Find the largest rectangular area possible in a given histogram where the largest rectangle can be made of a number of contiguous bars. For simplicity, assume that all bars have same width and the width is 1 unit.
For example, consider the following histogram with 7 bars of heights {6, 2, 5, 4, 5, 1, 6}. The largest possible rectangle possible is 12 (see the below figure, the max area rectangle is highlighted in red)
A simple solution is to one by one consider all bars as starting points and calculate area of all rectangles starting with every bar. Finally return maximum of all possible areas. Time complexity of this solution would be O(n^2).
We can use Divide and Conquer to solve this in O(nLogn) time. The idea is to find the minimum value in the given array. Once we have index of the minimum value, the max area is maximum of following three values.
- Maximum area in left side of minimum value (Not including the min value)
- Maximum area in right side of minimum value (Not including the min value)
- Number of bars multiplied by minimum value.
The areas in left and right of minimum value bar can be calculated recursively. If we use linear search to find the minimum value, then the worst case time complexity of this algorithm becomes O(n^2). In worst case, we always have (n-1) elements in one side and 0 elements in other side and if the finding minimum takes O(n) time, we get the recurrence similar to worst case of Quick Sort.
How to find the minimum efficiently? Range Minimum Query using Segment Tree can be used for this. We build segment tree of the given histogram heights. Once the segment tree is built, all range minimum queries take O(Logn) time. So over all complexity of the algorithm becomes.
Overall Time = Time to build Segment Tree + Time to recursively find maximum area Time to build segment tree is O(n). Let the time to recursively find max area be T(n). It can be written as following. T(n) = O(Logn) + T(n-1) The solution of above recurrence is O(nLogn). So overall time is O(n) + O(nLogn) which is O(nLogn). Following is C++ implementation of the above algorithm.
Implementation:
C++
// A Divide and Conquer Program to find maximum rectangular area in a histogram#include <bits/stdc++.h>using namespace std;// A utility function to find minimum of three integersint max(int x, int y, int z){ return max(max(x, y), z); }// A utility function to get minimum of two numbers in hist[]int minVal(int *hist, int i, int j){ if (i == -1) return j; if (j == -1) return i; return (hist[i] < hist[j])? i : j;}// A utility function to get the middle index from corner indexes.int getMid(int s, int e){ return s + (e -s)/2; }/* A recursive function to get the index of minimum value in a given range of indexes. The following are parameters for this function. hist --> Input array for which segment tree is built st --> Pointer to segment tree index --> Index of current node in the segment tree. Initially 0 is passed as root is always at index 0 ss & se --> Starting and ending indexes of the segment represented by current node, i.e., st[index] qs & qe --> Starting and ending indexes of query range */int RMQUtil(int *hist, int *st, int ss, int se, int qs, int qe, int index){ // If segment of this node is a part of given range, then return the // min of the segment if (qs <= ss && qe >= se) return st[index]; // If segment of this node is outside the given range if (se < qs || ss > qe) return -1; // If a part of this segment overlaps with the given range int mid = getMid(ss, se); return minVal(hist, RMQUtil(hist, st, ss, mid, qs, qe, 2*index+1), RMQUtil(hist, st, mid+1, se, qs, qe, 2*index+2));}// Return index of minimum element in range from index qs (query start) to// qe (query end). It mainly uses RMQUtil()int RMQ(int *hist, int *st, int n, int qs, int qe){ // Check for erroneous input values if (qs < 0 || qe > n-1 || qs > qe) { cout << "Invalid Input"; return -1; } return RMQUtil(hist, st, 0, n-1, qs, qe, 0);}// A recursive function that constructs Segment Tree for hist[ss..se].