Knight in n, part 3: rings

Previously in this series:

In this third installment, we will look at how to use various types as numbers, i.e. how to make them an instance of the Num type class. The solution the Knight-moves-problem will emerge at the end, almost as if by magic. :)

Tangent: Things as numbers

Many types can be used as if they are numbers. Haskell-wise this means they can be an instance of the Num type class. Mathematically it means that these types are rings.

Pairs as numbers

Let's start with a Num instance for pairs (α,β). In general, our only choice is to do everything pointwise. So for all operations ⊗ (i.e. (+), (-) and (*):

In ring theory this is called the direct product. In Haskell we can write it as:

instance (Num α, Num β) => Num (α,β) where
    (a,b) + (c,d) = (a+c,b+d)
    (a,b) - (c,d) = (a-c,b-d)
    (a,b) * (c,d) = (a*c,b*d)
    fromInteger i = (fromInteger i, fromInteger i)
    abs     (a,b) = (abs    a, abs    b)
    signum  (a,b) = (signum a, signum b)

We could also make instances for triples, quadruples and other tuples this way, but those are not needed for the rest of the story.

Arrays as numbers

A more general kind of tuple is an array; which is somewhat like a tuple of arbitrary size. Of course, that is not quite true, since two arrays with the same type can have a different size. One way around this problem is to treat all arrays as if they are infinite, by taking values outside the bounds to be equal to 0. So

-- EXAMPLE
listArray (0,0) [1] == listArray (-∞,∞) [..,0,0,1,0,0,..] -- pseudocode

That way we can still do addition pointwise,

The accumArray function can help with the missing elements by setting them to 0 by default:

addArray a b = accumArray (+) 0 (min alo blo, max ahi bhi) (assocs a ++ assocs b)
  where (alo,ahi) = bounds a
        (blo,bhi) = bounds b

Next up is the fromInteger function. fromInteger 0 is easy; there are two options for other values

.# fromInteger i is an infinite array of values i. .# fromInteger i is an array with values i at some single point.

The first choice mimics the definition for tuples, fromInteger i = (fromInteger i, fromInteger i). But for arrays this has the slight problem of requiring an infinite array. For the second alternative we need to pick the index where to put the number i. The obvious choice is to put i at 'the origin', index 0:

intArray i = listArray (0,0) [fromInteger i]

Finally, multiplication. As you have learned in school, multiplication can be seen as repeated addition, In our Haskell world that means that we expect the law a + a = fromInteger 2 * a to hold.

If we had used the first choice for fromInteger then multiplication could be done pointwise as it was for tuples. But we have made a different choice, so now fromInteger 2 is an array that contains the value 2 at index 0 (and is implicitly zero everywhere else). When calculating fromInteger 2 * a, this 2 should by multiplied with all elements of the array a.

The operation that does the right thing is convolution. It looks like this:

So for each element v at index i in the first array, we shift a copy of the second array so that its origin becomes i. This copy is multiplied by v and all these copies are added. If one of the arrays is fromInteger v (i.e. a scalar), then this corresponds to multiplying all elements in the other array by v; exactly what we wanted.

Convolution can be implemented with accumArray as:

mulArray a b
    = accumArray (+) 0 (bounds a + bounds b)
      [ (i+j, x*y) | (i,x) <- assocs a, (j,y) <- assocs b ]

Notice that we use the Num (α,β) instance for the bounds, and that this definition is nicely symmetrical.

Putting it all together, we get the following instance:

instance (Ix i, Num i, Num a) => Num (Array i a) where
    fromInteger = intArray
    (+)         = addArray
    (*)         = mulArray
    negate      = fmap negate
    abs         = fmap abs
    signum      = fmap signum

In mathematical terms, what we constructed here is called a group ring. There is a group ring G[R] for any group G and ring R, which corresponds to an instance Num (Array g r) when g is a group (i.e. an instance of Num) and r is a ring (also an instance of Num).

