Given a tree, and the weights of all the node. Each query contains two integers u and v, the task is to find the minimum and maximum weight on the simple path between u and v (both inclusive).
Examples:
Input:
Query=[{1, 3}, {2, 4}, {3, 5}]
Output:
-1 5
3 5
-2 5
Explanation:
Weight on path 1 to 3 is [-1, 5, -1]. Hence, the minimum and maximum weight is -1 and 5 respectively.
Weight on path 2 to 4 is [5, 3]. Hence, the minimum and maximum weight is 3 and 5 respectively.
Weight on path 2 to 4 is [-1, 5, -1, -2]. Hence, the minimum and maximum weight is -2 and 5 respectively.
Approach: The idea is to use LCA in a tree using Binary Lifting Technique.
- Binary Lifting is a Dynamic Programming approach where we pre-compute an array lca[i][j] where i = [1, n], j = [1, log(n)] and lca[i][j] contains 2j-th ancestor of node i.
- For computing the values of lca[][], the following recursion may be used
lca[i][j] = parent[i] if j = 0 and
lca[i][j] = lca[lca[i][j – 1]][j – 1] if j > 0.
- As we will compute the lca[][] array we will also calculate the MinWeight[][] and MaxWeight[][] where MinWeight[i][j] contains the minimum weight from node i to its 2j-th ancestor, and MaxWeight[i][j] contains the maximum weight from node i to its 2j-th ancestor
- For computing the values of MinWeight[][] and MaxWeight[], the following recursion may be used.
![MinWeight[i][j] =\begin{cases} min (weight[i], weight[parent[i]]) & \text{ ;if } j=0 \\ min( MinWeight[i][j-1], MinWeight[lca[i][j - 1]][j - 1]) & \text{ ;if } j>0 \end{cases}](https://geeksforgeeks.org/wp-content/uploads/2023/10/neveropen_quicklatex.com-41eb73da4d8c4003ba48630fee9df2b0_l3.png "Rendered by QuickLaTeX.com")
[Tex]MaxWeight[i][j] =\begin{cases} max (weight[i], weight[parent[i]]) & \text{ ;if } j=0 \\ max( MaxWeight[i][j-1], MaxWeight[lca[i][j – 1]][j – 1]) & \text{ ;if } j>0 \end{cases} [/Tex]
- After precomputation we find the minimum and maximum weight between (u, v) as we find the least common ancestor of (u, v).
Below is the implementation of the above approach:
C++
// C++ Program to find the maximum and// minimum weight between two nodes// in the given tree using LCA#include <bits/stdc++.h>using namespace std;#define MAX 1000#define log 10 // log2(MAX)// Array to store the level// of each nodeint level[MAX];int lca[MAX][log];int minWeight[MAX][log];int maxWeight[MAX][log];// Vector to store treevector<int> graph[MAX];// Array to store weight of nodesint weight[MAX];void addEdge(int u, int v){ graph[u].push_back(v); graph[v].push_back(u);}// Pre-Processing to calculate// values of lca[][], MinWeight[][]// and MaxWeight[][]void dfs(int node, int parent, int h){ // Using recursion formula to // calculate the values // of lca[][] lca[node][0] = parent; // Storing the level of // each node level[node] = h; if (parent != -1) { minWeight[node][0] = min(weight[node], weight[parent]); maxWeight[node][0] = max(weight[node], weight[parent]); } for (int i = 1; i < log; i++) { if (lca[node][i - 1] != -1) { // Using recursion formula to // calculate the values of lca[][], // MinWeight[][] and MaxWeight[][] lca[node][i] = lca[lca[node][i - 1]][i - 1]; minWeight[node][i] = min(minWeight[node][i - 1], minWeight[lca[node][i - 1]][i - 1]); maxWeight[node][i] = max(maxWeight[node][i - 1], maxWeight[lca[node][i - 1]][i - 1]); } } for (int i : graph[node]) { if (i == parent) continue; dfs(i, node, h + 1); }}// Function to find the minimum and// maximum weights in the given rangevoid findMinMaxWeight(int u, int v){ int minWei = INT_MAX; int maxWei = INT_MIN; // The node which is present // farthest from the root node // is taken as v If u is // farther from root node // then swap the two if (level[u] > level[v]) swap(u, v); // Finding