff = ba_sub(enc,ba_mul(pow,cv))
enc = ba_idiv(diff,fv)
decoded &= c
end for
return decoded
end function
constant tests = {"DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"},
radix = 10
for i=1 to length(tests) do
string str = tests[i]
{bigatom enc, integer pow, sequence freq} = arithmethic_coding(str, radix)
string dec = arithmethic_decoding(enc, radix, pow, freq),
ok = iff(str=dec?"(ok)":"***ERROR***"),
encs = ba_sprint(enc)
printf(1,"%-25s=> %19s * %d^%d %s\n",{str, encs, radix, pow, ok})
end for</lang>
- Output:
DABDDB => 251 * 10^2 (ok) DABDDBBDDBA => 167351 * 10^6 (ok) ABRACADABRA => 7954170 * 10^4 (ok) TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15 (ok)
Python
<lang python>from collections import Counter
def cumulative_freq(freq):
cf = {}
total = 0
for b in range(256):
if b in freq:
cf[b] = total
total += freq[b]
return cf
def arithmethic_coding(bytes, radix):
# The frequency characters freq = Counter(bytes)
# The cumulative frequency table cf = cumulative_freq(freq)
# Base base = len(bytes)
# Lower bound lower = 0
# Product of all frequencies pf = 1
# Each term is multiplied by the product of the
# frequencies of all previously occurring symbols
for b in bytes:
lower = lower*base + cf[b]*pf
pf *= freq[b]
# Upper bound upper = lower+pf
pow = 0
while True:
pf //= radix
if pf==0: break
pow += 1
enc = (upper-1) // radix**pow return enc, pow, freq
def arithmethic_decoding(enc, radix, pow, freq):
# Multiply enc by radix^pow enc *= radix**pow;
# Base base = sum(freq.values())
# Create the cumulative frequency table cf = cumulative_freq(freq)
# Create the dictionary
dict = {}
for k,v in cf.items():
dict[v] = k
# Fill the gaps in the dictionary
lchar = None
for i in range(base):
if i in dict:
lchar = dict[i]
elif lchar is not None:
dict[i] = lchar
# Decode the input number
decoded = bytearray()
for i in range(base-1, -1, -1):
pow = base**i
div = enc//pow
c = dict[div]
fv = freq[c]
cv = cf[c]
rem = (enc - pow*cv) // fv
enc = rem
decoded.append(c)
# Return the decoded output return bytes(decoded)
radix = 10 # can be any integer greater or equal with 2
for str in b'DABDDB DABDDBBDDBA ABRACADABRA TOBEORNOTTOBEORTOBEORNOT'.split():
enc, pow, freq = arithmethic_coding(str, radix) dec = arithmethic_decoding(enc, radix, pow, freq)
print("%-25s=> %19s * %d^%s" % (str, enc, radix, pow))
if str != dec:
raise Exception("\tHowever that is incorrect!")</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Ruby
<lang ruby>def cumulative_freq(freq)
cf = {}
total = 0
freq.keys.sort.each do |b|
cf[b] = total
total += freq[b]
end
return cf
end
def arithmethic_coding(bytes, radix)
# The frequency characters
freq = Hash.new(0)
bytes.each { |b| freq[b] += 1 }
# The cumulative frequency table cf = cumulative_freq(freq)
# Base base = bytes.size
# Lower bound lower = 0
# Product of all frequencies pf = 1
# Each term is multiplied by the product of the # frequencies of all previously occurring symbols bytes.each do |b| lower = lower*base + cf[b]*pf pf *= freq[b] end
# Upper bound upper = lower+pf
pow = 0 loop do pf /= radix break if pf==0 pow += 1 end
enc = ((upper-1) / radix**pow) [enc, pow, freq]
end
def arithmethic_decoding(enc, radix, pow, freq)
# Multiply enc by radix^pow enc *= radix**pow;
# Base base = freq.values.reduce(:+)
# Create the cumulative frequency table cf = cumulative_freq(freq)
# Create the dictionary
dict = {}
cf.each_pair do |k,v|
dict[v] = k
end
# Fill the gaps in the dictionary
lchar = nil
(0...base).each do |i|
if dict.has_key?(i)
lchar = dict[i]
elsif lchar != nil
dict[i] = lchar
end
end
# Decode the input number decoded = [] (0...base).reverse_each do |i| pow = base**i div = enc/pow
c = dict[div] fv = freq[c] cv = cf[c]
rem = ((enc - pow*cv) / fv)
