LCA for n-ary Tree | Constant Query O(1)

We have seen various methods with different Time Complexities to calculate LCA in n-ary tree:-

Method 1 : Naive Method ( by calculating root to node path) | O(n) per query
Method 2 :Using Sqrt Decomposition | O(sqrt H)
Method 3 : Using Sparse Matrix DP approach | O(logn)

Lets study another method which has faster query time than all the above methods. So, our aim will be to calculate LCA in constant time ~ O(1). Let’s see how we can achieve it.

Method 4 : Using Range Minimum Query

We have discussed LCA and RMQ for binary tree. Here we discuss LCA problem to RMQ problem conversion for n-ary tree.

Pre-requisites:- LCA in Binary Tree using RMQ
                 RMQ using sparse table

Key Concept : In this method, we will be reducing our LCA problem to RMQ(Range Minimum Query) problem over a static array. Once, we do that then we will relate the Range minimum queries to the required LCA queries.

The first step will be to decompose the tree into a flat linear array. To do this we can apply the Euler walk . The Euler walk will give the pre-order traversal of the graph. So we will perform a Euler Walk on the tree and store the nodes in an array as we visit them. This process reduces the tree data-structure to a simple linear array.

Consider the below tree and the euler walk over it :-

16933693_1309372792480521_1797138248_n

Now lets think in general terms : Consider any two nodes on the tree. There will be exactly one path connecting both the nodes and the node that has the smallest depth value in the path will be the LCA of the two given nodes.

Now take any two distinct node say u and v in the Euler walk array. Now all the elements in the path from u to v will lie in between the index of nodes u and v in the Euler walk array. Therefore, we just need to calculate the node with the minimum depth between the index of node u and node v in the euler array.

For this we will maintain another array that will contain the depth of all the nodes corresponding to their position in the Euler walk array so that we can Apply our RMQ algorithm on it.

Given below is the euler walk array parallel to its depth track array.

16901489_1309372785813855_1903972436_n

Example :- Consider two nodes node 6 and node 7 in the euler array. To calculate the LCA of node 6 and node 7 we look the the smallest depth value for all the nodes in between node 6 and node 7 .
Therefore, node 1 has the smallest depth value = 0 and hence, it is the LCA for node 6 and node 7.

16934185_1309372782480522_1333490382_n

Implementation :-

We will be maintaining three arrays 1)Euler Path   
                                    2)Depth array   
                                    3)First Appearance Index

Euler Path and Depth array are the same as described above

First Appearance Index FAI[] : The First Appearance index Array will store the index for the first position of every node in the Euler Path array. FAI[i] = First appearance of ith node in Euler Walk array.

The C++ Implementation for the above method is given below:-

// C++ program to demonstrate LCA of n-ary tree
// in constant time.
#include "bits/stdc++.h"
using namespace std;
#define sz 101

vector < int > adj[sz];    // stores the tree
vector < int > euler;      // tracks the eulerwalk
vector < int > depthArr;   // depth for each node corresponding
                           // to eulerwalk

int FAI[sz];     // stores first appearence index of every node
int level[sz];   // stores depth for all nodes in the tree
int ptr;         // pointer to euler walk
int dp[sz][18];  // sparse table
int logn[sz];    // stores log values
int p2[20];      // stores power of 2

void buildSparseTable(int n)
{
    // initializing sparse table
    memset(dp,-1,sizeof(dp));

    // filling base case values
    for (int i=1; i<n; i++)
        dp[i-1][0] = (depthArr[i]>depthArr[i-1])?i-1:i;

    // dp to fill sparse table
    for (int l=1; l<15; l++)
      for (int i=0; i<n; i++)
        if (dp[i][l-1]!=-1 and dp[i+p2[l-1]][l-1]!=-1)
          dp[i][l] =
            (depthArr[dp[i][l-1]]>depthArr[dp[i+p2[l-1]][l-1]])?
             dp[i+p2[l-1]][l-1] : dp[i][l-1];
        else
             break;
}

int query(int l,int r)
{
    int d = r-l;
    int dx = logn[d];
    if (l==r) return l;
    if (depthArr[dp[l][dx]] > depthArr[dp[r-p2[dx]][dx]])
        return dp[r-p2[dx]][dx];
    else
        return dp[l][dx];
}

void preprocess()
{
    // memorizing powers of 2
    p2[0] = 1;
    for (int i=1; i<18; i++)
        p2[i] = p2[i-1]*2;

    // memorizing all log(n) values
    int val = 1,ptr=0;
    for (int i=1; i<sz; i++)
    {
        logn[i] = ptr-1;
        if (val==i)
        {
            val*=2;
            logn[i] = ptr;
            ptr++;
        }
    }
}

/**
 * Euler Walk ( preorder traversal)
 * converting tree to linear depthArray
 * Time Complexity : O(n)
 * */
void dfs(int cur,int prev,int dep)
{
    // marking FAI for cur node
    if (FAI[cur]==-1)
        FAI[cur] = ptr;

    level[cur] = dep;

    // pushing root to euler walk
    euler.push_back(cur);

    // incrementing euler walk pointer
    ptr++;

    for (auto x:adj[cur])
    {
        if (x != prev)
        {
            dfs(x,cur,dep+1);

            // pushing cur again in backtrack
            // of euler walk
            euler.push_back(cur);

            // increment euler walk pointer
            ptr++;
        }
    }
}

// Create Level depthArray corresponding
// to the Euler walk Array
void makeArr()
{
    for (auto x : euler)
        depthArr.push_back(level[x]);
}

int LCA(int u,int v)
{
    // trival case
    if (u==v)
       return u;

    if (FAI[u] > FAI[v])
       swap(u,v);

    // doing RMQ in the required range
    return euler[query(FAI[u], FAI[v])];
}

void addEdge(int u,int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}

int main(int argc, char const *argv[])
{
    // constructing the described tree
    int numberOfNodes = 8;
    addEdge(1,2);
    addEdge(1,3);
    addEdge(2,4);
    addEdge(2,5);
    addEdge(2,6);
    addEdge(3,7);
    addEdge(3,8);

    // performing required precalculations
    preprocess();

    // doing the Euler walk
    ptr = 0;
    memset(FAI,-1,sizeof(FAI));
    dfs(1,0,0);

    // creating depthArray corresponding to euler[]
    makeArr();

    // building sparse table
    buildSparseTable(depthArr.size());

    cout << "LCA(6,7) : " << LCA(6,7) << "\n";
    cout << "LCA(6,4) : " << LCA(6,4) << "\n";

    return 0;
}

Output:

LCA(6,7) : 1
LCA(6,4) : 2

Note : We are precalculating all the required power of 2’s and also precalculating the all the required log values to ensure constant time complexity per query. Else if we did log calculation for every query operation our Time complexity would have not been constant.

Time Complexity: The Conversion process from LCA to RMQ is done by Euler Walk that takes O(n) time.
Pre-processing for the sparse table in RMQ takes O(nlogn) time and answering each Query is a Constant time process. Therefore, overall Time Complexity is O(nlogn) – preprocessing and O(1) for each query.

This article is contributed by Nitish Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

GATE CS Corner    Company Wise Coding Practice





Writing code in comment? Please use ide.geeksforgeeks.org, generate link and share the link here.