On Linear Solution of "Cherry Pickup II". Max Weight of Two Disjoint Paths in Node-Weighted Gridlike DAG
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"Minimum Falling Path Sum" (MFPS) is classic question in programming - "Given a grid of size N×N with integers in cells, return the minimum sum of a falling path through grid. A falling path starts at any cell in the first row and ends in last row, with the rule of motion - the next element after the cell (i,j) is one of the cells (i+1,j-1),(i+1,j) and (i+1,j+1)". This problem has linear solution (LS) (i.e. O(N^2)) using dynamic programming method (DPM). There is an Multi-Agent version of MFPS called "Cherry Pickup II" (CP2). CP2 is a search for the maximum sum of 2 falling paths started from top corners, where each covered cell summed up one time. All known fast solutions of CP2 uses DPM, but have O(N^3) time complexity on grid N×N. Here we offer a LS of CP2 (also using DPM) as finding maximum total weight of 2 vertex-disjoint paths. Also, we extend this LS for some extended version of CP2 with wider motion rules.
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