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Problem solution · C++

Palindromic Path Queries in a Tree

Palindromic Path Queries in a Tree: a C++ solution using segment tree or range structure. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Segment tree or range structure
Source
Kamyu LeetCode Solutions
Length
295 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Segment tree or range structure

For Palindromic Path Queries in a Tree, the implementation stores interval information in a range-query data structure so updates and queries avoid rescanning the full input.

  1. Choose the aggregate stored for each interval or prefix.
  2. Build or initialize the structure from the input.
  3. Apply updates and combine the affected nodes to answer each query.

Code notes

  • 295 lines of C++ from the credited upstream file palindromic-path-queries-in-a-tree.cpp.
  • The implementation visibly relies on sequence storage.
  • 19 loop blocks detected, together with recursive traversal.

Complexity

Count the build once, then multiply the logarithmic update or query path by the number of operations.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codePalindromic Path Queries in a Tree · C++C++
Use this to learn the idea, then write your own version.
// Time:  O((n + q) * logn)// Space: O(n) // hld, lca, fenwick treeclass Solution {public:    vector<bool> palindromePath(int n, vector<vector<int>>& edges, string s, vector<string>& queries) {        const auto& build_hld = [](const auto& adj, const auto& cb) {            vector<int> parent(size(adj), -1), depth(size(adj), 0), sz(size(adj), 1), heavy(size(adj), -1), head(size(adj));            iota(begin(head), end(head), 0);            vector<tuple<int, int, int>> stk = {{1, 0, -1}};            while (!empty(stk)) {                const auto [step, u, p] = stk.back(); stk.pop_back();                if (step == 1) {                    cb(u, p);                    parent[u] = p;                    depth[u] = (p == -1 ? 0 : depth[p] + 1);                    stk.emplace_back(2, u, p);                    for (const auto& v : adj[u]) {                        if (v == p) {                            continue;                        }                        stk.emplace_back(1, v, u);                    }                } else if (step == 2) {                    for (const auto& v : adj[u]) {                        if (v == parent[u]) {                            continue;                        }                        sz[u] += sz[v];                        if (heavy[u] == -1 || sz[v] > sz[heavy[u]]) {                            heavy[u] = v;                        }                    }                }            }            int idx = -1;            vector<int> left(size(adj), -1), right(size(adj), -1);            stk = {{1, 0, 0}};            while (!empty(stk)) {                const auto [step, u, h] = stk.back(); stk.pop_back();                if (step == 1) {                    head[u] = h;                    left[u] = ++idx;                    stk.emplace_back(2, u, h);                    for (const auto& v : adj[u]) {                        if (v == parent[u] || v == heavy[u]) {                            continue;                        }                        stk.emplace_back(1, v, v);                    }                    if (heavy[u] != -1) {                        stk.emplace_back(1, heavy[u], h);                    }                } else if (step == 2) {                    right[u] = idx;                }            }            return tuple(parent, depth, head, left, right);        };         vector<int> prefix(n);        const auto& callback = [&](int u, int p) {            prefix[u] = (p != -1 ? prefix[p] : 0) ^ (1 << (s[u] - 'a'));        };         vector<vector<int>> adj(n);        for (const auto& e : edges) {            adj[e[0]].emplace_back(e[1]);            adj[e[1]].emplace_back(e[0]);        }        const auto& [parent, depth, head, left, right] = build_hld(adj, callback);        const auto& lca = [&](int u, int v) {            while (head[u] != head[v]) {                if (depth[head[u]] < depth[head[v]]) {                    swap(u, v);                }                u = parent[head[u]];            }            return depth[u] < depth[v] ? u : v;        };         BIT bit(n + 1);        vector<bool> result;        for (const auto& q : queries) {            istringstream iss(q);            string op;            int u;            iss >> op >> u;            if (op == "update") {                char c;                iss >> c;                const auto& diff = (1 << (s[u] - 'a')) ^ (1 << (c - 'a'));                if (!diff) {                    continue;                }                s[u] = c;                bit.add(left[u], diff);                bit.add(right[u] + 1, diff);            } else {                int v;                iss >> v;                const auto& l = lca(u, v);                const auto& mask = (prefix[u] ^ bit.query(left[u])) ^ (prefix[v] ^ bit.query(left[v])) ^ (1 << (s[l] - 'a'));                