We are given 2*N – 1 integers. We need to check whether it is possible to construct a Range Minimum Query segment tree for an array of N distinct integers from these integers. If so, we must output the segment tree array. N is given to be a power of 2.
An RMQ segment tree is a binary tree where each node is equal to the minimum value of its children. This type of tree is used to efficiently find the minimum value of elements in a given range.
Input : 1 1 1 1 2 2 3 3 3 4 4 5 6 7 8
Output : 1 1 3 1 2 3 4 1 5 2 6 3 7 4 8
The segment tree is shown below
Input : -381 -460 -381 95 -460 855 -242
405 -460 982 -381 -460 95 981 855
Output : -460 -460 -381 -460 95 -381 855
-460 -242 95 405 -381 981 855 982
By constructing a segment tree from the output,
we can see that it a valid tree for RMQ and the
leaves are all distinct integers.
What we first do is iterate through the given integers counting the number of occurrences of each number, then sorting them by value. In C++, we can use the data structure map, which stores elements in sorted order.
Now we maintain a queue for each possible level of the segment tree. We put the initial root of the tree (array index 0) into the queue for the max level. We then insert the smallest element into the leftmost nodes. We then detach these nodes from the main tree. As we detach a node, we create a new tree of height h – 1, where h is the height of the current node. We can see this in figure 2. We insert the root node of this new tree into the appropriate queue based on its height.
We go through each element, getting a tree of the appropriate height based on the number of occurrences of that element. If at any point such a tree does not exist, then it is not possible to create a segment tree.
CPP
// C++ Program to Create RMQ Segment Tree#include <bits/stdc++.h>using namespace std;// Returns true if it is possible to construct// a range minimum segment tree from given array.bool createTree(int arr[], int N){ // Store the height of the final tree const int height = log2(N) + 1; // Container to sort and store occurrences of elements map<int, int> multi; // Insert elements into the container for (int i = 0; i < 2 * N - 1; ++i) ++multi[arr[i]]; // Used to store new subtrees created set<int> Q[height]; // Insert root into set Q[height - 1].emplace(0); // Iterate through each unique element in set for (map<int, int>::iterator it = multi.begin(); it != multi.end(); ++it) { // Number of occurrences is greater than height // Or, no subtree exists that can accommodate it if (it->second > height || Q[it->second - 1].empty()) return false; // Get the appropriate subtree int node = *Q[it->second - 1].begin(); // Delete the subtree we grabbed Q[it->second - 1].erase(Q[it->second - 1].begin()); int level = 1; for (int i = node; i < 2 * N - 1; i = 2 * i + 1, ++level) { // Insert new subtree created into root if (2 * i + 2 < 2 * N - 1) Q[it->second - level - 1].emplace(2 * i + 2); // Insert element into array at position arr[i] = it->first; } } return true;}// Driver programint main(){ int N = 8; int arr[2 * N - 1] = {1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8}; if (createTree(arr, N)) { cout << "YES\n"; for (int i = 0; i < 2 * N - 1; ++i) cout << arr[i] << " "; } else cout << "NO\n"; return 0;} |
Java
import java.util.*;public class RMQSegmentTree { // Returns true if it is possible to construct // a range minimum segment tree from given array. public static boolean createTree(int[] arr, int N) { // Store the height of the final tree final int height = (int)(Math.log(N) / Math.log(2)) + 1; // Container to sort and store occurrences of // elements Map<Integer, Integer> multi = new HashMap<>(); // Insert elements into the container for (int i = 0; i < 2 * N - 1; ++i) multi.put(arr[i], multi.getOrDefault(arr[i], 0) + 1); // Used to store new subtrees created Set<Integer>[] Q = new HashSet[height]; for (int i = 0; i < height; i++) { Q[i] = new HashSet<Integer>(); } // Insert root into set Q[height - 1].add(0); // Iterate through each unique element in set for (Map.Entry<Integer, Integer> entry : multi.entrySet()) { int key = entry.getKey(); int value = entry.getValue(); // Number of occurrences is greater than height // Or, no subtree exists that can accommodate it if (value > height || Q[value - 1].isEmpty()) return false; // Get the appropriate subtree int node = Q[value - 1].iterator().next(); // Delete the subtree we grabbed Q[value - 1].remove(node); int level = 1; for (int i = node; i < 2 * N - 1; i = 2 * i + 1, ++level) { // Insert new subtree created into root if (2 * i + 2 < 2 * N - 1) Q[value - level - 1].add(2 * i + 2); // Insert element into array at position arr[i] = key; } } return true; } // Driver program public static void main(String[] args) { int N = 8; int[] arr = { 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8 }; if (createTree(arr, N)) { System.out.println("YES"); for (int i = 0; i < 2 * N - 1; ++i) System.out.print(arr[i] + " "); } else { System.out.println("NO"); } }} |
Python3