Arrays as polynomials

Another way to interpret the above instance, is by treating arrays as polynomials over some variable x. The array array [(i,a),(j,b),(k,c),..] then represents the polynomial axi+bxj+cxk+.... The addition and multiplication defined above now have the expected meaning, for example:

> let a = listArray (0,2) [2,3,4]  --  2 + 3x + 4x^2
> let b = listArray (1,2) [5,6]    --  5x + 6x^2
> a + b
array (0,2) [(0,2),(1,8),(2,10)]   --  2 + 8x + 10x^2
> a * b
array (1,4) [(1,10),(2,27),(3,38),(4,24)]  --  10x + 27x^2 + 38x^3 + 24x^4

We can make this even more suggestive by defining:

x = listArray (1,1) [1]
> (2 + 3*x + 4*x^2) * (5*x + 6*x^2) == 10*x + 27*x^2 + 38*x^3 + 24*x^4
True

If you are interested in this interpretation, sigfpe wrote an interesting blog post about convolutions, polynomials and power series.

It's magic!

Now, let's go back to our original problem, the moves of a chess knight.

The positions reachable in a single move can be put into a two dimensional array (i.e. a matrix).

moveMatrix :: Array (Int,Int) Integer
moveMatrix = accumArray (+) 0 ((-2,-2),(2,2)) [ (m,1) | m <- moves ]

This is the familiar move matrix, which we already saw in part 1.

Knight3> printMatrix moveMatrix
    0 1 0 1 0
    1 0 0 0 1
    0 0 0 0 0
    1 0 0 0 1
    0 1 0 1 0

Now the magic. We defined multiplication of two arrays a and b as adding copies of b for each value in a. If we use the move matrix as b, then this means we add all possible destinations of a knight making one move from each place it can reach. Repeating this n times gives us our answer. Since repeated multiplication is exponentiation:

allPathsconv n = moveMatrix ^ n

For example, for n=2:
moveMatrix * moveMatrix

If we are interested in just a single point there is the array indexing operator (!!) to help us,

pathsconv n ij
    | inRange (bounds m) ij = m ! ij
    | otherwise             = 0
  where m = allPathsconv n

This convolutional algorithm can count the number of paths in O(n3), but not just for a single end point, but for all end points at once! The program is also a lot simpler than the

The pathsconv algorithm is pretty good, but we can still do better. Next time I will show how the algorithm from part 3 can be improved further, and curiously, how it will start to look more like the algorithm from part 2.

Comments

I like the convolution approach; it's very natural once you think of it :) However, it has O(n^2) memory requirement, which might become problematic for n bigger than, say 10.000. (I haven't looked in detail at the memory requirement of the other solutions, but I think they would be lower.)

All of the other solutions I discussed so far require O(n) memory. Either for a lookup table of factorials, or for stack space during recursion.

The part 4 algorithm contains an optimalization of the memory requirements. The first version I wrote needed O(n^2), but using a trick this could be reduced to O(n).

Finally, note that the memory requirement for pathsconv might actually be O(n^3) in this implementation, because of lazy evaluation. Depending on the order in which things are evaluated, the garbage collector might be required to keep all smaller arrays. All this in turn depends on what the exponential operator, (^), does. If it uses repeated multiplcation in a dumb way (x^4 = x*(x*(x*x)))) then there are O(n) intermediary matrices. However, if it uses a smarter algorithm (which it does), then x^4=(x*x)*(x*x), and there are only O(log n) intermediary matrices. So, while the time complexity is not affected by how exponentiation is done, space complexity is.

smuckersx
this definition is a nicely symmetrical.

should probably leave out the a.

Benx

Last picture sum?

I'm pretty sure that the last picture of the post is subtly wrong: while each summand is right, the sum is wrong. The middle line, for instance, should be 2 0 2 0 8 0 2 0 2, as in your first post...

HerbertDate: 2011-04-08T12:08Zx

This solution is really nice.

HerbertDate: 2011-04-18T09:13Zx

btw, how is a concolution of two matricies of nxn (since the matrix will get this big linear) O(n²)? it is O(n⁴). So the algorithm is O(n⁵) of O(n⁴log(n)) if you do the power thing by using that x^(2n) = (x^n)². What am I missing here?

Twan van LaarhovenDate: 2011-04-18T19:20Zx

Herbert: You don't need to calculate the convolution of two n*n matrices, but of one n*n matrix, and a constant matrix with 8 non-zero entries. That takes O(n2) time. Then you do n of these steps for a total of O(n3).

HerbertDate: 2011-04-18T19:34Zx

ah, right, for my suggestion using log(n) powers, you would need to calculate it for larger matrices.

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