@algorithm.ts/bellman-ford

A typescript implementation of the bellman-ford algorithm.

The following definition is quoted from Wikipedia (https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm):

The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. The algorithm was first proposed by Alfonso Shimbel (1955), but is instead named after Richard Bellman and Lester Ford Jr., who published it in 1958 and 1956, respectively. Edward F. Moore also published a variation of the algorithm in 1959, and for this reason it is also sometimes called the Bellman–Ford–Moore algorithm.

Install

• npm

  npm install --save @algorithm.ts/bellman-ford

• yarn

  yarn add @algorithm.ts/bellman-ford

Usage

• Simple

  import type { IBellmanFordGraph } from '@algorithm.ts/bellman-ford'
  import bellmanFord from '@algorithm.ts/bellman-ford'
  
  const graph: IBellmanFordGraph<number> = {
    N: 4,
    source: 0,
    edges: [
      { to: 1, cost: 2 },
      { to: 2, cost: 2 },
      { to: 3, cost: 2 },
      { to: 3, cost: 1 },
    ],
    G: [[0], [1, 2], [3], []],
  }
  
  const result = bellmanFord(graph)
  /**
   * {
   *   hasNegativeCycle: false, // there is no negative-cycle.
   *   INF: 4503599627370494,
   *   source: 0,
   *   bestFrom: ,
   *   dist: [0, 2, 4, 4]
   * }
   *
   * For dist:
   *    0 --> 0: cost is 0
   *    0 --> 1: cost is 2
   *    0 --> 2: cost is 4
   *    0 --> 3: cost is 4
   */

• Options

  | Name | Type | Required | Description | | :---: | :-----: | :------: | :---------- | ------------------------------------------------ | | INF | number | bigint | false | A big number, representing the unreachable cost. |

Example

• Get shortest path.

  import type { IBellmanFordGraph } from '@algorithm.ts/bellman-ford'
  import bellmanFord from '@algorithm.ts/bellman-ford'
  import { getShortestPath } from '@algorithm.ts/graph'
  
  const A = 0
  const B = 1
  const C = 2
  const D = 3
  
  const graph: IBellmanFordGraph<number> = {
    N: 4,
    source: A,
    edges: [
      // Nodes: [A, B, C, D]
      { to: B, cost: 1 },       // A-B (1)
      { to: A, cost: -1 },      // B-A (-1)
      { to: C, cost: 0.87 },    // B-C (0.87)
      { to: B, cost: -0.87 },   // C-B (-0.87)
      { to: D, cost: 5 },       // C-D (5)
      { to: C, cost: -5 },      // D-C (-5)
    ],
    G: [[0], [1, 2], [3, 4], [5]],
  }
  
  const result = _bellmanFord.bellmanFord(graph)
  assert(result.negativeCycle === false)
  
  getShortestPath(result.bestFrom, Nodes.A, Nodes.A) // [Nodes.A]
  getShortestPath(result.bestFrom, Nodes.A, Nodes.B) // [Nodes.A, Nodes.B]
  getShortestPath(result.bestFrom, Nodes.A, Nodes.C) // [Nodes.A, Nodes.B, Nodes.C]
  getShortestPath(result.bestFrom, Nodes.A, Nodes.D) // [Nodes.A, Nodes.B, Nodes.C, Nodes.D])

• A solution for leetcode "Number of Ways to Arrive at Destination" (https://leetcode.com/problems/number-of-ways-to-arrive-at-destination/):

  import type { IBellmanFordEdge, IBellmanFordGraph } from '@algorithm.ts/bellman-ford'
  import { bellmanFord } from '@algorithm.ts/bellman-ford'
  
  const MOD = 1e9 + 7
  export function countPaths(N: number, roads: number[][]): number {
    const edges: Array<IBellmanFordEdge<number>> = []
    const G: number[][] = new Array(N)
    for (let i = 0; i < N; ++i) G[i] = []
    for (const [from, to, cost] of roads) {
      G[from].push(edges.length)
      edges.push({ to, cost })
  
      G[to].push(edges.length)
      edges.push({ to: from, cost })
    }
  
    const source = 0
    const target = N - 1
    const graph: IBellmanFordGraph<number> = { N, source: target, edges, G }
    const result = bellmanFord(graph, { INF: 1e12 })
    if (result.hasNegativeCycle) return -1
  
    const { dist } = result
    const dp: number[] = new Array(N).fill(-1)
    return dfs(source)
  
    function dfs(o: number): number {
      if (o === target) return 1
  
      let answer = dp[o]
      if (answer !== -1) return answer
  
      answer = 0
      const d = dist[o]
      for (const idx of G[o]) {
        const e: IBellmanFordEdge<number> = edges[idx]
        if (dist[e.to] + e.cost === d) {
          const t = dfs(e.to)
          answer = modAdd(answer, t)
        }
      }
      return dp[o] = answer
    }
  }
  
  function modAdd(x: number, y: number): number {
    const z: number = x + y
    return z < MOD ? z : z - MOD
  }

Related

• 《算法竞赛入门经典（第 2 版）》（刘汝佳）： P363 Bellman-Ford 算法
• bellman-ford | Wikipedia
• @algorithm.ts/queue