the ancestor of v // which is at same level as u for (int i = log - 1; i >= 0; i--) { if (lca[v][i] != -1 && level[lca[v][i]] >= level[u]) { // Calculating Minimum and // Maximum Weight of node // v till its 2^i-th ancestor minWei = min(minWei, minWeight[v][i]); maxWei = max(maxWei, maxWeight[v][i]); v = lca[v][i]; } } // If u is the ancestor of v // then u is the LCA of u and v if (v == u) { cout << minWei << " " << maxWei << endl; } else { // Finding the node closest to the // root which is not the common // ancestor of u and v i.e. a node // x such that x is not the common // ancestor of u and v but lca[x][0] is for (int i = log - 1; i >= 0; i--) { if (lca[v][i] != lca[u][i]) { // Calculating the minimum of // MinWeight of v to its 2^i-th // ancestor and MinWeight of u // to its 2^i-th ancestor minWei = min(minWei, min(minWeight[v][i], minWeight[u][i])); // Calculating the maximum of // MaxWeight of v to its 2^i-th // ancestor and MaxWeight of u // to its 2^i-th ancestor maxWei = max(maxWei, max(maxWeight[v][i], maxWeight[u][i])); v = lca[v][i]; u = lca[u][i]; } } // Calculating the Minimum of // first ancestor of u and v minWei = min(minWei, min(minWeight[v][0], minWeight[u][0])); // Calculating the maximum of // first ancestor of u and v maxWei = max(maxWei, max(maxWeight[v][0], maxWeight[u][0])); cout << minWei << " " << maxWei << endl; }}// Driver Codeint main(){ // Number of nodes int n = 5; // Add edges addEdge(1, 2); addEdge(1, 5); addEdge(2, 4); addEdge(2, 3); weight[1] = -1; weight[2] = 5; weight[3] = -1; weight[4] = 3; weight[5] = -2; // Initialising lca values with -1 // Initialising MinWeight values // with INT_MAX // Initialising MaxWeight values // with INT_MIN for (int i = 1; i <= n; i++) { for (int j = 0; j < log; j++) { lca[i][j] = -1; minWeight[i][j] = INT_MAX; maxWeight[i][j] = INT_MIN; } } // Perform DFS dfs(1, -1, 0); // Query 1: {1, 3} findMinMaxWeight(1, 3); // Query 2: {2, 4} findMinMaxWeight(2, 4); // Query 3: {3, 5} findMinMaxWeight(3, 5); return 0;} |
Java
// Java Program to find the maximum and// minimum weight between two nodes// in the given tree using LCAimport java.util.*;class GFG{static final int MAX = 1000;// Math.log(MAX)static final int log = 10 ;// Array to store the // level of each nodestatic int []level = new int[MAX];static int [][]lca = new int[MAX][log];static int [][]minWeight = new int[MAX][log];static int [][]maxWeight = new int[MAX][log];// Vector to store treestatic Vector<Integer> []graph = new Vector[MAX]; // Array to store// weight of nodesstatic int []weight = new int[MAX]; private static void swap(int x, int y) { int temp = x; x = y; y = temp;}static void addEdge(int u, int v){ graph[u].add(v); graph[v].add(u);}// Pre-Processing to // calculate values of // lca[][], MinWeight[][]// and MaxWeight[][]static void dfs(int node, int parent, int h){ // Using recursion formula to // calculate the values // of lca[][] lca[node][0] = parent; // Storing the level of // each node level[node] = h; if (parent != -1) { minWeight[node][0] = Math.min(weight[node], weight[parent]); maxWeight[node][0] = Math.max(weight[node], weight[parent]); } for (int i = 1; i < log; i++) { if (lca[node][i - 1] != -1) { // Using recursion formula to // calculate the values of lca[][], // MinWeight[][] and MaxWeight[][] lca[node][i] = lca[lca[node][i - 1]][i - 1]; minWeight[node][i] = Math.min(minWeight[node][i - 1], minWeight[lca[node][i - 1]][i - 1]); maxWeight[node][i] = Math.max(maxWeight[node][i - 1], maxWeight[lca[node][i - 1]][i - 1]); } } for (int i : graph[node]) { if (i == parent) continue; dfs(i, node, h + 1); }}// Function to find the minimum and// maximum weights in the given