// si is index of current node in segment tree stint constructSTUtil(int hist[], int ss, int se, int *st, int si){ // If there is one element in array, store it in current node of // segment tree and return if (ss == se) return (st[si] = ss); // If there are more than one elements, then recur for left and // right subtrees and store the minimum of two values in this node int mid = getMid(ss, se); st[si] = minVal(hist, constructSTUtil(hist, ss, mid, st, si*2+1), constructSTUtil(hist, mid+1, se, st, si*2+2)); return st[si];}/* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */int *constructST(int hist[], int n){ // Allocate memory for segment tree int x = (int)(ceil(log2(n))); //Height of segment tree int max_size = 2*(int)pow(2, x) - 1; //Maximum size of segment tree int *st = new int[max_size]; // Fill the allocated memory st constructSTUtil(hist, 0, n-1, st, 0); // Return the constructed segment tree return st;}// A recursive function to find the maximum rectangular area.// It uses segment tree 'st' to find the minimum value in hist[l..r]int getMaxAreaRec(int *hist, int *st, int n, int l, int r){ // Base cases if (l > r) return INT_MIN; if (l == r) return hist[l]; // Find index of the minimum value in given range // This takes O(Logn)time int m = RMQ(hist, st, n, l, r); /* Return maximum of following three possible cases a) Maximum area in Left of min value (not including the min) a) Maximum area in right of min value (not including the min) c) Maximum area including min */ return max(getMaxAreaRec(hist, st, n, l, m-1), getMaxAreaRec(hist, st, n, m+1, r), (r-l+1)*(hist[m]) );}// The main function to find max areaint getMaxArea(int hist[], int n){ // Build segment tree from given array. This takes // O(n) time int *st = constructST(hist, n); // Use recursive utility function to find the // maximum area return getMaxAreaRec(hist, st, n, 0, n-1);}// Driver program to test above functionsint main(){ int hist[] = {6, 1, 5, 4, 5, 2, 6}; int n = sizeof(hist)/sizeof(hist[0]); cout << "Maximum area is " << getMaxArea(hist, n); return 0;} |
Java
// A Divide and Conquer Program to find maximum rectangular area in a histogramimport java.util.*;class GFG{ static int[] hist; static int[] st; // A utility function to find minimum of three integers static int max(int x, int y, int z) { return Math.max(Math.max(x, y), z); } // A utility function to get minimum of two numbers in hist[] static int minVal(int i, int j) { if (i == -1) return j; if (j == -1) return i; return (hist[i] < hist[j])? i : j; } // A utility function to get the middle index from corner indexes. static int getMid(int s, int e) { return s + (e -s)/2; } /* A recursive function to get the index of minimum value in a given range of indexes. The following are parameters for this function. hist -. Input array for which segment tree is built st -. Pointer to segment tree index -. Index of current node in the segment tree. Initially 0 is passed as root is always at index 0 ss & se -. Starting and ending indexes of the segment represented by current node, i.e., st[index] qs & qe -. Starting and ending indexes of query range */ static int RMQUtil( int ss, int se, int qs, int qe, int index) { // If segment of this node is a part of given range, then return the // min of the segment if (qs <= ss && qe >= se) return st[index]; // If segment of this node is outside the given range if (se < qs || ss > qe) return -1; // If a part of this segment overlaps with the given range int mid = getMid(ss, se); return minVal( RMQUtil(ss, mid, qs, qe, 2*index+1), RMQUtil( mid+1, se, qs, qe, 2*index+2)); } // Return index of minimum element in range from index qs (query start) to // qe (query end). It mainly uses RMQUtil() static int RMQ( int n, int qs, int qe) { // Check for erroneous input values if (qs < 0 || qe > n-1 || qs > qe) { System.out.print("Invalid Input"); return -1; } return RMQUtil( 0, n-1, qs, qe, 0); } // A recursive function that constructs Segment Tree for hist[ss..se]. // si is index of current node in segment tree st static int constructSTUtil(int ss, int se, int si) { // If there is one element in array, store it in current node of // segment tree and return if (ss == se) return (st[si] = ss); // If there are more than one elements, then recur for left and // right subtrees and store the minimum of two values in this node int mid = getMid(ss, se); st[si] = minVal( constructSTUtil( ss, mid, si*2+1), constructSTUtil( mid+1, se, si*2+2)); return st[si]; } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */ static void constructST(int n) { // Allocate memory for segment tree int x = (int)(Math.ceil(Math.log(n))); //Height of segment