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1766. Tree of Coprimes

SourceBiweekly Contest 46 Q4DifficultyHardRating2231

Description

There is a tree (i.e., a connected, undirected graph that has no cycles) consisting of n nodes numbered from 0 to n - 1 and exactly n - 1 edges. Each node has a value associated with it, and the root of the tree is node 0.

To represent this tree, you are given an integer array nums and a 2D array edges. Each nums[i] represents the ith node's value, and each edges[j] = [uj, vj] represents an edge between nodes uj and vj in the tree.

Two values x and y are coprime if gcd(x, y) == 1 where gcd(x, y) is the greatest common divisor of x and y.

An ancestor of a node i is any other node on the shortest path from node i to the root. A node is not considered an ancestor of itself.

Return an array ans of size n, where ans[i] is the closest ancestor to node i such that nums[i] and nums[ans[i]] are coprime, or -1 if there is no such ancestor.

 

Example 1:

Input: nums = [2,3,3,2], edges = [[0,1],[1,2],[1,3]]
Output: [-1,0,0,1]
Explanation: In the above figure, each node's value is in parentheses.
- Node 0 has no coprime ancestors.
- Node 1 has only one ancestor, node 0. Their values are coprime (gcd(2,3) == 1).
- Node 2 has two ancestors, nodes 1 and 0. Node 1's value is not coprime (gcd(3,3) == 3), but node 0's
  value is (gcd(2,3) == 1), so node 0 is the closest valid ancestor.
- Node 3 has two ancestors, nodes 1 and 0. It is coprime with node 1 (gcd(3,2) == 1), so node 1 is its
  closest valid ancestor.

Example 2:

Input: nums = [5,6,10,2,3,6,15], edges = [[0,1],[0,2],[1,3],[1,4],[2,5],[2,6]]
Output: [-1,0,-1,0,0,0,-1]

 

Constraints:

  • nums.length == n
  • 1 <= nums[i] <= 50
  • 1 <= n <= 105
  • edges.length == n - 1
  • edges[j].length == 2
  • 0 <= uj, vj < n
  • uj != vj

Solutions

Solution 1: Preprocessing + Enumeration + Stack + Backtracking

Thinking

For each node we want the closest ancestor whose value is coprime with it. \(n\le 10^5\) but values lie in \([1,50]\), so one stack per value is enough.

Precompute coprime pairs in \(1..50\). During DFS, look at the stack top of every value coprime to \(nums[i]\) and take the deepest.

Push \((i,\textit{depth})\) onto the stack of \(nums[i]\) before recursing and pop afterward, so the top is always the nearest ancestor.

Since the range of \(nums[i]\) in the problem is \([1, 50]\), we can preprocess all the coprime numbers for each number and record them in the array \(f\), where \(f[i]\) represents all the coprime numbers of \(i\).

Next, we can use a backtracking method to traverse the entire tree from the root node. For each node \(i\), we can get all the coprime numbers of \(nums[i]\) through the array \(f\). Then we enumerate all the coprime numbers of \(nums[i]\), find the ancestor node \(t\) that has appeared and has the maximum depth, which is the nearest coprime ancestor node of \(i\). Here we can use a stack array \(stks\) of length \(51\) to get each appeared value \(v\) and its depth. The top element of each stack \(stks[v]\) is the nearest ancestor node with the maximum depth.

The time complexity is \(O(n \times M)\), and the space complexity is \(O(M^2 + n)\). Where \(n\) is the number of nodes, and \(M\) is the maximum value of \(nums[i]\), in this problem \(M = 50\).

