Question

Suppose I had this Directed Acyclic Graph (DAG), where there is a directed edge from each node (other than the nodes in the bottom row) to the two nodes below it:

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

I need to find a path through this DAG where the sum of the nodes’ weights is maximized. You can only move diagonally down-left or down-right from a node in this tree. So for example, 7, 3, 8, 7, 5, would give the maximum path in this tree.

The input file contains the DAG formatted in this way

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

My question is, what algorithm would be best to find maximum path and also how would this tree be represented in C++?

The node weights are non-negative.

Answer 1

I’d represent this triangle with a vector of vectors of ints.

Start at the bottom row and compare each adjanced pair of numbers. Take the bigger one and add it to the number above the pair:

 5 3             13  3
  \
7 8 6  becomes  7  8  6
^ ^

                  13 3               13 11
                     /
Next iteration   7  8  6   becomes  7  8  6  etc.
                    ^  ^

Work your way to the top and when you’re done, the tip of the triangle will contain the largest sum.