enc = rem decoded << c end
# Return the decoded output return decoded
end
radix = 10 # can be any integer greater or equal with 2
%w(DABDDB DABDDBBDDBA ABRACADABRA TOBEORNOTTOBEORTOBEORNOT).each do |str|
enc, pow, freq = arithmethic_coding(str.bytes, radix)
dec = arithmethic_decoding(enc, radix, pow, freq).map{|b| b.chr }.join
printf("%-25s=> %19s * %d^%s\n", str, enc, radix, pow)
if str != dec raise "\tHowever that is incorrect!" end
end</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Sidef
<lang ruby>func cumulative_freq(freq) {
var cf = Hash()
var total = 0
256.range.each { |b|
if (freq.contains(b)) {
cf{b} = total
total += freq{b}
}
}
return cf
}
func arithmethic_coding(bytes, radix=10) {
# The frequency characters
var freq = Hash()
bytes.each { |b| freq{b} := 0 ++ }
# The cumulative frequency table var cf = cumulative_freq(freq)
# Base var base = bytes.len
# Lower bound var L = 0
# Product of all frequencies var pf = 1
# Each term is multiplied by the product of the
# frequencies of all previously occurring symbols
bytes.each { |b|
L = (L*base + cf{b}*pf)
pf *= freq{b}
}
# Upper bound var U = L+pf
var pow = pf.log(radix).int var enc = ((U-1) // radix**pow)
return (enc, pow, freq)
}
func arithmethic_decoding(enc, radix, pow, freq) {
# Multiply enc by radix^pow enc *= radix**pow;
# Base var base = freq.values.sum
# Create the cumulative frequency table var cf = cumulative_freq(freq);
# Create the dictionary
var dict = Hash()
cf.each_kv { |k,v|
dict{v} = k
}
# Fill the gaps in the dictionary
var lchar =
base.range.each { |i|
if (dict.contains(i)) {
lchar = dict{i}
}
elsif (!lchar.is_empty) {
dict{i} = lchar
}
}
# Decode the input number
var decoded = []
base.range.reverse.each { |i|
var pow = base**i;
var div = enc//pow
var c = dict{div}
var fv = freq{c}
var cv = cf{c}
var rem = ((enc - pow*cv) // fv)
enc = rem
decoded << c
}
# Return the decoded output return decoded
}
var radix = 10; # can be any integer greater or equal with 2
%w(DABDDB DABDDBBDDBA ABRACADABRA TOBEORNOTTOBEORTOBEORNOT).each { |str|
var (enc, pow, freq) = arithmethic_coding(str.bytes, radix)
var dec = arithmethic_decoding(enc, radix, pow, freq).join_bytes('UTF-8')
printf("%-25s=> %19s * %d^%s\n", str, enc, radix, pow);
if (str != dec) {
die "\tHowever that is incorrect!"
}
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
Visual Basic .NET
<lang vbnet>Imports System.Numerics Imports System.Text Imports Freq = System.Collections.Generic.Dictionary(Of Char, Long) Imports Triple = System.Tuple(Of System.Numerics.BigInteger, Integer, System.Collections.Generic.Dictionary(Of Char, Long))
Module Module1
Function CumulativeFreq(freq As Freq) As Freq
Dim total As Long = 0
Dim cf As New Freq
For i = 0 To 255
Dim c = Chr(i)
If freq.ContainsKey(c) Then
Dim v = freq(c)
cf(c) = total
total += v
End If
Next
Return cf
End Function
Function ArithmeticCoding(str As String, radix As Long) As Triple
'The frequency of characters
Dim freq As New Freq
For Each c In str
If freq.ContainsKey(c) Then
freq(c) += 1
Else
freq(c) = 1
End If
Next
'The cumulative frequency
Dim cf = CumulativeFreq(freq)
' Base
Dim base As BigInteger = str.Length
' Lower bound
Dim lower As BigInteger = 0
' Product of all frequencies
Dim pf As BigInteger = 1
' Each term is multiplied by the product of the
' frequencies of all previously occuring symbols
For Each c In str
Dim x = cf(c)
lower = lower * base + x * pf
pf = pf * freq(c)
Next
' Upper bound
Dim upper = lower + pf
Dim powr = 0
Dim bigRadix As BigInteger = radix
While True
pf = pf / bigRadix
If pf = 0 Then
Exit While
End If
powr = powr + 1
End While
Dim diff = (upper - 1) / (BigInteger.Pow(bigRadix, powr))
Return New Triple(diff, powr, freq)
End Function