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class Solution:
    def getCoprimes(self, nums: List[int], edges: List[List[int]]) -> List[int]:
        def dfs(i, fa, depth):
            t = k = -1
            for v in f[nums[i]]:
                stk = stks[v]
                if stk and stk[-1][1] > k:
                    t, k = stk[-1]
            ans[i] = t
            for j in g[i]:
                if j != fa:
                    stks[nums[i]].append((i, depth))
                    dfs(j, i, depth + 1)
                    stks[nums[i]].pop()

        g = defaultdict(list)
        for u, v in edges:
            g[u].append(v)
            g[v].append(u)
        f = defaultdict(list)
        for i in range(1, 51):
            for j in range(1, 51):
                if gcd(i, j) == 1:
                    f[i].append(j)
        stks = defaultdict(list)
        ans = [-1] * len(nums)
        dfs(0, -1, 0)
        return ans

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class Solution {
    private List<Integer>[] g;
    private List<Integer>[] f;
    private Deque<int[]>[] stks;
    private int[] nums;
    private int[] ans;

    public int[] getCoprimes(int[] nums, int[][] edges) {
        int n = nums.length;
        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);
        }
        f = new List[51];
        stks = new Deque[51];
        Arrays.setAll(f, k -> new ArrayList<>());
        Arrays.setAll(stks, k -> new ArrayDeque<>());
        for (int i = 1; i < 51; ++i) {
            for (int j = 1; j < 51; ++j) {
                if (gcd(i, j) == 1) {
                    f[i].add(j);
                }
            }
        }
        this.nums = nums;
        ans = new int[n];
        dfs(0, -1, 0);
        return ans;
    }

    private void dfs(int i, int fa, int depth) {
        int t = -1, k = -1;
        for (int v : f[nums[i]]) {
            var stk = stks[v];
            if (!stk.isEmpty() && stk.peek()[1] > k) {
                t = stk.peek()[0];
                k = stk.peek()[1];
            }
        }
        ans[i] = t;
        for (int j : g[i]) {
            if (j != fa) {
                stks[nums[i]].push(new int[] {i, depth});
                dfs(j, i, depth + 1);
                stks[nums[i]].pop();
            }
        }
    }

    private int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }
}

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class Solution {
public:
    vector<int> getCoprimes(vector<int>& nums, vector<vector<int>>& edges) {
        int n = nums.size();
        vector<vector<int>> g(n);
        vector<vector<int>> f(51);
        vector<stack<pair<int, int>>> stks(51);
        for (auto& e : edges) {
            int u = e[0], v = e[1];
            g[u].emplace_back(v);
            g[v].emplace_back(u);
        }
        for (int i = 1; i < 51; ++i) {
            for (int j = 1; j < 51; ++j) {
                if (__gcd(i, j) == 1) {
                    f[i].emplace_back(j);
                }
            }
        }
        vector<int> ans(n);
        function<void(int, int, int)> dfs = [&](int i, int fa, int depth) {
            int t = -1, k = -1;
            for (int v : f[nums[i]]) {
                auto& stk = stks[v];
                if (!stk.empty() && stk.top().second > k) {
                    t = stk.top().first;
                    k = stk.top().second;
                }
            }
            ans[i] = t;
            for (int j : g[i]) {
                if (j != fa) {
                    stks[nums[i]].push({i, depth});
                    dfs(j, i, depth + 1);
                    stks[nums[i]].pop();
                }
            }
        };
        dfs(0, -1, 0);
        return ans;
    }
};

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func getCoprimes(nums []int, edges [][]int) []int {
    n := len(nums)
    g := make([][]int, n)
    f := [51][]int{}
    type pair struct{ first, second int }
    stks := [51][]pair{}
    for _, e := range edges {
        u, v := e[0], e[1]
        g[u] = append(g[u], v)
        g[v] = append(g[v], u)
    }
    for i := 1; i < 51; i++ {
        for j := 1; j < 51; j++ {
            if gcd(i, j) == 1 {
                f[i] = append(f[i], j)
            }
        }
    }
    ans := make([]int, n)
    var dfs func(i, fa, depth int)
    dfs = func(i, fa, depth int) {
        t, k := -1, -1
        for _, v := range f[nums[i]] {
            stk := stks[v]
            if len(stk) > 0 && stk[len(stk)-1].second > k {
                t, k = stk[len(stk)-1].first, stk[len(stk)-1].second
            }
        }
        ans[i] = t
        for _, j := range g[i] {
            if j != fa {
                stks[nums[i]] = append(stks[nums[i]], pair{i, depth})
                dfs(j, i, depth+1)
                stks[nums[i]] = stks[nums[i]][:len(stks[nums[i]])-1]
            }
        }
    }
    dfs(0, -1, 0)
    return ans
}

func gcd(a, b int) int {
    if b == 0 {
        return a
    }
    return gcd(b, a%b)
}

Solution 2: Preprocessing + Enumeration + Explicit Stack

Thinking

For each node we want the closest ancestor whose value is coprime with it. With \(n\le 10^5\) and values in \([1,50]\), one ancestor stack per value is enough. Recursing into each child from the root is too deep: a chain makes the call depth \(n\).

