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2608. 图中的最短环

English Version

题目描述

现有一个含 n 个顶点的 双向 图，每个顶点按从 0 到 n - 1 标记。图中的边由二维整数数组 edges 表示，其中 edges[i] = [ui, vi] 表示顶点 ui 和 vi 之间存在一条边。每对顶点最多通过一条边连接，并且不存在与自身相连的顶点。

返回图中 最短 环的长度。如果不存在环，则返回 -1 。

环 是指以同一节点开始和结束，并且路径中的每条边仅使用一次。

 

示例 1：

输入：n = 7, edges = [[0,1],[1,2],[2,0],[3,4],[4,5],[5,6],[6,3]]
输出：3
解释：长度最小的循环是：0 -> 1 -> 2 -> 0

示例 2：

输入：n = 4, edges = [[0,1],[0,2]]
输出：-1
解释：图中不存在循环

 

提示：

  • 2 <= n <= 1000
  • 1 <= edges.length <= 1000
  • edges[i].length == 2
  • 0 <= ui, vi < n
  • ui != vi
  • 不存在重复的边

解法

方法一：枚举删除的边 + BFS

我们先根据数组 $edges$ 构建出邻接表 $g$，其中 $g[u]$ 表示顶点 $u$ 的所有邻接点。

接下来，我们枚举双向边 $(u, v)$，如果删除该边后，从顶点 $u$ 到顶点 $v$ 的路径依然存在，则包含该边的最短环的长度为 $dist[v] + 1$，其中 $dist[v]$ 表示从顶点 $u$ 到顶点 $v$ 的最短路径长度。我们取所有这样的环的最小值即可。

时间复杂度 $O(m^2)$，空间复杂度 $O(m + n)$。其中 $m$ 和 $n$ 分别为数组 $edges$ 的长度以及顶点数。

class Solution:
    def findShortestCycle(self, n: int, edges: List[List[int]]) -> int:
        def bfs(u: int, v: int) -> int:
            dist = [inf] * n
            dist[u] = 0
            q = deque([u])
            while q:
                i = q.popleft()
                for j in g[i]:
                    if (i, j) != (u, v) and (j, i) != (u, v) and dist[j] == inf:
                        dist[j] = dist[i] + 1
                        q.append(j)
            return dist[v] + 1

        g = defaultdict(set)
        for u, v in edges:
            g[u].add(v)
            g[v].add(u)
        ans = min(bfs(u, v) for u, v in edges)
        return ans if ans < inf else -1
class Solution {
    private List<Integer>[] g;
    private final int inf = 1 << 30;

    public int findShortestCycle(int n, int[][] edges) {
        g = new List[n];
        Arrays.setAll(g, k -> new ArrayList<>());
        for (var e : edges) {
            int u = e[0], v = e[1];
            g[u].add(v);
            g[v].add(u);
        }
        int ans = inf;
        for (var e : edges) {
            int u = e[0], v = e[1];
            ans = Math.min(ans, bfs(u, v));
        }
        return ans < inf ? ans : -1;
    }