rangestatic void findMinMaxWeight(int u, int v){ int minWei = Integer.MAX_VALUE; int maxWei = Integer.MIN_VALUE; // The node which is present // farthest from the root node // is taken as v If u is // farther from root node // then swap the two if (level[u] > level[v]) swap(u, v); // Finding the ancestor of v // which is at same level as u for (int i = log - 1; i >= 0; i--) { if (lca[v][i] != -1 && level[lca[v][i]] >= level[u]) { // Calculating Minimum and // Maximum Weight of node // v till its 2^i-th ancestor minWei = Math.min(minWei, minWeight[v][i]); maxWei = Math.max(maxWei, maxWeight[v][i]); v = lca[v][i]; } } // If u is the ancestor of v // then u is the LCA of u and v if (v == u) { System.out.print(minWei + " " + maxWei + "\n"); } else { // Finding the node closest to the // root which is not the common // ancestor of u and v i.e. a node // x such that x is not the common // ancestor of u and v but lca[x][0] is for (int i = log - 1; i >= 0; i--) { if(v == -1) v++; if (lca[v][i] != lca[u][i]) { // Calculating the minimum of // MinWeight of v to its 2^i-th // ancestor and MinWeight of u // to its 2^i-th ancestor minWei = Math.min(minWei, Math.min(minWeight[v][i], minWeight[u][i])); // Calculating the maximum of // MaxWeight of v to its 2^i-th // ancestor and MaxWeight of u // to its 2^i-th ancestor maxWei = Math.max(maxWei, Math.max(maxWeight[v][i], maxWeight[u][i])); v = lca[v][i]; u = lca[u][i]; } } // Calculating the Minimum of // first ancestor of u and v if(u == -1) u++; minWei = Math.min(minWei, Math.min(minWeight[v][0], minWeight[u][0])); // Calculating the maximum of // first ancestor of u and v maxWei = Math.max(maxWei, Math.max(maxWeight[v][0], maxWeight[u][0])); System.out.print(minWei + " " + maxWei + "\n"); }}// Driver Codepublic static void main(String[] args){ // Number of nodes int n = 5; for (int i = 0; i < graph.length; i++) graph[i] = new Vector<Integer>(); // Add edges addEdge(1, 2); addEdge(1, 5); addEdge(2, 4); addEdge(2, 3); weight[1] = -1; weight[2] = 5; weight[3] = -1; weight[4] = 3; weight[5] = -2; // Initialising lca values with -1 // Initialising MinWeight values // with Integer.MAX_VALUE // Initialising MaxWeight values // with Integer.MIN_VALUE for (int i = 1; i <= n; i++) { for (int j = 0; j < log; j++) { lca[i][j] = -1; minWeight[i][j] = Integer.MAX_VALUE; maxWeight[i][j] = Integer.MIN_VALUE; } } // Perform DFS dfs(1, -1, 0); // Query 1: {1, 3} findMinMaxWeight(1, 3); // Query 2: {2, 4} findMinMaxWeight(2, 4); // Query 3: {3, 5} findMinMaxWeight(3, 5);}}// This code is contributed by Rajput-Ji |
Python3
# Python3 Program to find the# maximum and minimum weight # between two nodes in the # given tree using LCAimport sysMAX = 1000 # log2(MAX)log = 10 # Array to store the level# of each nodelevel = [0 for i in range(MAX)];# Initialising lca values with -1# Initialising MinWeight values# with INT_MAX# Initialising MaxWeight values# with INT_MINlca = [[-1 for j in range(log)] for i in range(MAX)]minWeight = [[sys.maxsize for j in range(log)] for i in range(MAX)]maxWeight = [[-sys.maxsize for j in range(log)] for i in range(MAX)] # Vector to store treegraph = [[] for i in range(MAX)]# Array to store weight of nodesweight = [0 for i in range(MAX)]def addEdge(u, v): graph[u].append(v); graph[v].append(u);# Pre-Processing to calculate# values of lca[][], MinWeight[][]# and MaxWeight[][]def dfs(node, parent, h): # Using recursion formula to # calculate the values # of lca[][] lca[node][0] = parent; # Storing the level of # each node level[node] = h; if (parent != -1): minWeight[node][0] = (min(weight[node], weight[parent])); maxWeight[node][0] = (max(weight[node], weight[parent])); for i in range(1, log): if (lca[node][i - 1] != -1): # Using recursion formula to # calculate the values of lca[][], # MinWeight[][] and MaxWeight[][] lca[node][i] = lca[lca[node][i - 1]][i - 1]; minWeight[node][i] = min(minWeight[node][i - 1], minWeight[lca[node][i - 1]][i - 1]); maxWeight[node][i] = max(maxWeight[node][i - 1], maxWeight[lca[node][i - 1]][i - 1]); for i in graph[node]: if (i == parent): continue; dfs(i, node, h + 1);# Function to find the minimum# and maximum weights in the # given rangedef findMinMaxWeight(u, v): minWei = sys.maxsize maxWei = -sys.maxsize # The node which is present # farthest from the root node # is taken as v If u is # farther from root node # then swap the two if (level[u] > level[v]): u, v = v, u # Finding the ancestor of v # which is at same level as u for i in range(log - 1, -1, -1): if (lca[v][i] != -1 and level[lca[v][i]] >= level[u]): # Calculating Minimum and # Maximum Weight of node # v till its 2^i-th ancestor minWei = min(minWei, minWeight[v][i]); maxWei = max(maxWei, maxWeight[v][i]); v = lca[v][i]; # If u is the ancestor of v # then u is the LCA of u and v if (v == u): print(str(minWei) + ' ' + str(maxWei)) else: # Finding the node closest to the # root which is not the common # ancestor of u and v i.e. a node # x such that x is not the common # ancestor of u and v but lca[x][0] is for i in range(log - 1, -1, -1): if (lca[v][i] != lca[u][i]): # Calculating the minimum of # MinWeight of v to its 2^i-th # ancestor and MinWeight of u # to its 2^i-th ancestor minWei = (min(minWei, min(minWeight[v][i], minWeight[u][i]))); # Calculating the maximum of # MaxWeight of v to its 2^i-th # ancestor and MaxWeight of u # to its 2^i-th ancestor maxWei = max(maxWei, max(maxWeight[v][i], maxWeight[u][i])); v = lca[v][i]; u = lca[u][i]; # Calculating the Minimum of # first ancestor of u and v minWei = min(minWei, min(minWeight[v][0], minWeight[u][0])); # Calculating the maximum of # first ancestor of u and v maxWei = max(maxWei, max(maxWeight[v][0], maxWeight[u][0])); print(str(minWei) + ' ' + str(maxWei)) # Driver codeif __name__ == "__main__": # Number of nodes n = 5; # Add edges addEdge(1, 2); addEdge(1, 5); addEdge(2, 4); addEdge(2, 3); weight[1] = -1; weight[2] = 5; weight[3] = -1; weight[4] = 3; weight[5] = -2; # Perform DFS dfs(1, -1, 0); # Query 1: {1, 3} findMinMaxWeight(1, 3); # Query 2: {2, 4} findMinMaxWeight(2, 4); # Query 3: {3, 5} findMinMaxWeight(3, 5); # This code is contributed by Rutvik_56 |
C#
// C# Program to find the // maximum and minimum // weight between two nodes// in the given tree using LCAusing System;using System.Collections.Generic;class GFG{static readonly int MAX = 1000;// Math.Log(MAX)static readonly int log = 10 ;// Array to store the // level of each nodestatic int []level = new int[MAX];static int [,]lca = new int[MAX, log];static int [,]minWeight = new int[MAX, log];static int [,]maxWeight = new int[MAX, log];// List to store treestatic List<int> []graph = new List<int>[MAX]; // Array to store// weight of nodesstatic int []weight = new int[MAX]; private static void swap(int x, int y) { int temp = x; x = y; y = temp;}static void addEdge(int u, int v){ graph[u].Add(v); graph[v].Add(u);}// Pre-Processing to // calculate values of // lca[,], MinWeight[,]// and MaxWeight[,]static void dfs(int node, int parent, int h){ // Using recursion formula to // calculate the values // of lca[,] lca[node, 0] = parent; // Storing the level of // each node level[node] = h; if (parent != -1) { minWeight[node, 0] = Math.Min(weight[node], weight[parent]); maxWeight[node, 0] = Math.Max(weight[node], weight[parent]); } for (int i = 1; i < log; i++) { if (lca[node, i - 1] != -1) { // Using recursion formula to // calculate