Function ArithmeticDecoding(num As BigInteger, radix As Long, pwr As Integer, freq As Freq) As String
Dim powr As BigInteger = radix
Dim enc = num * BigInteger.Pow(powr, pwr)
Dim base = freq.Values.Sum()
' Create the cumulative frequency table
Dim cf = CumulativeFreq(freq)
' Create the dictionary
Dim dict As New Dictionary(Of Long, Char)
For Each key In cf.Keys
Dim value = cf(key)
dict(value) = key
Next
' Fill the gaps in the dictionary
Dim lchar As Long = -1
For i As Long = 0 To base - 1
If dict.ContainsKey(i) Then
lchar = AscW(dict(i))
Else
dict(i) = ChrW(lchar)
End If
Next
' Decode the input number
Dim decoded As New StringBuilder
Dim bigBase As BigInteger = base
For i As Long = base - 1 To 0 Step -1
Dim pow = BigInteger.Pow(bigBase, i)
Dim div = enc / pow
Dim c = dict(div)
Dim fv = freq(c)
Dim cv = cf(c)
Dim diff = enc - pow * cv
enc = diff / fv
decoded.Append(c)
Next
' Return the decoded ouput
Return decoded.ToString()
End Function
Sub Main()
Dim radix As Long = 10
Dim strings = {"DABDDB", "DABDDBBDDBA", "ABRACADABRA", "TOBEORNOTTOBEORTOBEORNOT"}
For Each St In strings
Dim encoded = ArithmeticCoding(St, radix)
Dim dec = ArithmeticDecoding(encoded.Item1, radix, encoded.Item2, encoded.Item3)
Console.WriteLine("{0,-25}=> {1,19} * {2}^{3}", St, encoded.Item1, radix, encoded.Item2)
If St <> dec Then
Throw New Exception(vbTab + "However that is incorrect!")
End If
Next
End Sub
End Module</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15
zkl
Uses libGMP (GNU MP Bignum Library) <lang zkl>var [const] BN=Import("zklBigNum"); // libGMP
fcn cumulativeFreq(freqHash){
total,cf := 0,Dictionary();
foreach b in (256){ if(v:=freqHash.find(b)){ cf[b]=total; total+=v; } }
cf
}
fcn arithmethicCoding(str, radix){
bytes :=str.split("").apply("toAsc"); // string to bytes: "0"-->0x31
freqHash:=Dictionary(); bytes.pump(Void,freqHash.incV); // frequency chars
cf :=cumulativeFreq(freqHash); // The cumulative frequency
base,lower:=bytes.len(), BN(0); // Lower bound
pf:=BN(1); // Product of all frequencies
// Each term is multiplied by the product of the
// frequencies of all previously occurring symbols
foreach b in (bytes){
lower.mul(base).add(pf*cf[b]); // gets quite large
pf.mul(freqHash[b]); // gets big
}
upper,powr := lower + pf, 0;
while(1){
pf.div(radix); // in place BigInt math, no garbage
if(pf==0) break;
powr+=1;
}
enc:=(upper - 1)/BN(radix).pow(powr);
return(enc,powr,freqHash);
}</lang> <lang zkl>fcn arithmethicDecoding(enc, radix, powr, freqHash){
enc*=radix.pow(powr); base:=freqHash.values.sum(0); cf :=cumulativeFreq(freqHash); // Create the cumulative frequency table dict:=cf.pump(Dictionary(), // Invert/transpose cumulative table, keys are strings
fcn(kv){ kv.reverse().apply("toInt") });
// Fill the gaps in the dictionary
lchar:=Void;
foreach b in (base){
if(v:=dict.find(b)) lchar=v;
else if(lchar) dict[b]=lchar;
}
// Decode the input number
decoded:=Data(); // byte bucket
foreach n in ([base-1..0, -1]){
pow:=BN(base).pow(n); // a big number
div:=(enc/pow).toInt(); // a small number, convert from BigInt
c,fv,cv := dict[div],freqHash[c],cf[c];
decoded.append(c.toChar());
enc.sub(pow*cv).div(fv); // in place BigInt math, no garbage
}
decoded.text // Return the decoded output
}</lang> <lang zkl>radix:=10; testStrings:=T(
"DABDDB",
"DABDDBBDDBA",
"ABRACADABRA",
"TOBEORNOTTOBEORTOBEORNOT",);
foreach str in (testStrings){
enc,pow,freq := arithmethicCoding(str,radix);
dec:=arithmethicDecoding(enc, radix, pow, freq);
print("%-25s=> %19s * %d^%s\n".fmt(str,enc,radix,pow));
if(str!=dec) println("\tHowever that is incorrect!");
}</lang>
- Output:
DABDDB => 251 * 10^2 DABDDBBDDBA => 167351 * 10^6 ABRACADABRA => 7954170 * 10^4 TOBEORNOTTOBEORTOBEORNOT => 1150764267498783364 * 10^15