    private int bfs(int u, int v) {
        int[] dist = new int[g.length];
        Arrays.fill(dist, inf);
        dist[u] = 0;
        Deque<Integer> q = new ArrayDeque<>();
        q.offer(u);
        while (!q.isEmpty()) {
            int i = q.poll();
            for (int j : g[i]) {
                if ((i == u && j == v) || (i == v && j == u) || dist[j] != inf) {
                    continue;
                }
                dist[j] = dist[i] + 1;
                q.offer(j);
            }
        }
        return dist[v] + 1;
    }
}
class Solution {
public:
    int findShortestCycle(int n, vector<vector<int>>& edges) {
        vector<vector<int>> g(n);
        for (auto& e : edges) {
            int u = e[0], v = e[1];
            g[u].push_back(v);
            g[v].push_back(u);
        }
        const int inf = 1 << 30;
        auto bfs = [&](int u, int v) -> int {
            int dist[n];
            fill(dist, dist + n, inf);
            dist[u] = 0;
            queue<int> q{{u}};
            while (!q.empty()) {
                int i = q.front();
                q.pop();
                for (int j : g[i]) {
                    if ((i == u && j == v) || (i == v && j == u) || dist[j] != inf) {
                        continue;
                    }
                    dist[j] = dist[i] + 1;
                    q.push(j);
                }
            }
            return dist[v] + 1;
        };
        int ans = inf;
        for (auto& e : edges) {
            int u = e[0], v = e[1];
            ans = min(ans, bfs(u, v));
        }
        return ans < inf ? ans : -1;
    }
};
func findShortestCycle(n int, edges [][]int) int {
	g := make([][]int, n)
	for _, e := range edges {
		u, v := e[0], e[1]
		g[u] = append(g[u], v)
		g[v] = append(g[v], u)
	}
	const inf = 1 << 30
	bfs := func(u, v int) int {
		dist := make([]int, n)
		for i := range dist {
			dist[i] = inf
		}
		dist[u] = 0
		q := []int{u}
		for len(q) > 0 {
			i := q[0]
			q = q[1:]
			for _, j := range g[i] {
				if (i == u && j == v) || (i == v && j == u) || dist[j] != inf {
					continue
				}
				dist[j] = dist[i] + 1
				q = append(q, j)
			}
		}
		return dist[v] + 1
	}
	ans := inf
	for _, e := range edges {
		u, v := e[0], e[1]
		ans = min(ans, bfs(u, v))
	}
	if ans < inf {
		return ans
	}
	return -1
}
function findShortestCycle(n: number, edges: number[][]): number {
    const g: number[][] = new Array(n).fill(0).map(() => []);
    for (const [u, v] of edges) {
        g[u].push(v);
        g[v].push(u);
    }
    const inf = 1 << 30;
    let ans = inf;
    const bfs = (u: number, v: number) => {
        const dist: number[] = new Array(n).fill(inf);
        dist[u] = 0;
        const q: number[] = [u];
        while (q.length) {
            const i = q.shift()!;
            for (const j of g[i]) {
                if ((i == u && j == v) || (i == v && j == u) || dist[j] != inf) {
                    continue;
                }
                dist[j] = dist[i] + 1;
                q.push(j);
            }
        }
        return 1 + dist[v];
    };
    for (const [u, v] of edges) {
        ans = Math.min(ans, bfs(u, v));
    }
    return ans < inf ? ans : -1;
}

方法二：枚举点 + BFS

与方法一类似，我们先根据数组 $edges$ 构建出邻接表 $g$，其中 $g[u]$ 表示顶点 $u$ 的所有邻接点。

接下来，我们枚举顶点 $u$，如果从顶点 $u$ 出发，有两条路径都到达了顶点 $v$，说明当前找到了一个环，长度为两条路径的长度之和。我们取所有这样的环的最小值即可。

时间复杂度 $O(m \times n)$，空间复杂度 $O(m + n)$。其中 $m$ 和 $n$ 分别为数组 $edges$ 的长度以及顶点数。

class Solution:
    def findShortestCycle(self, n: int, edges: List[List[int]]) -> int:
        def bfs(u: int) -> int:
            dist = [-1] * n
            dist[u] = 0
            q = deque([(u, -1)])
            ans = inf
            while q:
                u, fa = q.popleft()
                for v in g[u]:
                    if dist[v] < 0:
                        dist[v] = dist[u] + 1
                        q.append((v, u))
                    elif v != fa:
                        ans = min(ans, dist[u] + dist[v] + 1)
            return ans

        g = defaultdict(list)
        for u, v in edges:
            g[u].append(v)
            g[v].append(u)
        ans = min(bfs(i) for i in range(n))
        return ans if ans < inf else -1
class Solution {
    private List<Integer>[] g;
    private final int inf = 1 << 30;

    public int findShortestCycle(int n, int[][] edges) {
        g = new List[n];
        Arrays.setAll(g, k -> new ArrayList<>());
        for (var e : edges) {
            int u = e[0], v = e[1];
            g[u].add(v);
            g[v].add(u);
        }
        int ans = inf;
        for (int i = 0; i < n; ++i) {
            ans = Math.min(ans, bfs(i));
        }
        return ans < inf ? ans : -1;
    }