tree int max_size = 2*(int)Math.pow(2, x) - 1; //Maximum size of segment tree st = new int[max_size*2]; // Fill the allocated memory st constructSTUtil( 0, n-1, 0); // Return the constructed segment tree // return st; } // A recursive function to find the maximum rectangular area. // It uses segment tree 'st' to find the minimum value in hist[l..r] static int getMaxAreaRec( int n, int l, int r) { // Base cases if (l > r) return Integer.MIN_VALUE; if (l == r) return hist[l]; // Find index of the minimum value in given range // This takes O(Logn)time int m = RMQ( n, l, r); /* Return maximum of following three possible cases a) Maximum area in Left of min value (not including the min) a) Maximum area in right of min value (not including the min) c) Maximum area including min */ return max(getMaxAreaRec( n, l, m - 1), getMaxAreaRec( n, m + 1, r), (r - l + 1)*(hist[m]) ); } // The main function to find max area static int getMaxArea( int n) { // Build segment tree from given array. This takes // O(n) time constructST(n); // Use recursive utility function to find the // maximum area return getMaxAreaRec( n, 0, n - 1); } // Driver program to test above functions public static void main(String[] args) { int[] a = {6, 1, 5, 4, 5, 2, 6}; int n = a.length; hist = new int[n]; hist = a; System.out.print("Maximum area is " + getMaxArea(n)); }}// This code is contributed by Rajput-Ji |
Python3
# Python3 program for range minimum # query using segment tree# modified to return index of minimum instead of minimum itself# for further reference link#-------------------------------------------------------------------------from math import ceil,log2; # A utility function to get # minimum of two numbers def minVal(hist,x, y) : if x==-1: return y if y==-1: return x return x if (hist[x] < hist[y]) else y; # A utility function to get the # middle index from corner indexes. def getMid(s, e) : return s + (e - s) // 2; """ A recursive function to get the minimum value in a given range of array indexes. The following are parameters for this function. st --> Pointer to segment tree index --> Index of current node in the segment tree. Initially 0 is passed as root is always at index 0 ss & se --> Starting and ending indexes of the segment represented by current node, i.e., st[index] qs & qe --> Starting and ending indexes of query range """def RMQUtil( hist,st, ss, se, qs, qe, index) : # If segment of this node is a part # of given range, then return # the min of the segment if (qs <= ss and qe >= se) : return st[index]; # If segment of this node # is outside the given range if (se < qs or ss > qe) : return -1; # If a part of this segment # overlaps with the given range mid = getMid(ss, se); return minVal(hist,RMQUtil(hist,st, ss, mid, qs, qe, 2 * index + 1), RMQUtil(hist,st, mid + 1, se, qs, qe, 2 * index + 2)); # Return minimum of elements in range # from index qs (query start) to # qe (query end). It mainly uses RMQUtil() def RMQ( hist,st, n, qs, qe) : # Check for erroneous input values if (qs < 0 or qe > n - 1 or qs > qe) : print("Invalid Input"); return -1; return RMQUtil(hist,st, 0, n - 1, qs, qe, 0); # A recursive function that constructs # Segment Tree for array[ss..se]. # si is index of current node in segment tree st def constructSTUtil(hist, ss, se, st, si) : # If there is one element in array, # store it in current node of # segment tree and return if (ss == se) : st[si] = ss; return st[si]; # If there are more than one elements, # then recur for left and right subtrees # and store the minimum of two values in this node mid = getMid(ss, se); st[si] = minVal(hist,constructSTUtil(hist, ss, mid, st, si * 2 + 1), constructSTUtil(hist, mid + 1, se, st, si * 2 + 2)); return st[si]; """Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory """def constructST( hist, n) : # Allocate memory for segment tree # Height of segment tree x = (int)(ceil(log2(n))); # Maximum size of segment tree max_size = 2 * (int)(2**x) - 1; st = [0] * (max_size); # Fill the allocated memory st constructSTUtil(hist, 0, n - 1, st, 0); # Return the constructed segment tree return st; #----------------------------------------------------------------# main program# Python3 program using Divide and Conquer# to find maximum rectangular area under a histogramdef max_area_histogram(hist): area=0 #initialize area st = constructST(hist, len(hist)) # construct the segment tree try: # try except block is generally used in this way # to suppress all type of exceptions raised. def fun(left,right): # this function "fun" calculates area # recursively between indices left and right nonlocal area # global area won't work here as # variable area is defined inside function # not in main(). if left==right: return # the recursion has reached end index = RMQ(hist,st, len(hist), left, right-1) # RMQ function returns index # of minimum value # in the range of [left,right-1] # can also be found by using min() but # results in O(n) instead of O(log n) for traversing area=max(area,hist[index]*(right-left)) # calculate area with minimum above fun(index+1,right) fun(left,index) # initiate further recursion return fun(0,len(hist)) # initializes the recursion return(area) # return the max area to calling function # in this case "print" except: pass # Driver Code hist = [6, 2, 5, 4, 5, 1, 6] print("Maximum area is", max_area_histogram(hist)) # This code is contributed # by Vishnudev C. |
C#
// C# code to implement the approachusing System;using System.Numerics;using System.Collections.Generic;public class GFG { static int[] hist; static int[] st; // A utility function to find minimum of three integers static int max(int x, int y, int z) { return Math.Max(Math.Max(x, y), z); } // A utility function to get minimum of two numbers in hist[] static int minVal(int i, int j) { if (i == -1) return j; if (j == -1) return i; return (hist[i] < hist[j])? i : j; } // A utility function to get the middle index from corner indexes. static int getMid(int s, int e) { return s + (e -s)/2; } /* A recursive function to get the index of minimum value in a given range of indexes. The following are parameters for this function. hist -. Input array for which segment tree is built st -. Pointer to segment tree index -. Index of current node in the segment tree. Initially 0 is passed as root is always at index 0 ss & se -. Starting and ending indexes of the segment represented by current node, i.e., st[index] qs & qe -. Starting and ending indexes of query range */ static int RMQUtil( int ss, int se, int qs, int qe, int index) { // If segment of this node is a part of given range, then return the // min of the segment if (qs <= ss && qe >= se) return st[index]; // If segment of this node is outside the given range if (se < qs || ss > qe) return -1; // If a part of this segment overlaps with the given range int mid = getMid(ss, se); return minVal( RMQUtil(ss, mid, qs, qe, 2*index+1), RMQUtil( mid+1, se, qs, qe, 2*index+2)); } // Return index of minimum element in range from index qs (query start) to // qe (query end). It mainly uses RMQUtil() static int RMQ( int n, int qs, int qe) { // Check for erroneous input values if (qs < 0 || qe > n-1 || qs > qe) { Console.Write("Invalid Input"); return -1; } return RMQUtil( 0, n-1, qs, qe, 0); } // A recursive function that constructs Segment Tree for hist[ss..se]. // si is index of current node in segment tree st static int constructSTUtil(int ss, int se, int si) { // If there is one element in array, store it in current node of // segment tree and return if (ss == se) return (st[si] = ss); // If there are more than one elements, then recur for left and // right subtrees and store the minimum of two values in this node int mid = getMid(ss, se); st[si] = minVal( constructSTUtil( ss, mid, si*2+1), constructSTUtil( mid+1, se, si*2+2)); return st[si]; } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */ static void constructST(int n) { // Allocate memory for segment tree int x = (int)(Math.Ceiling(Math.Log(n))); //Height of segment tree int max_size = 2*(int)Math.Pow(2, x) - 1; //Maximum size of segment tree st = new int[max_size*2]; // Fill the allocated memory st constructSTUtil( 0, n-1, 0); // Return the constructed segment tree // return st; } // A recursive function to find the maximum rectangular area. // It uses segment tree 'st' to find the minimum value in hist[l..r] static int getMaxAreaRec( int n, int l, int r) { // Base cases if (l > r) return Int32.MinValue; if (l == r) return hist[l]; // Find index of the minimum value in given range // This takes O(Logn)time int m = RMQ( n, l, r); /* Return maximum of following three possible cases a) Maximum area in Left of min value (not including the min) a) Maximum area in right of min value (not including the min) c) Maximum area including min */ return max(getMaxAreaRec( n, l, m - 1), getMaxAreaRec( n, m + 1, r), (r - l + 1)*(hist[m]) ); } // The main function to find max area static int getMaxArea( int n) { // Build segment tree from given array. This takes // O(n) time constructST(n); // Use recursive utility function to find the // maximum area return getMaxAreaRec( n, 0, n - 1); }// Driver Codepublic static void Main(string[] args){ int[] a = {6, 1, 5, 4, 5, 2, 6}; int n = a.Length; hist = new int[n]; hist = a; Console.WriteLine("Maximum area is " + getMaxArea(n));}} |
