Range sum query using Sparse Table

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We have an array arr[]. We need to find the sum of all the elements in the range L and R where 0 <= L <= R <= n-1. Consider a situation when there are many range queries.

Examples:

Input : 3 7 2 5 8 9
        query(0, 5)
        query(3, 5)
        query(2, 4)
Output : 34
         22
         15

Note : array is 0 based indexed
       and queries too.

Since there are no updates/modifications, we use the Sparse table to answer queries efficiently. In a sparse table, we break queries in powers of 2.

Suppose we are asked to compute sum of 
elements from arr[i] to arr[i+12]. 
We do the following:

// Use sum of 8 (or 2³) elements 
table[i][3] = sum(arr[i], arr[i + 1], ...
                               arr[i + 7]).

// Use sum of 4 elements
table[i+8][2] = sum(arr[i+8], arr[i+9], ..
                                arr[i+11]).

// Use sum of single element
table[i + 12][0] = sum(arr[i + 12]).

Our result is sum of above values.

Notice that it took only 4 actions to compute the result over a subarray of size 13.

Flowchart:

Flowchart

C++

// CPP program to find the sum in a given
// range in an array using sparse table.
 
#include <bits/stdc++.h>
using namespace std;
 
// Because 2^17 is larger than 10^5
const int k = 16;
 
// Maximum value of array
const int N = 1e5;
 
// k + 1 because we need to access
// table[r][k]
long long table[N][k + 1];
 
// it builds sparse table.
void buildSparseTable(int arr[], int n)
{
    for (int i = 0; i < n; i++)
        table[i][0] = arr[i];
 
    for (int j = 1; j <= k; j++)
        for (int i = 0; i <= n - (1 << j); i++)
            table[i][j] = table[i][j - 1] +
               table[i + (1 << (j - 1))][j - 1];
}
 
// Returns the sum of the elements in the range
// L and R.
long long query(int L, int R)
{
    // boundaries of next query, 0-indexed
    long long answer = 0;
    for (int j = k; j >= 0; j--) {
        if (L + (1 << j) - 1 <= R) {
            answer = answer + table[L][j];
 
            // instead of having L', we
            // increment L directly
            L += 1 << j;
        }
    }
    return answer;
}
 
// Driver program.
int main()
{
    int arr[] = { 3, 7, 2, 5, 8, 9 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    buildSparseTable(arr, n);
 
    cout << query(0, 5) << endl;
    cout << query(3, 5) << endl;
    cout << query(2, 4) << endl;
 
    return 0;
}

Java

// Java program to find the sum 
// in a given range in an array 
// using sparse table.
 
class GFG
{
     
// Because 2^17 is larger than 10^5
static int k = 16;
 
// Maximum value of array
static int N = 100000;
 
// k + 1 because we need
// to access table[r][k]
static long table[][] = new long[N][k + 1];
 
// it builds sparse table.
static void buildSparseTable(int arr[],
                             int n)
{
    for (int i = 0; i < n; i++)
        table[i][0] = arr[i];
 
    for (int j = 1; j <= k; j++)
        for (int i = 0; i <= n - (1 << j); i++)
            table[i][j] = table[i][j - 1] +
            table[i + (1 << (j - 1))][j - 1];
}
 
// Returns the sum of the 
// elements in the range L and R.
static long query(int L, int R)
{
    // boundaries of next query,
    // 0-indexed
    long answer = 0;
    for (int j = k; j >= 0; j--) 
    {
        if (L + (1 << j) - 1 <= R) 
        {
            answer = answer + table[L][j];
 
            // instead of having L', we
            // increment L directly
            L += 1 << j;
        }
    }
    return answer;
}
 
// Driver Code
public static void main(String args[])
{
    int arr[] = { 3, 7, 2, 5, 8, 9 };
    int n = arr.length;
 
    buildSparseTable(arr, n);
 
    System.out.println(query(0, 5));
    System.out.println(query(3, 5));
    System.out.println(query(2, 4));
}
}
 
// This code is contributed 
// by Kirti_Mangal

C#

// C# program to find the 
// sum in a given range
// in an array using
// sparse table.
 
using System;
 
class GFG
{
    // Because 2^17 is
    // larger than 10^5
    static int k = 16;
     
    // Maximum value 
    // of array
    static int N = 100000;
     
    // k + 1 because we 
    // need to access table[r,k]
    static long [,]table = 
           new long[N, k + 1];
     
