XOR of all the nodes in the sub-tree of the given node

Given an n-ary tree and Q queries where each query consists of an integer u which denotes a node. The task is to print the xor of all the values of nodes in the sub-tree of the given node.



q[] = {0, 1, 4, 5}
Query 1: (1 ^ 2 ^ 3 ^ 4 ^ 5 ^ 6 ^ 7) = 0
Query 2: (2 ^ 4 ^ 5) = 3
Query 3: (5) = 5
Query 4: (6) = 6

Naive approach: For each node we have to traverse the tree for every query find the xor of all the nodes of the sub-tree. So the complexity of the code will be O(N * Q).

Efficient approach: If we pre-compute the xor of all the nodes of the sub-tree by traversing the total tree once and first computing the xor of all the nodes of the sub-tree of the child node first and then using the value of the child node for computing the xor of all the nodes of the sub-tree of the parent node. In this way we can compute the xor in O(N) time and store them for queries. When a query is asked we will print the pre-computed value in O(1) time.

Below is the implementation of the above approach:





// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
// Adjacency list of the graph
vector<vector<int> > graph;
// Value of the node
vector<int> values, xor_values;
// Function to pre-compute the xor values
int pre_compute_xor(int i, int prev)
    // xor of the sub-tree
    int x = values[i];
    for (int j = 0; j < graph[i].size(); j++)
        if (graph[i][j] != prev) {
            // xor x with xor of the sub-tree
            // of it child nodes
            x ^= pre_compute_xor(graph[i][j], i);
    xor_values[i] = x;
    // Return the xor
    return x;
// Function to return the xor of
// the nodes of the sub-tree
// rooted at node u
int query(int u)
    return xor_values[u];
// Driver code
int main()
    int n = 7;
    // Create the graph
    // Set the values of the nodes
    // Pre-computation
    pre_compute_xor(0, -1);
    // Perform queries
    int queries[] = { 0, 1, 4, 5 };
    int q = sizeof(queries) / sizeof(queries[0]);
    for (int i = 0; i < q; i++)
        cout << query(queries[i]) << endl;
    return 0;




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Second year Department of Information Technology Jadavpur University

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