Javascript
// A utility function to get minimum of two numbersfunction minVal(hist, x, y) { if (x == -1) return y; if (y == -1) return x; return hist[x] < hist[y] ? x : y;}// A utility function to get the middle index from corner indexesfunction getMid(s, e) { return s + Math.floor((e - s) / 2);}/* A recursive function to get the minimum value in a given range of array indexes. The following are parameters for this function. st --> Pointer to segment tree index --> Index of current node in the segment tree. Initially 0 is passed as root is always at index 0 ss & se --> Starting and ending indexes of the segment represented by current node, i.e., st[index] qs & qe --> Starting and ending indexes of query range */function RMQUtil(hist, st, ss, se, qs, qe, index) { // If segment of this node is a part of given range, then return the min of the segment if (qs <= ss && qe >= se) return st[index]; // If segment of this node is outside the given range if (se < qs || ss > qe) return -1; // If a part of this segment overlaps with the given range const mid = getMid(ss, se); return minVal(hist, RMQUtil(hist, st, ss, mid, qs, qe, 2 * index + 1), RMQUtil(hist, st, mid + 1, se, qs, qe, 2 * index + 2));}// Return minimum of elements in range from index qs (query start) to qe (query end). It mainly uses RMQUtil()function RMQ(hist, st, n, qs, qe) { // Check for erroneous input values if (qs < 0 || qe > n - 1 || qs > qe) { console.log("Invalid Input"); return -1; } return RMQUtil(hist, st, 0, n - 1, qs, qe, 0);}// A recursive function that constructs Segment Tree for array[ss..se].// si is index of current node in segment tree stfunction constructSTUtil(hist, ss, se, st, si) { // If there is one element in array, store it in current node of segment tree and return if (ss == se) { st[si] = ss; return st[si]; } // If there are more than one elements, then recur for left and right subtrees // and store the minimum of two values in this node const mid = getMid(ss, se); st[si] = minVal(hist, constructSTUtil(hist, ss, mid, st, si * 2 + 1), constructSTUtil(hist, mid + 1, se, st, si * 2 + 2)); return st[si];}/* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */function constructST(hist, n) { // Allocate memory for segment tree // Height of segment tree const x = Math.ceil(Math.log2(n)); // Maximum size of segment tree const max_size = 2 * Math.pow(2, x) - 1;const st = new Array(max_size).fill(0);constructSTUtil(hist, 0, n - 1, st, 0);// Return the constructed segment treereturn st;}/**Python: def max_area_histogram(hist)JavaScript equivalent:*/function max_area_histogram(hist) {let area = 0;// initialize areaconst st = constructST(hist, hist.length);// construct the segment treefunction fun(left, right) {// this function "fun" calculates area// recursively between indices left and right if (left === right) { return; } // the recursion has reached end const index = RMQ(hist, st, hist.length, left, right - 1); // RMQ function returns index of minimum value // in the range of [left,right-1] // can also be found by using Math.min() but // results in O(n) instead of O(log n) for traversing area = Math.max(area, hist[index] * (right - left)); // calculate area with minimum above fun(index + 1, right); fun(left, index); // initiate further recursion}fun(0, hist.length);return area;}/* Example usage */console.log("Maximum area is: " +max_area_histogram([6, 2, 5, 4, 5, 1, 6])); // expected output: 12 |
Output
Maximum area is 12
Time Complexity: O(N log N)
Auxiliary Space: O(N)
This problem can be solved in linear time. See below set 2 for linear time solution.
Linear time solution for Largest Rectangular Area in a Histogram
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
