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3372. Maximize the Number of Target Nodes After Connecting Trees I

Description

There exist two undirected trees with n and m nodes, with distinct labels in ranges [0, n - 1] and [0, m - 1], respectively.

You are given two 2D integer arrays edges1 and edges2 of lengths n - 1 and m - 1, respectively, where edges1[i] = [ai, bi] indicates that there is an edge between nodes ai and bi in the first tree and edges2[i] = [ui, vi] indicates that there is an edge between nodes ui and vi in the second tree. You are also given an integer k.

Node u is target to node v if the number of edges on the path from u to v is less than or equal to k. Note that a node is always target to itself.

Return an array of n integers answer, where answer[i] is the maximum possible number of nodes target to node i of the first tree if you have to connect one node from the first tree to another node in the second tree.

Note that queries are independent from each other. That is, for every query you will remove the added edge before proceeding to the next query.

 

Example 1:

Input: edges1 = [[0,1],[0,2],[2,3],[2,4]], edges2 = [[0,1],[0,2],[0,3],[2,7],[1,4],[4,5],[4,6]], k = 2

Output: [9,7,9,8,8]

Explanation:

  • For i = 0, connect node 0 from the first tree to node 0 from the second tree.
  • For i = 1, connect node 1 from the first tree to node 0 from the second tree.
  • For i = 2, connect node 2 from the first tree to node 4 from the second tree.
  • For i = 3, connect node 3 from the first tree to node 4 from the second tree.
  • For i = 4, connect node 4 from the first tree to node 4 from the second tree.

Example 2:

Input: edges1 = [[0,1],[0,2],[0,3],[0,4]], edges2 = [[0,1],[1,2],[2,3]], k = 1

Output: [6,3,3,3,3]

Explanation:

For every i, connect node i of the first tree with any node of the second tree.

 

Constraints:

  • 2 <= n, m <= 1000
  • edges1.length == n - 1
  • edges2.length == m - 1
  • edges1[i].length == edges2[i].length == 2
  • edges1[i] = [ai, bi]
  • 0 <= ai, bi < n
  • edges2[i] = [ui, vi]
  • 0 <= ui, vi < m
  • The input is generated such that edges1 and edges2 represent valid trees.
  • 0 <= k <= 1000

Solutions

Solution 1: Enumeration + DFS

According to the problem description, to maximize the number of target nodes for node $i$, we must connect node $i$ to one of the nodes $j$ in the second tree. Therefore, the number of target nodes for node $i$ can be divided into two parts:

  • In the first tree, the number of nodes reachable from node $i$ within a depth of $k$.
  • In the second tree, the maximum number of nodes reachable from any node $j$ within a depth of $k - 1$.

Thus, we can first calculate the number of nodes reachable within a depth of $k - 1$ for each node in the second tree. Then, we enumerate each node $i$ in the first tree, calculate the sum of the two parts mentioned above, and take the maximum value.

The time complexity is $O(n^2 + m^2)$, and the space complexity is $O(n + m)$. Here, $n$ and $m$ are the number of nodes in the two trees, respectively.

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class Solution:
    def maxTargetNodes(
        self, edges1: List[List[int]], edges2: List[List[int]], k: int
    ) -> List[int]:
        def build(edges: List[List[int]]) -> List[List[int]]:
            n = len(edges) + 1
            g = [[] for _ in range(n)]
            for a, b in edges:
                g[a].append(b)
                g[b].append(a)
            return g

        def dfs(g: List[List[int]], a: int, fa: int, d: int) -> int:
            if d < 0:
                return 0
            cnt = 1
            for b in g[a]:
                if b != fa:
                    cnt += dfs(g, b, a, d - 1)
            return cnt

        g2 = build(edges2)
        m = len(edges2) + 1
        t = max(dfs(g2, i, -1, k - 1) for i in range(m))
        g1 = build(edges1)
        n = len(edges1) + 1
        return [dfs(g1, i, -1, k) + t for i in range(n)]

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class Solution {
    public int[] maxTargetNodes(int[][] edges1, int[][] edges2, int k) {
        var g2 = build(edges2);
        int m = edges2.length + 1;
        int t = 0;
        for (int i = 0; i < m; ++i) {
            t = Math.max(t, dfs(g2, i, -1, k - 1));
        }
        var g1 = build(edges1);
        int n = edges1.length + 1;
        int[] ans = new int[n];
        Arrays.fill(ans, t);
        for (int i = 0; i < n; ++i) {
            ans[i] += dfs(g1, i, -1, k);
        }
        return ans;
    }

    private List<Integer>[] build(int[][] edges) {
        int n = edges.length + 1;
        List<Integer>[] g = new List[n];
        Arrays.setAll(g, i -> new ArrayList<>());
        for (var e : edges) {
            int a = e[0], b = e[1];
            g[a].add(b);
            g[b].add(a);
        }
        return g;
    }

    private int dfs(List<Integer>[] g, int a, int fa, int d) {
        if (d < 0) {
            return 0;
        }
        int cnt = 1;
        for (int b : g[a]) {
            if (b != fa) {
                cnt += dfs(g, b, a, d - 1);
            }
        }
        return cnt;
    }
}

