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MinCut.ts
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126 lines (100 loc) · 2.82 KB
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/*
*
* 最小割算法 - 基于最大流
*
* 问题：找到容量最小的割，使源点和汇点不再连通
*
* 核心思想：
* - 使用最大流算法计算最大流
* - 在残差网络中从源点BFS标记可达顶点
* - 从可达集到不可达集的边即为最小割
*
* 时间复杂度: O(VE²)
* 空间复杂度: O(V + E)
*/
const MAX_V = 100;
let graph: number[][] = Array(MAX_V).fill(null).map(() => Array(MAX_V).fill(0));
let V: number = 0;
function bfs(rGraph: number[][], s: number, t: number, parent: number[]): boolean {
let visited: boolean[] = Array(V).fill(false);
let queue: number[] = [];
queue.push(s);
visited[s] = true;
parent[s] = -1;
while (queue.length > 0) {
let u = queue.shift()!;
for (let v = 0; v < V; v++) {
if (!visited[v] && rGraph[u][v] > 0) {
queue.push(v);
parent[v] = u;
visited[v] = true;
if (v === t) {
return true;
}
}
}
}
return false;
}
function maxFlow(rGraph: number[][], s: number, t: number): number {
let parent: number[] = Array(V).fill(-1);
let max_flow = 0;
while (bfs(rGraph, s, t, parent)) {
let path_flow = Infinity;
for (let v = t; v !== s; v = parent[v]) {
let u = parent[v];
path_flow = Math.min(path_flow, rGraph[u][v]);
}
for (let v = t; v !== s; v = parent[v]) {
let u = parent[v];
rGraph[u][v] -= path_flow;
rGraph[v][u] += path_flow;
}
max_flow += path_flow;
}
return max_flow;
}
function minCut(s: number, t: number): void {
let rGraph: number[][] = graph.map(row => [...row]);
let max_flow = maxFlow(rGraph, s, t);
console.log(`最大流量: ${max_flow}`);
let visited: boolean[] = Array(V).fill(false);
let queue: number[] = [];
queue.push(s);
visited[s] = true;
while (queue.length > 0) {
let u = queue.shift()!;
for (let v = 0; v < V; v++) {
if (!visited[v] && rGraph[u][v] > 0) {
queue.push(v);
visited[v] = true;
}
}
}
console.log("最小割边:");
for (let u = 0; u < V; u++) {
for (let v = 0; v < V; v++) {
if (visited[u] && !visited[v] && graph[u][v] > 0) {
console.log(` ${u} -> ${v} (容量: ${graph[u][v]})`);
}
}
}
}
function main(): void {
console.log("=== 最小割算法 ===");
V = 6;
graph[0][1] = 16;
graph[0][2] = 13;
graph[1][2] = 10;
graph[1][3] = 12;
graph[2][1] = 4;
graph[2][4] = 14;
graph[3][2] = 9;
graph[3][5] = 20;
graph[4][3] = 7;
graph[4][5] = 4;
let s = 0, t = 5;
minCut(s, t);
}
// 运行测试
main();