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Introduction to dynamic programming

In this series we will try to solve one problem in different ways.

Problem​

The problem we are going to solve is one of CodeWars katas and is called Pyramid Slide Down.

We are given a 2D array of integers and we are to find the slide down. Slide down is a maximum sum of consecutive numbers from the top to the bottom.

Let's have a look at few examples. Consider the following pyramid:

   3
  7 4
 2 4 6
8 5 9 3

This pyramid has following slide down:

   *3
  *7 4
 2 *4 6
8 5 *9 3

And its value is 23.

We can also have a look at a bigger example:

                75
               95 64
              17 47 82
             18 35 87 10
            20  4 82 47 65
            19  1 23  3 34
         88  2 77 73  7 63 67
        99 65  4 28  6 16 70 92
       41 41 26 56 83 40 80 70 33
      41 48 72 33 47 32 37 16 94 29
     53 71 44 65 25 43 91 52 97 51 14
    70 11 33 28 77 73 17 78 39 68 17 57
   91 71 52 38 17 14 91 43 58 50 27 29 48
 63 66  4 68 89 53 67 30 73 16 69 87 40 31
 4 62 98 27 23 9 70 98 73 93 38 53 60  4 23

Slide down in this case is equal to 1074.

Solving the problem​

caution

I will describe the following ways you can approach this problem and implement them in Java1.

For all of the following solutions I will be using basic main function that will output true/false based on the expected output of our algorithm. Any other differences will lie only in the solutions of the problem. You can see the main here:

public static void main(String[] args) {
    System.out.print("Test #1: ");
    System.out.println(longestSlideDown(new int[][] {
            { 3 },
            { 7, 4 },
            { 2, 4, 6 },
            { 8, 5, 9, 3 }
    }) == 23 ? "passed" : "failed");

    System.out.print("Test #2: ");
    System.out.println(longestSlideDown(new int[][] {
            { 75 },
            { 95, 64 },
            { 17, 47, 82 },
            { 18, 35, 87, 10 },
            { 20, 4, 82, 47, 65 },
            { 19, 1, 23, 75, 3, 34 },
            { 88, 2, 77, 73, 7, 63, 67 },
            { 99, 65, 4, 28, 6, 16, 70, 92 },
            { 41, 41, 26, 56, 83, 40, 80, 70, 33 },
            { 41, 48, 72, 33, 47, 32, 37, 16, 94, 29 },
            { 53, 71, 44, 65, 25, 43, 91, 52, 97, 51, 14 },
            { 70, 11, 33, 28, 77, 73, 17, 78, 39, 68, 17, 57 },
            { 91, 71, 52, 38, 17, 14, 91, 43, 58, 50, 27, 29, 48 },
            { 63, 66, 4, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31 },
            { 4, 62, 98, 27, 23, 9, 70, 98, 73, 93, 38, 53, 60, 4, 23 },
    }) == 1074 ? "passed" : "failed");
}

Footnotes​

  1. cause why not, right!? ↩