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class Solution {
public:
    vector<int> maxTargetNodes(vector<vector<int>>& edges1, vector<vector<int>>& edges2, int k) {
        auto g2 = build(edges2);
        int m = edges2.size() + 1;
        int t = 0;
        for (int i = 0; i < m; ++i) {
            t = max(t, dfs(g2, i, -1, k - 1));
        }

        auto g1 = build(edges1);
        int n = edges1.size() + 1;

        vector<int> ans(n, t);
        for (int i = 0; i < n; ++i) {
            ans[i] += dfs(g1, i, -1, k);
        }

        return ans;
    }

private:
    vector<vector<int>> build(const vector<vector<int>>& edges) {
        int n = edges.size() + 1;
        vector<vector<int>> g(n);
        for (const auto& e : edges) {
            int a = e[0], b = e[1];
            g[a].push_back(b);
            g[b].push_back(a);
        }
        return g;
    }

    int dfs(const vector<vector<int>>& g, int a, int fa, int d) {
        if (d < 0) {
            return 0;
        }
        int cnt = 1;
        for (int b : g[a]) {
            if (b != fa) {
                cnt += dfs(g, b, a, d - 1);
            }
        }
        return cnt;
    }
};

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func maxTargetNodes(edges1 [][]int, edges2 [][]int, k int) []int {
    g2 := build(edges2)
    m := len(edges2) + 1
    t := 0
    for i := 0; i < m; i++ {
        t = max(t, dfs(g2, i, -1, k-1))
    }

    g1 := build(edges1)
    n := len(edges1) + 1
    ans := make([]int, n)
    for i := 0; i < n; i++ {
        ans[i] = t + dfs(g1, i, -1, k)
    }
    return ans
}

func build(edges [][]int) [][]int {
    n := len(edges) + 1
    g := make([][]int, n)
    for _, e := range edges {
        a, b := e[0], e[1]
        g[a] = append(g[a], b)
        g[b] = append(g[b], a)
    }
    return g
}

func dfs(g [][]int, a, fa, d int) int {
    if d < 0 {
        return 0
    }
    cnt := 1
    for _, b := range g[a] {
        if b != fa {
            cnt += dfs(g, b, a, d-1)
        }
    }
    return cnt
}

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function maxTargetNodes(edges1: number[][], edges2: number[][], k: number): number[] {
    const g2 = build(edges2);
    const m = edges2.length + 1;
    let t = 0;
    for (let i = 0; i < m; i++) {
        t = Math.max(t, dfs(g2, i, -1, k - 1));
    }

    const g1 = build(edges1);
    const n = edges1.length + 1;
    const ans = Array(n).fill(t);

    for (let i = 0; i < n; i++) {
        ans[i] += dfs(g1, i, -1, k);
    }

    return ans;
}

function build(edges: number[][]): number[][] {
    const n = edges.length + 1;
    const g: number[][] = Array.from({ length: n }, () => []);
    for (const [a, b] of edges) {
        g[a].push(b);
        g[b].push(a);
    }
    return g;
}

function dfs(g: number[][], a: number, fa: number, d: number): number {
    if (d < 0) {
        return 0;
    }
    let cnt = 1;
    for (const b of g[a]) {
        if (b !== fa) {
            cnt += dfs(g, b, a, d - 1);
        }
    }
    return cnt;
}

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public class Solution {
    public int[] MaxTargetNodes(int[][] edges1, int[][] edges2, int k) {
        var g2 = Build(edges2);
        int m = edges2.Length + 1;
        int t = 0;

        for (int i = 0; i < m; i++) {
            t = Math.Max(t, Dfs(g2, i, -1, k - 1));
        }

        var g1 = Build(edges1);
        int n = edges1.Length + 1;
        var ans = new int[n];
        Array.Fill(ans, t);

        for (int i = 0; i < n; i++) {
            ans[i] += Dfs(g1, i, -1, k);
        }

        return ans;
    }

    private List<int>[] Build(int[][] edges) {
        int n = edges.Length + 1;
        var g = new List<int>[n];
        for (int i = 0; i < n; i++) {
            g[i] = new List<int>();
        }
        foreach (var e in edges) {
            int a = e[0], b = e[1];
            g[a].Add(b);
            g[b].Add(a);
        }
        return g;
    }

    private int Dfs(List<int>[] g, int a, int fa, int d) {
        if (d < 0) {
            return 0;
        }
        int cnt = 1;
        foreach (var b in g[a]) {
            if (b != fa) {
                cnt += Dfs(g, b, a, d - 1);
            }
        }
        return cnt;
    }
}

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