    private int bfs(int u) {
        int[] dist = new int[g.length];
        Arrays.fill(dist, -1);
        dist[u] = 0;
        Deque<int[]> q = new ArrayDeque<>();
        q.offer(new int[] {u, -1});
        int ans = inf;
        while (!q.isEmpty()) {
            var p = q.poll();
            u = p[0];
            int fa = p[1];
            for (int v : g[u]) {
                if (dist[v] < 0) {
                    dist[v] = dist[u] + 1;
                    q.offer(new int[] {v, u});
                } else if (v != fa) {
                    ans = Math.min(ans, dist[u] + dist[v] + 1);
                }
            }
        }
        return ans;
    }
}
class Solution {
public:
    int findShortestCycle(int n, vector<vector<int>>& edges) {
        vector<vector<int>> g(n);
        for (auto& e : edges) {
            int u = e[0], v = e[1];
            g[u].push_back(v);
            g[v].push_back(u);
        }
        const int inf = 1 << 30;
        auto bfs = [&](int u) -> int {
            int dist[n];
            memset(dist, -1, sizeof(dist));
            dist[u] = 0;
            queue<pair<int, int>> q;
            q.emplace(u, -1);
            int ans = inf;
            while (!q.empty()) {
                auto p = q.front();
                u = p.first;
                int fa = p.second;
                q.pop();
                for (int v : g[u]) {
                    if (dist[v] < 0) {
                        dist[v] = dist[u] + 1;
                        q.emplace(v, u);
                    } else if (v != fa) {
                        ans = min(ans, dist[u] + dist[v] + 1);
                    }
                }
            }
            return ans;
        };
        int ans = inf;
        for (int i = 0; i < n; ++i) {
            ans = min(ans, bfs(i));
        }
        return ans < inf ? ans : -1;
    }
};
func findShortestCycle(n int, edges [][]int) int {
	g := make([][]int, n)
	for _, e := range edges {
		u, v := e[0], e[1]
		g[u] = append(g[u], v)
		g[v] = append(g[v], u)
	}
	const inf = 1 << 30
	bfs := func(u int) int {
		dist := make([]int, n)
		for i := range dist {
			dist[i] = -1
		}
		dist[u] = 0
		q := [][2]int{{u, -1}}
		ans := inf
		for len(q) > 0 {
			p := q[0]
			u = p[0]
			fa := p[1]
			q = q[1:]
			for _, v := range g[u] {
				if dist[v] < 0 {
					dist[v] = dist[u] + 1
					q = append(q, [2]int{v, u})
				} else if v != fa {
					ans = min(ans, dist[u]+dist[v]+1)
				}
			}
		}
		return ans
	}
	ans := inf
	for i := 0; i < n; i++ {
		ans = min(ans, bfs(i))
	}
	if ans < inf {
		return ans
	}
	return -1
}
function findShortestCycle(n: number, edges: number[][]): number {
    const g: number[][] = new Array(n).fill(0).map(() => []);
    for (const [u, v] of edges) {
        g[u].push(v);
        g[v].push(u);
    }
    const inf = 1 << 30;
    let ans = inf;
    const bfs = (u: number) => {
        const dist: number[] = new Array(n).fill(-1);
        dist[u] = 0;
        const q: number[][] = [[u, -1]];
        let ans = inf;
        while (q.length) {
            const p = q.shift()!;
            u = p[0];
            const fa = p[1];
            for (const v of g[u]) {
                if (dist[v] < 0) {
                    dist[v] = dist[u] + 1;
                    q.push([v, u]);
                } else if (v !== fa) {
                    ans = Math.min(ans, dist[u] + dist[v] + 1);
                }
            }
        }
        return ans;
    };
    for (let i = 0; i < n; ++i) {
        ans = Math.min(ans, bfs(i));
    }
    return ans < inf ? ans : -1;
}