#3841Hard~50 min

Palindromic Path Queries in a Tree

Time O((n + q) * logn) · Space O(n) · Official statement on LeetCode

Solutions

// Time:  O((n + q) * logn)
// Space: O(n)

// hld, lca, fenwick tree
class 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 tree
class 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_;
    };
};

Beginner Explanation

What is Palindromic Path Queries in a Tree?

Palindromic Path Queries in a Tree (LeetCode #3841) is a Hard problem that primarily trains tree.

How to think about it

Restate the goal in your own words before coding.
Work a tiny example by hand so the invariant becomes obvious.
Identify the pattern — this problem aligns with dfs backtracking and fenwick tree.
Only then translate the idea into code.

Why this problem matters

Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: DFS, HLD, Heavy-Light Decomposition, LCA.

AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.

Interview Walkthrough

Interview approach for Palindromic Path Queries in a Tree

Opening (30–60 seconds)

Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
State a brute force so the interviewer knows you can solve it naively.
Propose the optimal direction tied to dfs backtracking and fenwick tree.

Core solution narrative

Define the state you track (pointers, DP cell, set membership, stack top, etc.).
Explain the transition when you process the next element.
Call out time (O((n + q) * logn)) and space (O(n)) before coding.
Code cleanly; narrate variable names.

What interviewers listen for

Correctness on edge cases
Complexity honesty
Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)

Follow-up questions they may ask

Can you solve it with less memory?
What if the input stream is infinite / doesn't fit in RAM?
How would tests look for adversarial inputs?

Optimized Approach

Optimized solution notes

The reference solutions on AlgoForge target O((n + q) * logn) time and O(n) space.

Pattern focus: dfs backtracking and fenwick tree

Use the pattern as a checklist:

dfs backtracking — confirm the invariant holds after each step
fenwick tree — confirm the invariant holds after each step

Multiple methods appear in the source solutions — compare them and explain when each is preferable.

Implementation tips

Prefer readable names over micro-optimizations in interviews.
Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
After AC-level logic, re-scan for off-by-one and null checks.

Complexity Analysis

Complexity

Measure	Bound
Time	*O((n + q) logn)**
Space	O(n)

How to justify this in an interview

Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
Space: include hash maps, recursion stack, and output allocation when the problem asks for it.

If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.

Common Mistakes

Common mistakes on Palindromic Path Queries in a Tree

Skipping edge cases — empty collections, single-element inputs, max constraints.
Wrong invariant for dfs backtracking and fenwick tree — updating state too early or too late.
Mutating input unexpectedly when the problem forbids it.
Off-by-one in windows, ranges, or binary search bounds.
Ignoring overflow / precision for integer arithmetic problems.
Overengineering — jumping to an advanced structure when a simpler approach works.

Alternative Approaches

Alternatives

The source file includes more than one method. Compare:

Primary optimized path — best complexity for typical interviews.
Secondary approach — often brute force, sorting-based, or space-optimized variant.

Practice articulating when you would pick each (constraints, readability, follow-ups).

Edge Cases

Edge cases checklist

Minimum input size
Maximum input size / time limits
Duplicates and already-sorted input
Negative numbers / zeros (if applicable)
Disconnected structures (graphs/trees)
Single path vs branching recursion depth

Pattern Recognition

Spotting this pattern

Signal phrases that point to dfs backtracking and fenwick tree:

Sorted input or ability to sort without changing the answer class
Need for contiguous subarray / substring → consider sliding window
Need for O(1) membership → hash set/map
Optimal substructure + overlapping subproblems → DP
Connectivity / components → graph DFS/BFS or Union-Find

Primary topics: tree.

Follow-up Interview Questions

Follow-ups

How does the solution change if the input is a stream?
Can you solve it in-place?
What if duplicates must be handled differently?
How would you parallelize the approach?
Design tests that would break a buggy implementation.

Practice Recommendations

What to practice next

Re-solve Palindromic Path Queries in a Tree in a second language (cpp, python).
Drill 3–5 more problems tagged tree.
Teach the solution out loud in under 5 minutes.
Add this problem to your revision calendar in 3 days and 14 days.

Visualization

Conceptual diagram for Palindromic Path Queries in a Tree: show input structure (tree), highlight the moving parts of the dfs backtracking and fenwick tree approach, and annotate each step with the maintained invariant and complexity.

Study checklist

Read the official problem statement on LeetCode
Solve on paper / whiteboard first
Implement the dfs backtracking and fenwick tree approach
Verify edge cases from the checklist
State time and space complexity aloud
Compare with the AlgoForge reference solution
Schedule a revision session

Revision notes

Palindromic Path Queries in a Tree (#3841) — Hard. Pattern: dfs backtracking and fenwick tree. Complexity: O((n + q) * logn) time / O(n) space. Re-derive the invariant before coding.

FAQs

What is the time complexity of Palindromic Path Queries in a Tree?+

The reference solutions aim for O((n + q) * logn) time and O(n) space. Always re-derive complexity from the code you write in the interview.

What pattern does Palindromic Path Queries in a Tree use?+

It primarily maps to dfs backtracking and fenwick tree, within the broader topic of tree.

Is Palindromic Path Queries in a Tree good for interviews?+

Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.

Where can I read the official statement?+

Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/palindromic-path-queries-in-a-tree/

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