I Think

Problem 18

My home pc has been getting a lot of makeover lately. First it was the os update, and yesterday I replaced my eight year old CRT monitor with a sleek 20″ led monitor. Also replaced the dusty keyboard with a new one. The pc looked so nice that I sat and did some coding for fun after a long time on this machine. Went to project euler and started solving the problems, in python. The first real test came in 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

The question also had a helpful tip warning against using brute force. After a few minutes of thought, the algorithm became clear. Given a point in the triangle (that is not on either end of the row), there are two ways of reaching it of which only one has the max sum, so I only need to maintain the maximum sum leading up to the point and discard the other. Here is the code.

#!/usr/bin/python
def problem18():
    a = [
        [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,]
        ]

    def flatten(row1, row2):
        flat = []
        firstrowlen = len(row1)
        secondrowlen = len(row2)
        for i, val in enumerate(row2):
            if i == 0:
                flat.append(val + row1[0])
            elif i == secondrowlen - 1 :
                flat.append(val + row1[firstrowlen - 1])
            else:
                flat.append(max(val + row1[i-1], (val + row1[i])))
        return flat

    finalflat = []
    numrows = len(a)
    for rownum, row in enumerate(a):
        if rownum == 0:
            finalflat = row
        if rownum + 1 <= numrows-1:
            finalflat = flatten(finalflat, a[rownum + 1])

    print reduce(max, finalflat)

if __name__ =="__main__":
    problem18()

*That was because I chose python, using python almost feels like cheating, for example to compute 2 power 1000 was tricky in C, but in python it is as simple as 2**1000.

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2 thoughts on “Problem 18

  1. NK says:

    dei actually this looks like a variant of viterbi decoding problem of convolutional code. anyway if u had thought abt it apriori, it is remarkable. (kitathatta viterbi rangeu ;))

  2. dei.. venam.. anda alavukku ithu illa.. i think this is a case of dynamic programming.. anyway i have seen similar(if not the same) problems elsewhere.. viterbia asinga paduttadha 😀

    anyway check that site.. the problems get progressively difficult.. nalla timepass

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