On Linear Solution of "Cherry Pickup II". Max Weight of Two Disjoint Paths in Node-Weighted Gridlike DAG

TL;DR

This paper introduces a linear-time dynamic programming solution for the Cherry Pickup II problem, optimizing the maximum sum of two disjoint falling paths in a grid, improving upon the previous cubic-time algorithms.

Contribution

The paper presents the first linear solution for Cherry Pickup II by reformulating it as a maximum weight of two vertex-disjoint paths problem, extending to wider motion rules.

Findings

Linear solution for Cherry Pickup II using dynamic programming

Reduction of complexity from O(N^3) to O(N^2)

Extension to broader motion rules in grid paths

Abstract

"Minimum Falling Path Sum" (MFPS) is classic question in programming - "Given a grid of size $N{\times}N$ 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 $N{\times}N$. 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.