    // it builds sparse table.
    static void buildSparseTable(int []arr, 
                                 int n)
    {
        for (int i = 0; i < n; i++)
            table[i, 0] = arr[i];
     
        for (int j = 1; j <= k; j++)
            for (int i = 0;     
                     i <= n - (1 << j); i++)
                table[i, j] = table[i, j - 1] +
                table[i + (1 << (j - 1)), j - 1];
    }     
     
    // Returns the sum of the
    // elements in the range
    // L and R.
    static long query(int L, int R)
    {
        // boundaries of next 
        // query, 0-indexed
        long answer = 0;
        for (int j = k; j >= 0; j--) 
        {
            if (L + (1 << j) - 1 <= R) 
            {
                answer = answer + 
                         table[L, j];
     
                // instead of having 
                // L', we increment 
                // L directly
                L += 1 << j;
            }
        }
        return answer;
    }
     
    // Driver Code
    static void Main()
    {
        int []arr = new int[]{3, 7, 2, 
                              5, 8, 9};
        int n = arr.Length;
     
        buildSparseTable(arr, n);
     
        Console.WriteLine(query(0, 5));
        Console.WriteLine(query(3, 5));
        Console.WriteLine(query(2, 4));
    }
}
 
// This code is contributed by 
// Manish Shaw(manishshaw1)

Python3

# Python3 program to find the sum in a given
# range in an array using sparse table.
 
# Because 2^17 is larger than 10^5
k = 16
 
# Maximum value of array
n = 100000
 
# k + 1 because we need to access
# table[r][k]
 
table = [[0 for j in range(k+1)] for i in range(n)]
 
# it builds sparse table
def buildSparseTable(arr, n):
    global table, k
    for i in range(n):
        table[i][0] = arr[i]
 
    for j in range(1,k+1):
        for i in range(0,n-(1<<j)+1):
            table[i][j] = table[i][j-1] + \
                          table[i + (1 << (j - 1))][j - 1]
 
# Returns the sum of the elements in the range
# L and R.
def query(L, R):
    global table, k
 
    # boundaries of next query, 0 - indexed
    answer = 0
    for j in range(k,-1,-1):
        if (L + (1 << j) - 1 <= R):
            answer = answer + table[L][j]
 
            # instead of having L ', we
            # increment L directly
            L+=1<<j
 
    return answer
 
# Driver program
if __name__ == '__main__':
    arr = [3, 7, 2, 5, 8, 9]
    n = len(arr)
 
    buildSparseTable(arr, n)
    print(query(0,5))
    print(query(3,5))
    print(query(2,4))
     
# This code is contributed by 
# chaudhary_19 (Mayank Chaudhary)

Javascript

<script>
// JavaScript program to find the sum in a given
// range in an array using sparse table.
 
// Because 2^17 is larger than 10^5
const k = 16;
 
// Maximum value of array
const N = 1e5;
 
// k + 1 because we need to access
// table[r][k]
const table = new Array(N).fill(0).map(() => new Array(k + 1).fill(0));
 
// it builds sparse table.
function buildSparseTable(arr, n)
{
    for (let i = 0; i < n; i++)
        table[i][0] = arr[i];
 
    for (let j = 1; j <= k; j++)
        for (let i = 0; i <= n - (1 << j); i++)
            table[i][j] = table[i][j - 1] +
            table[i + (1 << (j - 1))][j - 1];
}
 
// Returns the sum of the elements in the range
// L and R.
function query(L, R)
{
 
    // boundaries of next query, 0-indexed
    let answer = 0;
    for (let j = k; j >= 0; j--)
    {
        if (L + (1 << j) - 1 <= R)
        {
            answer = answer + table[L][j];
 
            // instead of having L', we
            // increment L directly
            L += 1 << j;
        }
    }
    return answer;
}
 
// Driver program.
    let arr = [ 3, 7, 2, 5, 8, 9 ];
    let n = arr.length;
 
    buildSparseTable(arr, n);
 
    document.write(query(0, 5) + "<br>");
    document.write(query(3, 5) + "<br>");
    document.write(query(2, 4) + "<br>");
 
// This code is contributed by Manoj.
</script>

Output:

34
22
15

This algorithm for answering queries with Sparse Table works in O(k), which is O(log(n)) because we choose minimal k such that 2^k+1 > n.
Time complexity of sparse table construction : Outer loop runs in O(k), inner loop runs in O(n). Thus, in total we get O(n * k) = O(n * log(n))

Auxiliary Space: O(n*k), since n*k extra space has been taken.

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Last Updated : 21 Aug, 2022