Knight's Traversal 2
Jan 26, 2012 - 09 p.m.


So remember a week ago, when I wrote a bit about some code I had written to solve the Knight's Keypad Traversal problem over at Programming Praxis? Sure you do. It's the last post I wrote. Go read it if you haven't.

Anyway, my solution used recursion, and was not the best solution because the complexity of getting the answer grew exponentially. I did not understand fully the better solution, so this week, I made it my business to understand it. I also decided it would be a good time to try and write some code in Ruby, for reasons which are clear to no one. (Notes on Ruby: Arrays seem far too difficult to create.)

The correct solution is the Dynamic Programming solution, which basically means we collapse the problem into smaller subproblems that are faster to solve. Bear with me here, as I've just finished grappling with this solution myself. I personally got a lot out of this stackoverflow answer. Like the recursive solution, we find the base case, which for this problem is the length of a single step path, and we solve it for every position on the key pad. By adding those single step path values together, we can determine the length of a two step path. When you start on one and step to either six or eight, you have one step left, and the know the length of a one step path from any number already, so you just add those together.
i.e. [two step path from one] = [one step path from 8] + [one step path from 6].
Here is a chart that illustrates the first two rows:

1 2 3 4 5 6 7 8 9 0
1 1 1 1 1 1 1 1 1 1
2 2 2 2 3 3 2 2 2 2

What makes this faster than the recursive solution is that we build up from the base case and each successive case is just a bit of quick addition with the previous steps answers. Here is some code.

paths = [
        [4, 6], #0
        [6,8], #1
        [7,9], #2
        [4,8], #3
        [0, 3, 9], #4
        [], #5
        [7, 1, 0], #6
        [6,2], #7
        [3,1], #8
        [4,2] #9

n = ARGV[0].to_i

counts = + 1) { { |i| i } }

(0..9).each do |i|
    counts[1][i] = 1

(2..n).each do |number|
    (0..9).each do |digit|
        sum = 0
        paths[digit].each do |from|
            sum += counts[number - 1][from]
        counts[number][digit] = sum

puts counts[n][1]
(on bitbucket)

I hope that it is fairly straightforward to see. I've found Ruby easier to read than to write so far. For every path length, we just check on step back in our array of previous answers and add together what we find. It is also quite zippy when compared to the recursive version. One is in python, the other ruby, so the comparison is sort of moot, but just for kicks:
[phil@philsmacbook:random-bits ]$ time python 10

real    0m0.346s
user    0m0.017s
sys     0m0.018s
[phil@philsmacbook:random-bits ]$ time ruby knight_keypad.rb 10

real    0m0.047s
user    0m0.003s
sys     0m0.004s

(I tried to run 100, but I think the recursive version is still going.) Pretty cool stuff. Let me know if this post doesn't make a lot of sense... I'm still learning to explain technical things, and since this is something it took me the better part of a day to understand, it might not be perfectly clear. Enjoy.