Question

NOTE: This problem is a significantly more challenging version of Problem 81.

In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to 2297.

Find the minimal path sum, in matrix.txt (right click and ‘Save Link/Target As…’), a 31K text file containing a 80 by 80 matrix, from the top left to the bottom right by moving left, right, up, and down.

JavaScript

const fs = require("fs")

function parseMatrix(matrix) {
  return matrix.toString().trim().split("\n").map((line) => {
    return line.split(",").map((c) => parseInt(c))
  })
}

function bfs(graph, root, target) {
  function neighbors([x, y]) {
    const candidates = [
      [x - 1, y],
      [x + 1, y],
      [x, y - 1],
      [x, y + 1]
    ]
    return candidates.filter(([x, y]) => {
      return x >= 0 && x < graph[0].length && y >= 0 && y < graph.length
    })
  }

  function evaluate(path) {
    return path.reduce((acc, [x, y]) => acc + graph[y][x], 0)
  }

  const start = [root]
  const frontier = [[evaluate(start), start]]
  const explored = new Set()
  while (frontier.length > 0) {
    let path = null
    let min = Infinity
    let index = -1
    frontier.forEach(([score, candidate], i) => {
      if (score < min) {
        min = score
        path = candidate
        index = i
      }
    })
    frontier.splice(index, 1)
    const node = path[path.length - 1]
    explored.add(node.toString())
    if (node.toString() === target.toString()) {
      return min
    }
    neighbors(node).forEach((neighbor) => {
      if (!explored.has(neighbor.toString())) {
        const newPath = path.slice()
        newPath.push(neighbor)
        frontier.push([evaluate(newPath), newPath])
      }
    })
  }
}

const graph = parseMatrix(fs.readFileSync(__dirname + "/matrix.txt"))
console.log(bfs(graph, [0, 0], [graph[0].length - 1, graph.length - 1]))

$ time node --use-strict path-sum.js
real   0m0.666s
user   0m0.895s
sys    0m0.055s

Python

#!/usr/bin/env python2
import os
import heapq

def parse(matfile):
    return [[int(n) for n in row.split(',')] for row in matfile]

def distance(a, b):
    return sum(abs(a[i] - b[i]) for i in range(len(a)))

def astar(matrix, targets, sources):
    def neighbors(position):
        (x, y) = position
        candidates = [(x - 1, y), (x + 1, y), (x, y - 1), (x, y + 1)]
        return [(x, y) for (x, y) in candidates if x >= 0 and x < len(matrix)
                                               and y >= 0 and y < len(matrix[0])]

    def evaluate(path):
        f = sum(matrix[y][x] for (x, y) in path)
        h = min(distance(path[-1], target) for target in targets)
        return f + h

    targets = set(targets)
    frontier = set(sources)
    explored = set()
    frontier_queue = []
    for source in sources:
        path = [source]
        heapq.heappush(frontier_queue, (evaluate(path), path))

    while frontier:
        (_, path) = heapq.heappop(frontier_queue)
        frontier.remove(path[-1])
        explored.add(path[-1])
        if path[-1] in targets:
            return path
        for neighbor in neighbors(path[-1]):
            if neighbor not in frontier | explored:
                frontier.add(neighbor)
                new_path = path + [neighbor]
                heapq.heappush(frontier_queue, (evaluate(new_path), new_path))

def main():
    with open(os.path.join(os.path.dirname(__file__), "matrix.txt")) as matfile:
        matrix = parse(matfile)
    targets = [(len(matrix) - 1, len(matrix[0]) - 1)]
    sources = [(0, 0)]
    print(sum(matrix[y][x] for (x, y) in astar(matrix, targets, sources)))

if __name__ == "__main__": main()

$ time python3 path-sum.py
real   0m1.971s
user   0m1.970s
sys    0m0.000s