Problem 018


Project Euler

Maximum path sum I

Problem 18

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

3
7 4
2 4 6
8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

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

NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)


See the problem statement from project euler since the format is a bit off here. Also, using the right approach, this problem and problem 67 can be solved at the same time. Notice from the example, from the bottom up, if 9 is greater than 5 at the bottom, then at level 3 (with the top denote as level 1), 4 should choose 9 over 5 whenever a route from the top arrive at 4 in level 3. Once the sum 4+9 is calculated, it also can be use for the calculation at both 7 and 4 at level 2. Thus, a recursive algorithm (finding the maximum at a smaller triangle first from bottom up) with dynamic programming (using memory to remember the maximum value to be reuse) can have the following pseudo-code (assume a 2D array is used to fill the triangle shown above):

sum(row, col)
  if result[row][col] > 0 then // already calculated
    return result[row][col]
  if row == (size - 1) then // at the bottom
    return result[row][col] = triangle[row][col]
  return result[row][col] = triangle[row][col] + 
                            max(sum(row+1, col), sum(row+1, col+1))
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