# Python Program to Create RMQ Segment Treefrom typing import Listfrom math import log2# Returns true if it is possible to construct# a range minimum segment tree from given array.def createTree(arr: List[int], N: int) -> bool: # Store the height of the final tree height = int(log2(N)) + 1 # Container to sort and store occurrences of elements multi = {} # Insert elements into the container for i in range(2 * N - 1): if arr[i] not in multi: multi[arr[i]] = 0 multi[arr[i]] += 1 # Used to store new subtrees created Q = [set() for _ in range(height)] # Insert root into set Q[height - 1].add(0) # Iterate through each unique element in set for k, v in multi.items(): # Number of occurrences is greater than height # Or, no subtree exists that can accommodate it if (v > height or len(Q[v - 1]) == 0): return False # Get the appropriate subtree node = sorted(Q[v - 1])[0] # Delete the subtree we grabbed Q[v - 1].remove(sorted(Q[v - 1])[0]) level = 1 # for (int i = node; i < 2 * N - 1; # i = 2 * i + 1, ++level) i = node while i < 2 * N - 1: # Insert new subtree created into root if (2 * i + 2 < 2 * N - 1): Q[v - level - 1].add(2 * i + 2) # Insert element into array at position arr[i] = k level += 1 i = 2 * i + 1 return True# Driver programif __name__ == "__main__": N = 8 arr = [1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8] if (createTree(arr, N)): print("YES") # for (int i = 0; i < 2 * N - 1; ++i) for i in range(2 * N - 1): print(arr[i], end=" ") else: print("No")# This code is contributed by sanjeev2552 |
C#
// C# Program to Create RMQ Segment Treeusing System;using System.Collections.Generic;public class RMQSegmentTree { // Returns true if it is possible to construct // a range minimum segment tree from given array. static bool CreateTree(int[] arr, int N) { // Store the height of the final tree int height = (int)(Math.Log(N) / Math.Log(2)) + 1; // Container to sort and store occurrences of // elements Dictionary<int, int> multi = new Dictionary<int, int>(); // Insert elements into the container for (int i = 0; i < 2 * N - 1; ++i) if (multi.ContainsKey(arr[i])) multi[arr[i]]++; else multi[arr[i]] = 1; // Used to store new subtrees created SortedSet<int>[] Q = new SortedSet<int>[ height ]; // Initialize each set in the array for (int i = 0; i < height; i++) Q[i] = new SortedSet<int>(); // Insert root into set Q[height - 1].Add(0); // Iterate through each unique element in set foreach(KeyValuePair<int, int> entry in multi) { int count = entry.Value; int key = entry.Key; // Number of occurrences is greater than height // Or, no subtree exists that can accommodate it if (count > height || Q[count - 1].Count == 0) return false; // Get the appropriate subtree int node = Q[count - 1].Min; // Delete the subtree we grabbed Q[count - 1].Remove(node); int level = 1; for (int i = node; i < 2 * N - 1; i = 2 * i + 1, ++level) { // Insert new subtree created into root if (2 * i + 2 < 2 * N - 1) Q[count - level - 1].Add(2 * i + 2); // Insert element into array at position arr[i] = key; } } return true; } // Driver program static public void Main() { int N = 8; int[] arr = { 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8 }; if (CreateTree(arr, N)) { Console.WriteLine("YES"); for (int i = 0; i < 2 * N - 1; ++i) Console.Write(arr[i] + " "); } else Console.WriteLine("NO"); Console.ReadLine(); }}// This code is contributed by prasad264 |
Javascript
// JavaScript Program to Create RMQ Segment Tree// Returns true if it is possible to construct// a range minimum segment tree from given array.function createTree(arr, N) {// Store the height of the final treeconst height = Math.floor(Math.log2(N)) + 1;// Container to sort and store occurrences of elementsconst multi = {};// Insert elements into the containerfor (let i = 0; i < 2 * N - 1; i++) {if (multi[arr[i]] == undefined) {multi[arr[i]] = 0;}multi[arr[i]] += 1;}// Used to store new subtrees createdconst Q = new Array(height);for (let i = 0; i < height; i++) {Q[i] = new Set();}// Insert root into setQ[height - 1].add(0);// Iterate through each unique element in setfor (const [k, v] of Object.entries(multi)) {// Number of occurrences is greater than height// Or, no subtree exists that can accommodate itif (v > height || Q[v - 1].size == 0) { return false;}// Get the appropriate subtreeconst node = Math.min(...Q[v - 1]);// Delete the subtree we grabbedQ[v - 1].delete(node);let level = 1;let i = node;while (i < 2 * N - 1) { // Insert new subtree created into root if (2 * i + 2 < 2 * N - 1) { Q[v - level - 1].add(2 * i + 2); } // Insert element into array at position arr[i] = k; level += 1; i = 2 * i + 1;}}return true;}// Driver programconst N = 8;const arr = [1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8];if (createTree(arr, N)) {console.log("YES");let ans="";for (let i = 0; i < 2 * N - 1; i++) {ans = ans + arr[i] + " ";}console.log(ans);} else {console.log("No");} |
Output:
YES 1 1 3 1 2 3 4 1 5 2 6 3 7 4 8
Main time complexity is caused by sorting elements.
Time Complexity: O(N log N)
Space Complexity: O(N)
Related Topic: Segment Tree
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