Square Root (Sqrt) Decomposition Algorithm

Square Root Decomposition Technique is one of the most common query optimization techniques used by competitive programmers. This technique helps us to reduce Time Complexity by a factor of sqrt(N) 

The key concept of this technique is to decompose a given array into small chunks specifically of size sqrt(N)

Follow the below steps to solve the problem:

• We have an array of n elements and we decompose this array into small chunks of size sqrt(N)
• We will be having exactly sqrt(N) such chunks provided that N is a perfect square
• Therefore, now our array of N elements is decomposed into sqrt(N) blocks, where each block contains sqrt(N) elements (assuming the size of the array is a perfect square)
• Let's consider these chunks or blocks as an individual array each of which contains sqrt(N) elements and you have computed your desired answer(according to your problem) individually for all the chunks
• Now, you need to answer certain queries asking you the answer for the elements in the range l to r(l and r are starting and ending indices of the array) in the original n-sized array

Naive Approach: To solve the problem follow the below idea:

Simply iterate over each element in the range l to r and calculate its corresponding answer. Therefore, the Time Complexity per query will be O(N)

Below is the code for above approach :

// C++ program to demonstrate working of simple
// array query using iteration
#include <bits/stdc++.h>
using namespace std;

#define MAXN 10000

int arr[MAXN]; // original array

// Time Complexity: O(r-l+1)
int query(int l, int r)
{
    int sum = 0;
    for (int i = l; i <= r; i++) {
        sum += arr[i];
    }
    return sum;
}

// Driver code
int main()
{
    // We have used separate array for input because
    // the purpose of this code is to explain simple
    // array query using iteration in competitive
    // programming where we have multiple inputs.
    int input[] = { 1, 5, 2, 4, 6, 1, 3, 5, 7, 10 };
    int n = sizeof(input) / sizeof(input[0]);

    // copying input[] to arr[]
    memcpy(arr, input, sizeof(input));

    cout << "query(3,8) : " << query(3, 8) << endl;
    cout << "query(1,6) : " << query(1, 6) << endl;
    arr[8] = 0; // updating arr[8] to 0
    cout << "query(8,8) : " << query(8, 8) << endl;
    return 0;
}

import java.util.Arrays;

public class GFG{
    static int[] arr = new int[10000]; // original array

    // Time Complexity: O(r-l+1)
    static int query(int l, int r) {
        int sum = 0;
        for (int i = l; i <= r; i++) {
            sum += arr[i];
        }
        return sum;
    }

    // Driver code
    public static void main(String[] args) {
        // We have used separate array for input because
        // the purpose of this code is to explain simple
        // array query using iteration in competitive
        // programming where we have multiple inputs.
        int[] input = {1, 5, 2, 4, 6, 1, 3, 5, 7, 10};
        int n = input.length;

        // copying input[] to arr[]
        System.arraycopy(input, 0, arr, 0, n);

        System.out.println("query(3,8) : " + query(3, 8));
        System.out.println("query(1,6) : " + query(1, 6));
        arr[8] = 0; // updating arr[8] to 0
        System.out.println("query(8,8) : " + query(8, 8));
    }
}

arr = [0] * 10000  # original array

# Time Complexity: O(r-l+1)
def query(l, r):
    _sum = 0
    for i in range(l, r + 1):
        _sum += arr[i]
    return _sum

# Driver code
if __name__ == '__main__':
    # We have used separate list for input because
    # the purpose of this code is to explain simple
    # array query using iteration in competitive
    # programming where we have multiple inputs.
    input_arr = [1, 5, 2, 4, 6, 1, 3, 5, 7, 10]

    # copying input_arr to arr
    arr[:len(input_arr)] = input_arr

    print("query(3,8) :", query(3, 8))
    print("query(1,6) :", query(1, 6))
    arr[8] = 0  # updating arr[8] to 0
    print("query(8,8) :", query(8, 8))

using System;

class Program
{
    const int MAXN = 10000;
    static int[] arr = new int[MAXN]; // original array

    // Time Complexity: O(r-l+1)
    static int Query(int l, int r)
    {
        int sum = 0;
        for (int i = l; i <= r; i++)
        {
            sum += arr[i];
        }
        return sum;
    }

    static void Main()
    {
        // We have used separate array for input because
        // the purpose of this code is to explain simple
        // array query using iteration in competitive
        // programming where we have multiple inputs.
        int[] input = { 1, 5, 2, 4, 6, 1, 3, 5, 7, 10 };
        int n = input.Length;

        // copying input[] to arr[]
        Array.Copy(input, arr, n);

