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684. 冗余连接

题目描述

树可以看成是一个连通且 无环 的 无向 图。

给定往一棵 n 个节点 (节点值 1～n) 的树中添加一条边后的图。添加的边的两个顶点包含在 1 到 n 中间，且这条附加的边不属于树中已存在的边。图的信息记录于长度为 n 的二维数组 edges ，edges[i] = [ai, bi] 表示图中在 ai 和 bi 之间存在一条边。

请找出一条可以删去的边，删除后可使得剩余部分是一个有着 n 个节点的树。如果有多个答案，则返回数组 edges 中最后出现的那个。

 

示例 1：

输入: edges = [[1,2], [1,3], [2,3]]
输出: [2,3]

示例 2：

输入: edges = [[1,2], [2,3], [3,4], [1,4], [1,5]]
输出: [1,4]

 

提示:

  • n == edges.length
  • 3 <= n <= 1000
  • edges[i].length == 2
  • 1 <= ai < bi <= edges.length
  • ai != bi
  • edges 中无重复元素
  • 给定的图是连通的 

解法

方法一：并查集

根据题意，我们需要找到一条可以删去的边，删除后剩余部分是一个有着 $n$ 个节点的树。我们可以遍历每一条边，判断这条边是否在同一个连通分量中。如果在同一个连通分量中，则说明这条边是多余的，可以删除，直接返回这条边即可。否则，我们将这条边所连接的两个节点合并到同一个连通分量中。

时间复杂度 $O(n \log n)$，空间复杂度 $O(n)$。其中 $n$ 为边的数量。

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class Solution:
    def findRedundantConnection(self, edges: List[List[int]]) -> List[int]:
        def find(x: int) -> int:
            if p[x] != x:
                p[x] = find(p[x])
            return p[x]

        p = list(range(len(edges)))
        for a, b in edges:
            pa, pb = find(a - 1), find(b - 1)
            if pa == pb:
                return [a, b]
            p[pa] = pb

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class Solution {
    private int[] p;

    public int[] findRedundantConnection(int[][] edges) {
        int n = edges.length;
        p = new int[n];
        for (int i = 0; i < n; ++i) {
            p[i] = i;
        }
        for (int i = 0;; ++i) {
            int pa = find(edges[i][0] - 1);
            int pb = find(edges[i][1] - 1);
            if (pa == pb) {
                return edges[i];
            }
            p[pa] = pb;
        }
    }

    private int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }
}

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class Solution {
public:
    vector<int> findRedundantConnection(vector<vector<int>>& edges) {
        int n = edges.size();
        vector<int> p(n);
        iota(p.begin(), p.end(), 0);
        function<int(int)> find = [&](int x) {
            return x == p[x] ? x : p[x] = find(p[x]);
        };
        for (int i = 0;; ++i) {
            int pa = find(edges[i][0] - 1);
            int pb = find(edges[i][1] - 1);
            if (pa == pb) {
                return edges[i];
            }
            p[pa] = pb;
        }
    }
};

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func findRedundantConnection(edges [][]int) []int {
    n := len(edges)
    p := make([]int, n)
    for i := range p {
        p[i] = i
    }
    var find func(int) int
    find = func(x int) int {
        if p[x] != x {
            p[x] = find(p[x])
        }
        return p[x]
    }
    for i := 0; ; i++ {
        pa, pb := find(edges[i][0]-1), find(edges[i][1]-1)
        if pa == pb {
            return edges[i]
        }
        p[pa] = pb
    }
}

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function findRedundantConnection(edges: number[][]): number[] {
    const n = edges.length;
    const p: number[] = Array.from({ length: n }, (_, i) => i);
    const find = (x: number): number => {
        if (p[x] !== x) {
            p[x] = find(p[x]);
        }
        return p[x];
    };
    for (let i = 0; ; ++i) {
        const pa = find(edges[i][0] - 1);
        const pb = find(edges[i][1] - 1);
        if (pa === pb) {
            return edges[i];
        }
        p[pa] = pb;
    }
}

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/**
 * @param {number[][]} edges
 * @return {number[]}
 */
var findRedundantConnection = function (edges) {
    const n = edges.length;
    const p = Array.from({ length: n }, (_, i) => i);
    const find = x => {
        if (p[x] !== x) {
            p[x] = find(p[x]);
        }
        return p[x];
    };
    for (let i = 0; ; ++i) {
        const pa = find(edges[i][0] - 1);
        const pb = find(edges[i][1] - 1);
        if (pa === pb) {
            return edges[i];
        }
        p[pa] = pb;
    }
};

方法二：并查集（模板做法）

这里给出一个并查集的模板做法，供大家参考。

时间复杂度 $O(n \alpha(n))$，空间复杂度 $O(n)$。其中 $n$ 为边的数量，而 $\alpha(n)$ 是阿克曼函数的反函数，可以认为是一个很小的常数。

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class UnionFind:
    __slots__ = "p", "size"

    def __init__(self, n: int):
        self.p: List[int] = list(range(n))
        self.size: List[int] = [1] * n

    def find(self, x: int) -> int:
        if self.p[x] != x:
            self.p[x] = self.find(self.p[x])
        return self.p[x]

    def union(self, a: int, b: int) -> bool:
        pa, pb = self.find(a), self.find(b)
        if pa == pb:
            return False
        if self.size[pa] > self.size[pb]:
            self.p[pb] = pa
            self.size[pa] += self.size[pb]
        else:
            self.p[pa] = pb
            self.size[pb] += self.size[pa]
        return True


class Solution:
    def findRedundantConnection(self, edges: List[List[int]]) -> List[int]:
        uf = UnionFind(len(edges))
        for a, b in edges:
            if not uf.union(a - 1, b - 1):
                return [a, b]

