Talk:Spiral matrix

Explanation of Python code

See Spiral. --Paddy3118 06:30, 5 August 2008 (UTC)

At least for the iterative solution. --Paddy3118 10:48, 5 August 2008 (UTC)

J

Unlike the concise solution to Talk:Zig Zag#epilogue, this J function works entirely on a list, until the very end, when it reshapes the 1D list into the 2D array.

So if SPIRAL:

   SPIRAL=:spiral 5

is our spiral array, then ,SPIRAL is the rows catenated together:

   ,SPIRAL
0 1 2 3 4 15 16 17 18 5 14 23 24 19 6 13 22 21 20 7 12 11 10 9 8

insight no. 1

Nothing about this list pops out at me, but if you read the original article, you'll see a smart J guy (Joey Tuttle) noticed that this list looks like the grade of another list.

aside: grading

As I said in my earlier exposition, a grade represents the permutation vector required to sort the list.

Let's say in some non-J language, you had a list of chars:

  ['E','C','D','F','A','B' ]

Well, if you take that list, and make it a 2D array where the chars are the first column, and the index is the second column:

  [ ['E',0],
    ['C',1],
    ['D',2],
    ['F',3],
    ['A',4],
    ['B',5] ]

And then sort that 2D array (either by the first column, or by both, which is equivalent), you'd get this:

 [ ['A',4],
   ['B',5],
   ['C',1],
   ['D',2],
   ['E',0] ]

Pulling out the second column from the matrix (the integers), you'd get this:

  [4,5,1,2,0]

Which is the grade.

Well, in J, you don't use sort to get the grade, you use grade to get the sort. For example, get the grade:

   /:'ECDFAB'
4 5 1 2 0 3

then use that grade to index into the array:

   4 5 1 2 0 3 { 'ECDFAB'
ABCDEF

(in J indexing is a function, not part of the syntax. So, for example, x[4] in C becomes 4{x in J).

chasing Joey

Ok, back to the spiral. When we left off, Joey had begun to suspect that ,SPIRAL was actually a permutation vector. That is, that ,SPIRAL was the result of grading some other list. More formally, he suspected:

  (,SPIRAL) = (/: something_else)

But how to find something_else? Well, it may not be obvious, but /: is self-inverse. That is, if you apply it twice, it "undoes" itself.

    /: 4 5 1 2 0 3
4 2 3 5 0 1
   
   /: /: 4 5 1 2 0 3
4 5 1 2 0 3
   
   4 5 1 2 0 3 = (/: /: 4 5 1 2 0 3)
1 1 1 1 1 1

So, since we know

   something_else = (/: /: something_else)

and

   (,SPIRAL) = (/: something_else)

then, by substitution, we know

   something_else = (/: , SPIRAL)

If you're still iffy on function inverse, don't worry, we'll return to it later, and find out a more formal way to ask J to invert a function for us.

In any case, let's see what Joey saw:

   /: , SPIRAL
0 1 2 3 4 9 14 19 24 23 22 21 20 15 10 5 6 7 8 13 18 17 16 11 12

insight no. 2

I don't see obvious patterns when I look at the result of /: , SPIRAL above. But then, I'm not as smart as Joey. When he looked at it, he had a second insight: this array looks like a running-sum.

Here's what I mean by running-sum:

   +/\ 1 2 3 4     NB. running sum
1 3 6 10
  
   (1),(1+2),(1+2+3),(1+2+3+4)
1 3 6 10

Formally, Joey suspected:

 (/: , SPIRAL) = (+/\ another_thing)

But obviously +/\ is not self-inverse. So how can we discover another_thing?

We ask J to do it for us.

aside: calculus of functions

J provides for a calculus of functions. What is a "calculus of functions"? Well, just like functions act on data to produce data, we can have meta-functions that operate on functions to produce functions.

Think back to math notation ¹:

  log x   # (1) log of x
  log x2  # (2) log of x * x
  log2 x  # (3) log of log of x

Read the (3) again. How is it different from (2)? In (2), x has been manipulated, but in (3) log has been manipulated!

That is, in (2), there are two instances of x (in x*x), but in (3) there are two instances of log (in log(log(x))).

Analogously in J:

  log =: 10&^.  NB.  "^." is the log function in J
  x   =: 100
  log x         NB. (1) log of x
2
  log x^2       NB. (2) log of x*x
4
  log^:2 x      NB. (3) log of log of x
0.30103

Now let's take this one step further. You'll remember the notation

  x-1  # (4)

meant 1/x, that is, the inverse of the number x. But what did

  log-1 # (5)

mean? By analogy to (4), (5) meant the inverse of the log function.

