Project Euler #18: Maximum path sum I

This article is part of a series where I'll be diving head first into the Project Euler puzzles. I want to document the challenge of solving such a puzzle and how I got to the answer. I want to prefix this by stating that I can't cheat for any of these challenges; with that I mean I can't look up any other implementations online. After the implementation, I will validate the answer by using this document or a similar sheet.

In this article I'll be solving: Project Euler #18.

Introduction

The puzzle starts out with explaining that there’s a triangle. We start from the top and go down until we find the route which produces the highest number.

The example they give is:

   3
  7 4
 2 4 6
8 5 9 3

3 + 7 + 4 + 9 = 23

The puzzle is to give a solution for a more complex triangle:

              75
             95 64
            17 47 82
           18 35 87 10
          20 04 82 47 65
         19 01 23 75 03 34
        88 02 77 73 07 63 67
       99 65 04 28 06 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 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

The first step while writing code, is going to be finding a structure that works for containing these numbers in such a way. My first thought is: “this should be a tree”, however I do think a two dimensional vector works just as well:

// Example with the small triangle:
let triangle = vec![
    vec![3],
    vec![7, 4],
    vec![2, 4, 6],
    vec![8, 5, 9, 3]
];

The wrong route The reason I’m saying this is because, you can still limit the amount of choices here be taking slices of length 2 from the array below each time. The way this would work is like this:

// Taking the 'triangle' variable above as 't':
fn problem_18(mut t: Vec<Vec<u32>>) -> u32 {
    let mut result = t.remove(0)[0]; // This value is: 3
    let mut offset = 0;

    for sub_t in t {
        let mut max = 0;
        // - for the first cycle, look at [7, 4]
        // as subset from [7, 4]
        // the highest index = 0
        // - for the next cycle look at subset [2, 4]
        // as subset from [2, 4, 6]
        // the highest index = 1
        // - for the next cycle look at the subset [5, 9]
        // from [8, 5, 9, 3]
        // the highest index = 2
        for (i, val) in sub_t[offset..offset + 2].iter().enumerate() {
            if max < *val {
                max = *val;
                offset = i;
            }
        }
        result += max
    }

    result
}

I’m not sure if this is how I should interpret this triangle, or if I should actually use a tree. Because the puzzle isn’t specific about this, it might be nice to explore both as possible answers. Putting the big triangle in the method above I get the answer 1036 while the answer says it’s 1074 (38 off), meaning it’s time to start debugging. The first thing I noticed is that the offset that’s being picked is wrong while it descends down the triangle; the offset should add i to offset.

for (i, val) in sub_t[offset..offset + 2].iter().enumerate() {
    if max < *val {
        max = *val;
        offset += i;
    }
}

This changes the answer from 1036 to 1064 (which is 10 off from 1074). I Googled my faulty answer and somebody proceeded to explain why, causing me to read the actual approach (aren’t there spoiler free hints anywhere?).

The right route To resolve this you actually have to start from the bottom of the triangle. Technically, I didn’t come up with this answer, which does feel a little bit like cheating. However, now that I know, let’s draw out how that works anyway:

   3
  7 4
 2 4 6
8 5 9 3

Do 8 + 2 <> 5 + 2 . swap 2 with 10
Do 5 + 4 <> 9 + 4 . swap 4 with 13
Do 9 + 6 <> 6 + 3 . swap 6 with 15

   3
  7 4
 10 13 15

etc.

Even though I cheated a little bit, writing the code was still a bit of a challenge in Rust because of the inline updating of the triangle vector. However after some carefully picked iter_mut()’s I came up with this:

let mut inbetween = triangle.pop().unwrap();

for layer in triangle.iter_mut().rev() {
    for (index, value) in layer.iter_mut().enumerate() {
        let x1 = inbetween.get(index).unwrap();
        let x2 = inbetween.get(index + 1).unwrap_or(&0);

        if *value + x1 > *value + x2 {
            *value += x1
        } else {
            *value += x2
        }
    }

    inbetween = layer.to_vec();
}

triangle[0][0]

The full solution is available on GitHub.

55545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321