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class UnionFind {
    private final int[] p;
    private final int[] size;

    public UnionFind(int n) {
        p = new int[n];
        size = new int[n];
        for (int i = 0; i < n; ++i) {
            p[i] = i;
            size[i] = 1;
        }
    }

    public int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }

    public boolean union(int a, int b) {
        int pa = find(a), pb = find(b);
        if (pa == pb) {
            return false;
        }
        if (size[pa] > size[pb]) {
            p[pb] = pa;
            size[pa] += size[pb];
        } else {
            p[pa] = pb;
            size[pb] += size[pa];
        }
        return true;
    }
}

class Solution {
    public int[] findRedundantConnection(int[][] edges) {
        UnionFind uf = new UnionFind(edges.length);
        for (int i = 0;; ++i) {
            if (!uf.union(edges[i][0] - 1, edges[i][1] - 1)) {
                return edges[i];
            }
        }
    }
}

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class UnionFind {
public:
    UnionFind(int n) {
        p = vector<int>(n);
        size = vector<int>(n, 1);
        iota(p.begin(), p.end(), 0);
    }

    bool unite(int a, int b) {
        int pa = find(a), pb = find(b);
        if (pa == pb) {
            return false;
        }
        if (size[pa] > size[pb]) {
            p[pb] = pa;
            size[pa] += size[pb];
        } else {
            p[pa] = pb;
            size[pb] += size[pa];
        }
        return true;
    }

    int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }

private:
    vector<int> p, size;
};

class Solution {
public:
    vector<int> findRedundantConnection(vector<vector<int>>& edges) {
        UnionFind uf(edges.size());
        for (int i = 0;; ++i) {
            if (!uf.unite(edges[i][0] - 1, edges[i][1] - 1)) {
                return edges[i];
            }
        }
    }
};

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type unionFind struct {
    p, size []int
}

func newUnionFind(n int) *unionFind {
    p := make([]int, n)
    size := make([]int, n)
    for i := range p {
        p[i] = i
        size[i] = 1
    }
    return &unionFind{p, size}
}

func (uf *unionFind) find(x int) int {
    if uf.p[x] != x {
        uf.p[x] = uf.find(uf.p[x])
    }
    return uf.p[x]
}

func (uf *unionFind) union(a, b int) bool {
    pa, pb := uf.find(a), uf.find(b)
    if pa == pb {
        return false
    }
    if uf.size[pa] > uf.size[pb] {
        uf.p[pb] = pa
        uf.size[pa] += uf.size[pb]
    } else {
        uf.p[pa] = pb
        uf.size[pb] += uf.size[pa]
    }
    return true
}

func findRedundantConnection(edges [][]int) []int {
    uf := newUnionFind(len(edges))
    for i := 0; ; i++ {
        if !uf.union(edges[i][0]-1, edges[i][1]-1) {
            return edges[i]
        }
    }
}

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class UnionFind {
    p: number[];
    size: number[];
    constructor(n: number) {
        this.p = Array.from({ length: n }, (_, i) => i);
        this.size = Array(n).fill(1);
    }

    find(x: number): number {
        if (this.p[x] !== x) {
            this.p[x] = this.find(this.p[x]);
        }
        return this.p[x];
    }

    union(a: number, b: number): boolean {
        const [pa, pb] = [this.find(a), this.find(b)];
        if (pa === pb) {
            return false;
        }
        if (this.size[pa] > this.size[pb]) {
            this.p[pb] = pa;
            this.size[pa] += this.size[pb];
        } else {
            this.p[pa] = pb;
            this.size[pb] += this.size[pa];
        }
        return true;
    }
}

function findRedundantConnection(edges: number[][]): number[] {
    const uf = new UnionFind(edges.length);
    for (let i = 0; ; ++i) {
        if (!uf.union(edges[i][0] - 1, edges[i][1] - 1)) {
            return edges[i];
        }
    }
}

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class UnionFind {
    constructor(n) {
        this.p = Array.from({ length: n }, (_, i) => i);
        this.size = Array(n).fill(1);
    }

    find(x) {
        if (this.p[x] !== x) {
            this.p[x] = this.find(this.p[x]);
        }
        return this.p[x];
    }

    union(a, b) {
        const pa = this.find(a);
        const pb = this.find(b);
        if (pa === pb) {
            return false;
        }
        if (this.size[pa] > this.size[pb]) {
            this.p[pb] = pa;
            this.size[pa] += this.size[pb];
        } else {
            this.p[pa] = pb;
            this.size[pb] += this.size[pa];
        }
        return true;
    }
}

/**
 * @param {number[][]} edges
 * @return {number[]}
 */
var findRedundantConnection = function (edges) {
    const uf = new UnionFind(edges.length);
    for (let i = 0; i < edges.length; i++) {
        if (!uf.union(edges[i][0] - 1, edges[i][1] - 1)) {
            return edges[i];
        }
    }
};

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