Range maximum query using Sparse Table
Last Updated :
04 Jun, 2021
Given an array arr[], the task is to answer queries to find the maximum of all the elements in the index range arr[L...R].
Examples:
Input: arr[] = {6, 7, 4, 5, 1, 3}, q[][] = {{0, 5}, {3, 5}, {2, 4}}
Output:
7
5
5
Input: arr[] = {3, 34, 1}, q[][] = {{1, 2}}
Output:
34
Approach: A similar problem to answer range minimum queries has been discussed here. The same approach can be modified to answer range maximum queries. Below is the modification:
// Maximum of single element subarrays is same
// as the only element
lookup[i][0] = arr[i]
// If lookup[0][2] ? lookup[4][2],
// then lookup[0][3] = lookup[0][2]
If lookup[i][j-1] ? lookup[i+2j-1-1][j-1]
lookup[i][j] = lookup[i][j-1]
// If lookup[0][2] < lookup[4][2],
// then lookup[0][3] = lookup[4][2]
Else
lookup[i][j] = lookup[i+2j-1-1][j-1]
Below is the implementation of the above approach:
C++
// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
#define MAX 500
// lookup[i][j] is going to store maximum
// value in arr[i..j]. Ideally lookup table
// size should not be fixed and should be
// determined using n Log n. It is kept
// constant to keep code simple.
int lookup[MAX][MAX];
// Fills lookup array lookup[][] in bottom up manner
void buildSparseTable(int arr[], int n)
{
// Initialize M for the intervals with length 1
for (int i = 0; i < n; i++)
lookup[i][0] = arr[i];
// Compute values from smaller to bigger intervals
for (int j = 1; (1 << j) <= n; j++) {
// Compute maximum value for all intervals with
// size 2^j
for (int i = 0; (i + (1 << j) - 1) < n; i++) {
// For arr[2][10], we compare arr[lookup[0][7]]
// and arr[lookup[3][10]]
if (lookup[i][j - 1] > lookup[i + (1 << (j - 1))][j - 1])
lookup[i][j] = lookup[i][j - 1];
else
lookup[i][j] = lookup[i + (1 << (j - 1))][j - 1];
}
}
}
// Returns maximum of arr[L..R]
int query(int L, int R)
{
// Find highest power of 2 that is smaller
// than or equal to count of elements in given
// range
// For [2, 10], j = 3
int j = (int)log2(R - L + 1);
// Compute maximum of last 2^j elements with first
// 2^j elements in range
// For [2, 10], we compare arr[lookup[0][3]] and
// arr[lookup[3][3]]
if (lookup[L][j] >= lookup[R - (1 << j) + 1][j])
return lookup[L][j];
else
return lookup[R - (1 << j) + 1][j];
}
// Driver program
int main()
{
int a[] = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };
int n = sizeof(a) / sizeof(a[0]);
buildSparseTable(a, n);
cout << query(0, 4) << endl;
cout << query(4, 7) << endl;
cout << query(7, 8) << endl;
return 0;
}
// Java implementation of the approach
import java.util.*;
class GFG {
static final int MAX = 500;
// lookup[i][j] is going to store maximum
// value in arr[i..j]. Ideally lookup table
// size should not be fixed and should be
// determined using n Log n. It is kept
// constant to keep code simple.
static int lookup[][] = new int[MAX][MAX];
// Fills lookup array lookup[][] in bottom up manner
static void buildSparseTable(int arr[], int n)
{
// Initialize M for the intervals with length 1
for (int i = 0; i < n; i++)
lookup[i][0] = arr[i];
// Compute values from smaller to bigger intervals
for (int j = 1; (1 << j) <= n; j++) {
// Compute maximum value for all intervals with
// size 2^j
for (int i = 0; (i + (1 << j) - 1) < n; i++) {
// For arr[2][10], we compare arr[lookup[0][7]]
// and arr[lookup[3][10]]
if (lookup[i][j - 1] > lookup[i + (1 << (j - 1))][j - 1])
lookup[i][j] = lookup[i][j - 1];
else
lookup[i][j] = lookup[i + (1 << (j - 1))][j - 1];
}
}
}
// Returns maximum of arr[L..R]
static int query(int L, int R)
{
// Find highest power of 2 that is smaller
// than or equal to count of elements in given
// range
// For [2, 10], j = 3
int j = (int)Math.log(R - L + 1);
// Compute maximum of last 2^j elements with first
// 2^j elements in range
// For [2, 10], we compare arr[lookup[0][3]] and
// arr[lookup[3][3]]
if (lookup[L][j] >= lookup[R - (1 << j) + 1][j])
return lookup[L][j];
else
return lookup[R - (1 << j) + 1][j];
}
// Driver program
public static void main(String args[])
{
int a[] = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };
int n = a.length;
buildSparseTable(a, n);
System.out.println(query(0, 4));
System.out.println(query(4, 7));
System.out.println(query(7, 8));
