Question

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

The minimal path sum in the 5 by 5 matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red and bold; the sum is equal to 994.

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 left column to the right column.

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, sources, targets) {
  function neighbors([x, y]) {
    const candidates = [
      [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 frontier = []
  sources.forEach((source) => {
    const start = [source]
    frontier.push([evaluate(start), start])
  })
  const destinations = new Set()
  targets.forEach((target) => {
    destinations.add(target.toString())
  })
  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 (destinations.has(node.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"))
const sources = []
const targets = []
for (let i = 0; i < graph.length; i++) {
  sources.push([0, i])
  targets.push([graph[0].length - 1, i])
}
console.log(bfs(graph, sources, targets))

$ time node --use-strict path-sum.js
real   0m0.951s
user   0m1.133s
sys    0m0.047s

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, y - 1), (x, y + 1), (x + 1, y)]
        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(row) - 1, y) for y, row in enumerate(matrix)]
    sources = [(0, y) for y, row in enumerate(matrix)]
    print(sum(matrix[y][x] for (x, y) in astar(matrix, targets, sources)))

if __name__ == "__main__": main()

$ time python3 path-sum.py
real   0m1.978s
user   0m1.970s
sys    0m0.008s