Analogously in J:

   x^_1                   NB.  (4)
0.01
   log x   
2
   log_inverse =: log^:_1 NB.  (5)

   log_inverse log x  
100

Neat, huh?

caught him!

So what does all this have to do with our hero, Joey? Well, if you remember, he had the insight that (/: , SPIRAL) =(+/\ another_thing). But he was stumped: +/\ is not self-inverse, so how could he recover another_thing?

Well, now we know how he proceeded: +/\^:_1. Let's follow him:

   +/\^:_1 /: ,SPIRAL
0 1 1 1 1 5 5 5 5 _1 _1 _1 _1 _5 _5 _5 1 1 1 5 5 _1 _1 _5 1

Eureka! This list clearly shows a pattern. We can simply replicate the pattern, then undo (invert) the operations that generated it from SPIRAL, and we'll have our SPIRAL back.

home again

Put another way: if we can generate these ones-and-fives, we can generate SPIRAL. How? Well, since we generated the ones-and-fives from SPIRAL using:

   ones_and_fives =. +/\^:_1 /: , SPIRAL

Then we need to do the opposite of that to recover SPIRAL. To break this procedure down, we have:

   func3 =. +/\^:_1
   func2 =. /:
   func1 =. ,

   ones_and_fives =. func3 func2 func1 SPIRAL

Obviously if you're doing the "opposite" of something you proceed LIFO. That is, given y = func3(func2(func1(x))), then x =  func1^-1(func2^-1func3^-1(y))) ².

That is:

   SPIRAL =. func1^:_1 func2^:_1 func3^:_1 ones_and_fives

and we know:

1. The inverse of an inverse is the thing itself, so func3^:_1 is merely +/\.
2. The function /: is self-inverse, so func2^:_1 is merely /: (though /:^:_1 would work as well).
3. The function , (ravel) is not invertible: it loses information (it is possible that (,x)=(,y) is true but x=y is not). But that's ok, we have the information that it lost: it's the original shape of SPIRAL, which is the input to our function! So we can undo the ravel.

conclusion

Putting that all together, we conclude:

   SPIRAL =. (input) reshape /: +/\ ones_and_fives 

In English:

Joey discovered a very simple pattern underlying the spiral arrays. The spiral itself is merely the grade of the running-sum of this pattern, reshaped into a matrix.

Generating the underlying pattern (ones_and_fives) is left as an exercise for the reader. But, if you get stuck, it's spelled out in the original article.

notes

1. For simplicity, here log means log-base-10, or log₁₀.
2. In fact, LIFO of inverses is exactly how J inverts composite function.

DanBron 19:40, 5 August 2008 (UTC)

original J exposition

The J solution was:

   spiral =. ,~ $ [: /: }.@(2 # >:@i.@-) +/\@# <:@+: $ (, -)@(1&,)

Here are some hints that will allow you to reimplement it in your language:

   counts   =:  }.@(2 # >:@i.@-)
   counts 5
5 4 4 3 3 2 2 1 1
   
   values   =:  <:@:+: $ (, -)@(1&,)
   values 5
1 5 _1 _5 1 5 _1 _5 1
   
   copy     =:  #
   3 copy 9
9 9 9
   
   (counts copy values) 5
1 1 1 1 1 5 5 5 5 _1 _1 _1 _1 _5 _5 _5 1 1 1 5 5 _1 _1 _5 1
   
   sumscan  =:  +/\       NB.  Cumulative sum
   sumscan 0 1 2 3 4
0 1 3 6 10
   
   (counts sumscan@copy values) 5
1 2 3 4 5 10 15 20 25 24 23 22 21 16 11 6 7 8 9 14 19 18 17 12 13
   
   grade    =:  /:  NB.  Permutation which tells us how to sort
   grade 5 2 3 1 0 4
4 3 1 2 5 0
   
   (counts grade@sumscan@copy values) 5
0 1 2 3 4 15 16 17 18 5 14 23 24 19 6 13 22 21 20 7 12 11 10 9 8
   
   dup      =:  ,~
   dup 5
5 5
   
   reshape  =:  $   NB. Reshape an array
   3 4 reshape 'hello'
hell
ohel
lohe
   
   (dup reshape counts grade@sumscan@copy values) 5
 0  1  2  3 4
15 16 17 18 5
14 23 24 19 6
13 22 21 20 7
12 11 10  9 8

For a fuller explanation, see the original source.

Yet another J solution that looks both interesting and impenetrable to me. at least for Zig Zag a Haskel person had reimplemented the J solution and left me the clue that it involved a sort :-)

--Paddy3118 16:56, 5 August 2008 (UTC)

This one includes a sort, too :) That's one of the neatest parts! Alright, let me write out the algo real quick (I'll gloss over details and may fib a bit, to get the idea across quickly).