## 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.

 131 673 234 103 18 201 96 342 965 150 630 803 746 422 111 537 699 497 121 956 805 732 524 37 331

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) => {
})
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]
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 --harmony-destructuring path-sum.js real 0m13.790s user 0m13.516s sys 0m0.120s ## 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   0m2.066s
user   0m1.996s
sys    0m0.060s