Precompute coprime pairs in \(1..50\). On a visit, look at the stack top of every value coprime to \(nums[i]\) and take the deepest.

The explicit stack stores \((node, parent, depth, next child)\). Before walking to a child we push \((i,\textit{depth})\) onto the stack of \(nums[i]\) and pop it after that child is finished, so the top stays the nearest ancestor.

Since the range of \(nums[i]\) in the problem is \([1, 50]\), we can preprocess all the coprime numbers for each number and record them in the array \(f\), where \(f[i]\) represents all the coprime numbers of \(i\).

Next, an explicit stack walks the tree in preorder from the root. For each node \(i\), enumerate the values coprime to \(nums[i]\) and take the ancestor \(t\) of maximum depth among those stack tops; that is the nearest coprime ancestor. The stack array \(stks\) has length \(51\), and \(stks[v]\) records nodes of value \(v\) on the current root path together with their depths. Before walking to a child, push \((i,\textit{depth})\) onto \(stks[nums[i]]\), and pop it after that child is finished.

The time complexity is \(O(n \times M)\), and the space complexity is \(O(M^2 + n)\). Where \(n\) is the number of nodes, and \(M\) is the maximum value of \(nums[i]\), in this problem \(M = 50\).

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class Solution:
    def getCoprimes(self, nums: List[int], edges: List[List[int]]) -> List[int]:
        n = len(nums)
        g = [[] for _ in range(n)]
        for u, v in edges:
            g[u].append(v)
            g[v].append(u)
        f = [[] for _ in range(51)]
        for i in range(1, 51):
            for j in range(1, 51):
                if gcd(i, j) == 1:
                    f[i].append(j)
        stks = [[] for _ in range(51)]
        ans = [-1] * n
        stk = [(0, -1, 0, 0)]
        while stk:
            i, fa, depth, k = stk[-1]
            if k == 0:
                t = mx = -1
                for v in f[nums[i]]:
                    cur = stks[v]
                    if cur and cur[-1][1] > mx:
                        t, mx = cur[-1]
                ans[i] = t
            else:
                jprev = g[i][k - 1]
                if jprev != fa:
                    stks[nums[i]].pop()
            while k < len(g[i]) and g[i][k] == fa:
                k += 1
            if k == len(g[i]):
                stk.pop()
                continue
            j = g[i][k]
            stk[-1] = (i, fa, depth, k + 1)
            stks[nums[i]].append((i, depth))
            stk.append((j, i, depth + 1, 0))
        return ans

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class Solution {
    public int[] getCoprimes(int[] nums, int[][] edges) {
        int n = nums.length;
        List<Integer>[] 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);
        }
        List<Integer>[] f = new List[51];
        Deque<int[]>[] stks = new Deque[51];
        Arrays.setAll(f, k -> new ArrayList<>());
        Arrays.setAll(stks, k -> new ArrayDeque<>());
        for (int i = 1; i < 51; ++i) {
            for (int j = 1; j < 51; ++j) {
                if (gcd(i, j) == 1) {
                    f[i].add(j);
                }
            }
        }
        int[] ans = new int[n];
        Deque<int[]> stk = new ArrayDeque<>();
        stk.push(new int[] {0, -1, 0, 0});
        while (!stk.isEmpty()) {
            int[] cur = stk.peek();
            int i = cur[0], fa = cur[1], depth = cur[2], k = cur[3];
            if (k == 0) {
                int t = -1, mx = -1;
                for (int v : f[nums[i]]) {
                    Deque<int[]> s = stks[v];
                    if (!s.isEmpty() && s.peek()[1] > mx) {
                        t = s.peek()[0];
                        mx = s.peek()[1];
                    }
                }
                ans[i] = t;
            } else {
                int jprev = g[i].get(k - 1);
                if (jprev != fa) {
                    stks[nums[i]].pop();
                }
            }
            while (k < g[i].size() && g[i].get(k) == fa) {
                ++k;
            }
            if (k == g[i].size()) {
                stk.pop();
                continue;
            }
            int j = g[i].get(k);
            cur[3] = k + 1;
            stks[nums[i]].push(new int[] {i, depth});
            stk.push(new int[] {j, i, depth + 1, 0});
        }
        return ans;
    }