}
}
# Python3 implementation of the approach
from math import log
MAX = 500
# lookup[i][j] is going to store maximum
# value in arr[i..j]. Ideally lookup table
# size should not be fixed and should be
# determined using n Log n. It is kept
# constant to keep code simple.
lookup = [[0 for i in range(MAX)]
for i in range(MAX)]
# Fills lookup array lookup[][]
# in bottom up manner
def buildSparseTable(arr, n):
# Initialize M for the intervals
# with length 1
for i in range(n):
lookup[i][0] = arr[i]
# Compute values from smaller
# to bigger intervals
i, j = 0, 1
while (1 << j) <= n:
# Compute maximum value for
# all intervals with size 2^j
while (i + (1 << j) - 1) < n:
# For arr[2][10], we compare arr[lookup[0][7]]
# and arr[lookup[3][10]]
if (lookup[i][j - 1] >
lookup[i + (1 << (j - 1))][j - 1]):
lookup[i][j] = lookup[i][j - 1]
else:
lookup[i][j] = lookup[i + (1 << (j - 1))][j - 1]
i += 1
j += 1
# Returns maximum of arr[L..R]
def query(L, R):
# Find highest power of 2 that is smaller
# than or equal to count of elements in given
# range
# For [2, 10], j = 3
j = int(log(R - L + 1))
# Compute maximum of last 2^j elements with first
# 2^j elements in range
# For [2, 10], we compare arr[lookup[0][3]] and
# arr[lookup[3][3]]
if (lookup[L][j] >= lookup[R - (1 << j) + 1][j]):
return lookup[L][j]
else:
return lookup[R - (1 << j) + 1][j]
# Driver Code
a = [7, 2, 3, 0, 5, 10, 3, 12, 18]
n = len(a)
buildSparseTable(a, n);
print(query(0, 4))
print(query(4, 7))
print(query(7, 8))
# This code is contributed by Mohit Kumar
// Java implementation of the approach
using System;
class GFG {
static int MAX = 500;
// lookup[i][j] is going to store maximum
// value in arr[i..j]. Ideally lookup table
// size should not be fixed and should be
// determined using n Log n. It is kept
// constant to keep code simple.
static int[, ] lookup = new int[MAX, MAX];
// Fills lookup array lookup[][] in bottom up manner
static void buildSparseTable(int[] arr, int n)
{
// Initialize M for the intervals with length 1
for (int i = 0; i < n; i++)
lookup[i, 0] = arr[i];
// Compute values from smaller to bigger intervals
for (int j = 1; (1 << j) <= n; j++) {
// Compute maximum value for all intervals with
// size 2^j
for (int i = 0; (i + (1 << j) - 1) < n; i++) {
// For arr[2][10], we compare arr[lookup[0][7]]
// and arr[lookup[3][10]]
if (lookup[i, j - 1] > lookup[i + (1 << (j - 1)), j - 1])
lookup[i, j] = lookup[i, j - 1];
else
lookup[i, j] = lookup[i + (1 << (j - 1)), j - 1];
}
}
}
// Returns maximum of arr[L..R]
static int query(int L, int R)
{
// Find highest power of 2 that is smaller
// than or equal to count of elements in given
// range
// For [2, 10], j = 3
int j = (int)Math.Log(R - L + 1);
// Compute maximum of last 2^j elements with first
// 2^j elements in range
// For [2, 10], we compare arr[lookup[0][3]] and
// arr[lookup[3][3]]
if (lookup[L, j] >= lookup[R - (1 << j) + 1, j])
return lookup[L, j];
else
return lookup[R - (1 << j) + 1, j];
}
// Driver program
public static void Main(String[] args)
{
int[] a = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };
int n = a.Length;
buildSparseTable(a, n);
Console.WriteLine(query(0, 4));
Console.WriteLine(query(4, 7));
Console.WriteLine(query(7, 8));
}
}
<script>
// Javascript implementation of the approach
let MAX = 500;
// lookup[i][j] is going to store maximum
// value in arr[i..j]. Ideally lookup table
// size should not be fixed and should be
// determined using n Log n. It is kept
// constant to keep code simple.
let lookup = new Array();
for(let i = 0; i < MAX; i++)
{
let temp = [];
for(let j = 0; j < MAX; j++)
{
temp.push([])
}
lookup.push(temp)
}
// Fills lookup array lookup[][] in bottom up manner
function buildSparseTable(arr, n)
{
// Initialize M for the letervals with length 1
for (let i = 0; i < n; i++)
lookup[i][0] = arr[i];
// Compute values from smaller to bigger letervals
for (let j = 1; (1 << j) <= n; j++) {
// Compute maximum value for all letervals with
// size 2^j
for (let i = 0; (i + (1 << j) - 1) < n; i++) {
// For arr[2][10], we compare arr[lookup[0][7]]
// and arr[lookup[3][10]]
if (lookup[i][j - 1] > lookup[i + (1 << (j - 1))][j - 1])
lookup[i][j] = lookup[i][j - 1];
else
lookup[i][j] = lookup[i + (1 << (j - 1))][j - 1];
}
}
}
// Returns maximum of arr[L..R]
function query(L, R)
{
// Find highest power of 2 that is smaller
// than or equal to count of elements in given
// range
// For [2, 10], j = 3
let j = Math.floor(Math.log2(R - L + 1));
// Compute maximum of last 2^j elements with first
// 2^j elements in range
// For [2, 10], we compare arr[lookup[0][3]] and
// arr[lookup[3][3]]
if (lookup[L][j] >= lookup[R - (1 << j) + 1][j])
return lookup[L][j];
else
return lookup[R - (1 << j) + 1][j];
}
// Driver program
let a = [ 7, 2, 3, 0, 5, 10, 3, 12, 18 ];
let n = a.length;
buildSparseTable(a, n);
document.write(query(0, 4) + "<br>");
document.write(query(4, 7) + "<br>");
document.write(query(7, 8) + "<br>");
// This code is contributed by _saurabh_jaiswal.
</script>
So sparse table method supports query operation in O(1) time with O(n Log n) preprocessing time and O(n Log n) space.
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