Queries for M-th node in the DFS of subtree

Given a tree of N nodes and N-1 edges. Also given an integer M and a node, the task is to print the M-th node in the DFS of the subtree of a given node for multiple queries.

Note: M will not be greater than the number of nodes in the subtree of the given node.

Input: M = 3, node = 1
Output: 4
In the above example if 1 is given as the node, then the DFS of subtree will be 1 2 4 6 7 5 3, hence if M is 3, then the 3rd node is 4

Input: M = 4, node = 2
Output: 7
If 2 is given as the node, then the DFS of the subtree will be 2 4 6 7 5., hence if M is 4 then the 4th node is 7.

Approach:

  • Add the edges between the nodes in an adjacency list.
  • Call DFS function to generate the DFS of the complete tree.
  • Use an under[] array to store the height of the subtree under the given node including the node.
  • In the DFS function, keep incrementing the size of subtree on every recursive call.
  • Mark the node index in the DFS of complete using hashing.
  • Let index of given node in the DFS of the tree be ind, then the M-th node will be at index ind + M -1 as the DFS of a subtree of a node will always be a contiguous subarray starting from the node.

Below is the implementation of the above approach.

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// C++ program for Queries
// for DFS of subtree of a node in a tree
#include <bits/stdc++.h>
using namespace std;
const int N = 100000;
  
// Adjaceny list to store the
// tree nodes connection
vector<int> v[N];
  
// stores the index of node in DFS
unordered_map<int, int> mp;
  
// stores the index of node in
// original node
vector<int> a;
  
// Function to call DFS and count nodes
// under that subtree
void dfs(int under[], int child, int parent)
{
  
    // stores the DFS of tree
    a.push_back(child);
  
    // hieght of subtree
    under[child] = 1;
  
    // iterate for children
    for (auto it : v[child]) {
  
        // if not equal to parent
        // so that it does not traverse back
        if (it != parent) {
  
            // call DFS for subtree
            dfs(under, it, child);
  
            // add the heigth
            under[child] += under[it];
        }
    }
}
  
// Function to return the DFS of subtree of nodec
int printnodeDFSofSubtree(int node, int under[], int m)
{
    // index of node in the original DFS
    int ind = mp[node];
  
    // height of subtree of node
    return a[ind + m - 1];
}
  
// Function to add edges to a tree
void addEdge(int x, int y)
{
    v[x].push_back(y);
    v[y].push_back(x);
}
  
// Marks the index of node in original DFS
void markIndexDfs()
{
    int size = a.size();
  
    // marks the index
    for (int i = 0; i < size; i++) {
        mp[a[i]] = i;
    }
}
  
// Driver Code
int main()
{
    int n = 7;
  
    // add edges of a tree
    addEdge(1, 2);
    addEdge(1, 3);
    addEdge(2, 4);
    addEdge(2, 5);
    addEdge(4, 6);
    addEdge(4, 7);
  
    // array to store the height of subtree
    // of every node in a tree
    int under[n + 1];
  
    // Call the function DFS to generate the DFS
    dfs(under, 1, 0);
  
    // Function call to mark the index of node
    markIndexDfs();
  
    int m = 3;
  
    // Query 1
    cout << printnodeDFSofSubtree(1, under, m) << endl;
  
    // Query 2
    m = 4;
    cout << printnodeDFSofSubtree(2, under, m);
  
    return 0;
}

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Time Complexity: O(1), for processing each query.
Auxiliary Space: O(N)



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Striver(underscore)79 at Codechef and codeforces D

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