result.emplace_back((mask & (mask - 1)) == 0);            }        }        return result;    } private:    class BIT {    public:        BIT(int n) : bit_(n + 1) {  // 0-indexed        }                void add(int i, int val) {            ++i;            for (; i < size(bit_); i += lower_bit(i)) {                bit_[i] ^= val;            }        }         int query(int i) const {            ++i;            int total = 0;            for (; i > 0; i -= lower_bit(i)) {                total ^= bit_[i];            }            return total;        }     private:        inline int lower_bit(int i) const {            return i & -i;        }                vector<int> bit_;    };}; // Time:  O((n + q) * logn)// Space: O(nlogn)// dfs, lca, binary lifting, fenwick treeclass Solution2 {public:    vector<bool> palindromePath(int n, vector<vector<int>>& edges, string s, vector<string>& queries) {        vector<vector<int>> adj(n);        for (const auto& e : edges) {            adj[e[0]].emplace_back(e[1]);            adj[e[1]].emplace_back(e[0]);        }        TreeInfos tree_infos(adj);        BIT bit(n + 1);        for (int u = 0; u < n; ++u) {            const auto& diff = 1 << (s[u] - 'a');            bit.add(tree_infos.left(u), diff);            bit.add(tree_infos.right(u) + 1, diff);        }        vector<bool> result;        for (const auto& q : queries) {            istringstream iss(q);            string op;            int u;            iss >> op >> u;            if (op == "update") {                char c;                iss >> c;                const auto& diff = (1 << (s[u] - 'a')) ^ (1 << (c - 'a'));                if (!diff) {                    continue;                }                s[u] = c;                bit.add(tree_infos.left(u), diff);                bit.add(tree_infos.right(u) + 1, diff);            } else {                int v;                iss >> v;                const auto& l = tree_infos.lca(u, v);                const auto& mask = bit.query(tree_infos.left(u)) ^ bit.query(tree_infos.left(v)) ^ (1 << (s[l] - 'a'));                result.emplace_back(mask == 0 || (mask & (mask - 1)) == 0);            }        }        return result;    } private:    class BIT {    public:        BIT(int n) : bit_(n + 1) {  // 0-indexed        }                void add(int i, int val) {            ++i;            for (; i < size(bit_); i += lower_bit(i)) {                bit_[i] ^= val;            }        }         int query(int i) const {            ++i;            int total = 0;            for (; i > 0; i -= lower_bit(i)) {                total ^= bit_[i];            }            return total;        }     private:        inline int lower_bit(int i) const {            return i & -i;        }                vector<int> bit_;    };     class TreeInfos {    public:        TreeInfos(const vector<vector<int>>& adj)         : L_(size(adj))         , R_(size(adj))         , D_(size(adj))         , P_(size(adj)) {              const int N = size(adj);             int idx = -1;             vector<tuple<int, int, int>> stk = {{1, 0, -1}};             while (!empty(stk)) {                const auto [step, u, p] = stk.back(); stk.pop_back();                if (step == 1) {                    D_[u] = (p == -1) ? 1 : D_[p] + 1;                    if (p != -1) {                        P_[u].emplace_back(p);  // ancestors of the node i                    }                    for (int i = 0; i < size(P_[u]); ++i) {                        if (i >= size(P_[P_[u][i]])) {                            break;                        }                        P_[u].emplace_back(P_[P_[u][i]][i]);                    }                    L_[u] = ++idx;                     stk.emplace_back(2, u, -1);                    for (int i = size(adj[u]) -1; i >= 0; --i) {                        const auto& v = adj[u][i];                        if (v == p) {                            continue;                        }                        stk.emplace_back(1, v, u);                    }                } else if (step == 2) {                    R_[u] = idx;                }            }            assert(idx == N - 1);        }                bool is_ancestor(int a, int b) const {            return L_[a] <= L_[b] && R_[b] <= R_[a];        }         int lca(int a, int b) const {            if (D_[a] > D_[b]) {                swap(a, b);            }            if (is_ancestor(a, b)) {                return a;            }            for (int i = size(P_[a]) - 1; i >= 0; --i) {  // O(logN)                if (i < size(P_[a]) && !is_ancestor(P_[a][i], b)) {                    a = P_[a][i];                }            }            return P_[a][0];        }         int left(int a) const {            return L_[a];        }         int right(int a) const {            return R_[a];        }                int depth(int a) const {            return D_[a];        }        private:        vector<int> L_;        vector<int> R_;        vector<int> D_;        vector<vector<int>> P_;    };};

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