    private int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }
}

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class Solution {
public:
    vector<int> getCoprimes(vector<int>& nums, vector<vector<int>>& edges) {
        int n = nums.size();
        vector<vector<int>> g(n);
        vector<vector<int>> f(51);
        vector<vector<pair<int, int>>> stks(51);
        for (auto& e : edges) {
            int u = e[0], v = e[1];
            g[u].emplace_back(v);
            g[v].emplace_back(u);
        }
        for (int i = 1; i < 51; ++i) {
            for (int j = 1; j < 51; ++j) {
                if (__gcd(i, j) == 1) {
                    f[i].emplace_back(j);
                }
            }
        }
        vector<int> ans(n);
        vector<array<int, 4>> stk{{0, -1, 0, 0}};
        while (!stk.empty()) {
            auto& cur = stk.back();
            int i = cur[0], fa = cur[1], depth = cur[2], k = cur[3];
            if (k == 0) {
                int t = -1, mx = -1;
                for (int v : f[nums[i]]) {
                    auto& s = stks[v];
                    if (!s.empty() && s.back().second > mx) {
                        t = s.back().first;
                        mx = s.back().second;
                    }
                }
                ans[i] = t;
            } else {
                int jprev = g[i][k - 1];
                if (jprev != fa) {
                    stks[nums[i]].pop_back();
                }
            }
            while (k < (int) g[i].size() && g[i][k] == fa) {
                ++k;
            }
            if (k == (int) g[i].size()) {
                stk.pop_back();
                continue;
            }
            int j = g[i][k];
            cur[3] = k + 1;
            stks[nums[i]].emplace_back(i, depth);
            stk.push_back({j, i, depth + 1, 0});
        }
        return ans;
    }
};

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func getCoprimes(nums []int, edges [][]int) []int {
    n := len(nums)
    g := make([][]int, n)
    f := [51][]int{}
    type pair struct{ first, second int }
    stks := [51][]pair{}
    for _, e := range edges {
        u, v := e[0], e[1]
        g[u] = append(g[u], v)
        g[v] = append(g[v], u)
    }
    for i := 1; i < 51; i++ {
        for j := 1; j < 51; j++ {
            if gcd(i, j) == 1 {
                f[i] = append(f[i], j)
            }
        }
    }
    ans := make([]int, n)
    type frame struct{ i, fa, depth, k int }
    stk := []frame{{0, -1, 0, 0}}
    for len(stk) > 0 {
        cur := &stk[len(stk)-1]
        i, fa, depth, k := cur.i, cur.fa, cur.depth, cur.k
        if k == 0 {
            t, mx := -1, -1
            for _, v := range f[nums[i]] {
                s := stks[v]
                if len(s) > 0 && s[len(s)-1].second > mx {
                    t, mx = s[len(s)-1].first, s[len(s)-1].second
                }
            }
            ans[i] = t
        } else {
            jprev := g[i][k-1]
            if jprev != fa {
                stks[nums[i]] = stks[nums[i]][:len(stks[nums[i]])-1]
            }
        }
        for k < len(g[i]) && g[i][k] == fa {
            k++
        }
        if k == len(g[i]) {
            stk = stk[:len(stk)-1]
            continue
        }
        j := g[i][k]
        cur.k = k + 1
        stks[nums[i]] = append(stks[nums[i]], pair{i, depth})
        stk = append(stk, frame{j, i, depth + 1, 0})
    }
    return ans
}

func gcd(a, b int) int {
    if b == 0 {
        return a
    }
    return gcd(b, a%b)
}

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