the values of lca[,], // MinWeight[,] and MaxWeight[,] lca[node, i] = lca[lca[node, i - 1], i - 1]; minWeight[node, i] = Math.Min(minWeight[node, i - 1], minWeight[lca[node, i - 1], i - 1]); maxWeight[node, i] = Math.Max(maxWeight[node, i - 1], maxWeight[lca[node, i - 1], i - 1]); } } foreach (int i in graph[node]) { if (i == parent) continue; dfs(i, node, h + 1); }}// Function to find the minimum and// maximum weights in the given rangestatic void findMinMaxWeight(int u, int v){ int minWei = int.MaxValue; int maxWei = int.MinValue; // The node which is present // farthest from the root node // is taken as v If u is // farther from root node // then swap the two if (level[u] > level[v]) swap(u, v); // Finding the ancestor of v // which is at same level as u for (int i = log - 1; i >= 0; i--) { if (lca[v, i] != -1 && level[lca[v, i]] >= level[u]) { // Calculating Minimum and // Maximum Weight of node // v till its 2^i-th ancestor minWei = Math.Min(minWei, minWeight[v, i]); maxWei = Math.Max(maxWei, maxWeight[v, i]); v = lca[v, i]; } } // If u is the ancestor of v // then u is the LCA of u and v if (v == u) { Console.Write(minWei + " " + maxWei + "\n"); } else { // Finding the node closest to the // root which is not the common // ancestor of u and v i.e. a node // x such that x is not the common // ancestor of u and v but lca[x,0] is for (int i = log - 1; i >= 0; i--) { if(v == -1) v++; if (lca[v, i] != lca[u, i]) { // Calculating the minimum of // MinWeight of v to its 2^i-th // ancestor and MinWeight of u // to its 2^i-th ancestor minWei = Math.Min(minWei, Math.Min(minWeight[v, i], minWeight[u, i])); // Calculating the maximum of // MaxWeight of v to its 2^i-th // ancestor and MaxWeight of u // to its 2^i-th ancestor maxWei = Math.Max(maxWei, Math.Max(maxWeight[v, i], maxWeight[u, i])); v = lca[v, i]; u = lca[u, i]; } } // Calculating the Minimum of // first ancestor of u and v if(u == -1) u++; minWei = Math.Min(minWei, Math.Min(minWeight[v, 0], minWeight[u, 0])); // Calculating the maximum of // first ancestor of u and v maxWei = Math.Max(maxWei, Math.Max(maxWeight[v, 0], maxWeight[u, 0])); Console.Write(minWei + " " + maxWei + "\n"); }}// Driver Codepublic static void Main(String[] args){ // Number of nodes int n = 5; for (int i = 0; i < graph.Length; i++) graph[i] = new List<int>(); // Add edges addEdge(1, 2); addEdge(1, 5); addEdge(2, 4); addEdge(2, 3); weight[1] = -1; weight[2] = 5; weight[3] = -1; weight[4] = 3; weight[5] = -2; // Initialising lca values with -1 // Initialising MinWeight values // with int.MaxValue // Initialising MaxWeight values // with int.MinValue for (int i = 1; i <= n; i++) { for (int j = 0; j < log; j++) { lca[i, j] = -1; minWeight[i, j] = int.MaxValue; maxWeight[i, j] = int.MinValue; } } // Perform DFS dfs(1, -1, 0); // Query 1: {1, 3} findMinMaxWeight(1, 3); // Query 2: {2, 4} findMinMaxWeight(2, 4); // Query 3: {3, 5} findMinMaxWeight(3, 5);}}// This code is contributed by Rajput-Ji |
Javascript
<script> // JavaScript Program to find the maximum and // minimum weight between two nodes // in the given tree using LCA let MAX = 1000; // Math.log(MAX) let log = 10 ; // Array to store the // level of each node let level = new Array(MAX); level.fill(0); let lca = new Array(MAX); let minWeight = new Array(MAX); let maxWeight = new Array(MAX); // Vector to store tree let graph = new Array(MAX); // Array to store // weight of nodes let weight = new Array(MAX); weight.fill(0); function addEdge(u, v) { graph[u].push(v); graph[v].push(u); } // Pre-Processing to // calculate values of // lca[][], MinWeight[][] // and MaxWeight[][] function dfs(node, parent, h) { // Using recursion formula to // calculate the values // of lca[][] lca[node][0] = parent; // Storing the level of // each node level[node] = h; if (parent != -1) { minWeight[node][0] = Math.min(weight[node], weight[parent]); maxWeight[node][0] = Math.max(weight[node], weight[parent]); } for (let i = 1; i < log; i++) { if (lca[node][i - 1] != -1) { // Using recursion formula to // calculate the values of lca[][], // MinWeight[][] and MaxWeight[][] lca[node][i] = lca[lca[node][i - 1]][i - 1]; minWeight[node][i] = Math.min(minWeight[node][i - 1], minWeight[lca[node][i - 1]][i - 1]); maxWeight[node][i] = Math.max(maxWeight[node][i - 1], maxWeight[lca[node][i - 1]][i - 1]); } } for (let i = 0; i < graph[node].length; i++) { if (graph[node][i] == parent) continue; dfs(graph[node][i], node, h + 1); } } // Function to find the minimum and // maximum weights in the given range function findMinMaxWeight(u, v) { let minWei = Number.MAX_VALUE; let maxWei = Number.MIN_VALUE; // The node which is present // farthest from the root node // is taken as v If u is // farther from root node // then swap the two if (level[u] > level[v]) { let temp = u; u = v; v = temp; } // Finding the ancestor of v // which is at same level as u for (let i = log - 1; i >= 0; i--) { if (lca[v][i] != -1 && level[lca[v][i]] >= level[u]) { // Calculating Minimum and // Maximum Weight of node // v till its 2^i-th ancestor minWei = Math.min(minWei, minWeight[v][i]); maxWei = Math.max(maxWei, maxWeight[v][i]); v = lca[v][i]; } } // If u is the ancestor of v // then u is the LCA of u and v if (v == u) { document.write(minWei + " " + maxWei + "</br>"); } else { // Finding the node closest to the // root which is not the common // ancestor of u and v i.e. a node // x such that x is not the common // ancestor of u and v but lca[x][0] is for (let i = log - 1; i >= 0; i--) { if(v == -1) v++; if (lca[v][i] != lca[u][i]) { // Calculating the minimum of // MinWeight of v to its 2^i-th // ancestor and MinWeight of u // to its 2^i-th ancestor minWei = Math.min(minWei, Math.min(minWeight[v][i], minWeight[u][i])); // Calculating the maximum of // MaxWeight of v to its 2^i-th // ancestor and MaxWeight of u // to its 2^i-th ancestor maxWei = Math.max(maxWei, Math.max(maxWeight[v][i], maxWeight[u][i])); v = lca[v][i]; u = lca[u][i]; } } // Calculating the Minimum of // first ancestor of u and v if(u == -1) u++; minWei = Math.min(minWei, Math.min(minWeight[v][0], minWeight[u][0])); // Calculating the maximum of // first ancestor of u and v maxWei = Math.max(maxWei, Math.max(maxWeight[v][0], maxWeight[u][0])); document.write(minWei + " " + maxWei + "</br>"); } } // Number of nodes let n = 5; for (let i = 0; i < graph.length; i++) graph[i] = []; // Add edges addEdge(1, 2); addEdge(1, 5); addEdge(2, 4); addEdge(2, 3); weight[1] = -1; weight[2] = 5; weight[3] = -1; weight[4] = 3; weight[5] = -2; // Initialising lca values with -1 // Initialising MinWeight values // with Integer.MAX_VALUE // Initialising MaxWeight values // with Integer.MIN_VALUE for (let i = 1; i <= n; i++) { lca[i] = new Array(log); minWeight[i] = new Array(log); maxWeight[i] = new Array(log); for (let j = 0; j < log; j++) { lca[i][j] = -1; minWeight[i][j] = Number.MAX_VALUE; maxWeight[i][j] = Number.MIN_VALUE; } } // Perform DFS dfs(1, -1, 0); // Query 1: {1, 3} findMinMaxWeight(1, 3); // Query 2: {2, 4} findMinMaxWeight(2, 4); // Query 3: {3, 5} findMinMaxWeight(3, 5); </script> |
Output
-1 5 3 5 -2 5
Time Complexity: The time taken in pre-processing is O(N logN) and every query takes O(logN) time. So the overall time complexity of the solution is O(N logN).
Auxiliary Space: O(MAX*log), for storing lca, where MAX=1000 and log=10.
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