        Console.WriteLine("query(3,8) : " + Query(3, 8));
        Console.WriteLine("query(1,6) : " + Query(1, 6));
        arr[8] = 0; // updating arr[8] to 0
        Console.WriteLine("query(8,8) : " + Query(8, 8));
    }
}
// This code is contributed by uomkar369

// Function to calculate the sum of elements in the range [l, r] of the array arr
// Time Complexity: O(r - l + 1)
function query(l, r, arr) {
    let sum = 0;
    for (let i = l; i <= r; i++) {
        sum += arr[i];
    }
    return sum;
}

// Driver code
    // We have used a separate array for input because
    // the purpose of this code is to explain a simple
    // array query using iteration in competitive
    // programming where we have multiple inputs.
    const input = [1, 5, 2, 4, 6, 1, 3, 5, 7, 10];
    const n = input.length;

    // copying input[] to arr[]
    const arr = [...input];

    console.log("query(3,8) :", query(3, 8, arr));
    console.log("query(1,6) :", query(1, 6, arr));
    arr[8] = 0; // updating arr[8] to 0
    console.log("query(8,8) :", query(8, 8, arr));

query(3,8) : 26
query(1,6) : 21
query(8,8) : 0

Efficient Approach(Sqrt Decomposition Trick): To solve the problem follow the below idea: 

As we have already precomputed the answer for all individual chunks and now we need to answer the queries in range l to r. Now we can simply combine the answers of the chunks that lie in between the range l to r in the original array. So, if we see carefully here we are jumping sqrt(N) steps at a time instead of jumping 1 step at a time as done in the naive approach. Let's just analyze its Time Complexity and implementation considering the below problem:

Problem :

Given an array of n elements. We need to answer q 
queries telling the sum of elements in range l to 
r in the array. Also the array is not static i.e 
the values are changed via some point update query.

Range Sum Queries are of form : Q l r , 
where l is the starting index r is the ending index

Point update Query is of form : U idx val, 
where idx is the index to update val is the 
updated value

Below is the illustration of the above approach:

Let us consider that we have an array of 9 elements: A[] = {1, 5, 2, 4, 6, 1, 3, 5, 7}
1. Let's decompose this array into sqrt(9) blocks, where each block will contain the sum of elements lying in it. Therefore now our decomposed array would look like this:
 

2. Till now we have constructed the decomposed array of sqrt(9) blocks and now we need to print the sum of elements in a given range. 
So first let's see two basic types of overlap that a range query can have on our array:

Range Query of type 1 (Given Range is on Block Boundaries) :

In this type the query, the range may totally cover the continuous sqrt blocks. So we can easily answer the sum of values in this range as the sum of completely overlapped blocks. 

So the answer for the above query in the described image will be: ans = 11 + 15 = 26 

Time Complexity: O(sqrt(N)). In the worst case, our range can be 0 to N-1(where N is the size of the array and assuming N to be a perfect square). In this case, all the blocks are completely overlapped by our query range. Therefore, to answer this query we need to iterate over all the decomposed blocks for the array and we know that the number of blocks = sqrt(N). Hence, the complexity for this type of query will be O(sqrt(N)) in the worst case.

Range Query of type 2 (Given Range is NOT on boundaries):

We can deal with these types of queries by summing the data from the completely overlapped decomposed blocks lying in the query range and then summing elements one by one from the original array whose corresponding block is not completely overlapped by the query range. 

So the answer for the above query in the described image will be: ans = 5 + 2 + 11 + 3 = 21 

Time Complexity: O(sqrt(N)). Let's consider a query [l = 1 and r = n-2] (n is the size of the array and has 0-based indexing). Therefore, for this query exactly ( sqrt(n) - 2 ) blocks will be completely overlapped whereas the first and last blocks will be partially overlapped with just one element left outside the overlapping range. So, the completely overlapped blocks can be summed up in ( sqrt(n) - 2 ) ~ sqrt(n) iterations, whereas the first and last blocks are needed to be traversed one by one separately. But as we know that the number of elements in each block is at max sqrt(n), to sum up, the last block individually we need to make, 
(sqrt(n)-1) ~ sqrt(n) iterations and same for the last block. 

So, the overall Complexity = O(sqrt(n)) + O(sqrt(n)) + O(sqrt(n)) = O(3*sqrt(N)) = O(sqrt(N))
 

Update Queries(Point update):

In this query, we simply find the block in which the given index lies, then subtract its previous value and add the new updated value as per the point update query. 

Time Complexity: O(1) 

Below is the implementation of the above approach:

// C++ program to demonstrate working of Square Root
// Decomposition.
#include <bits/stdc++.h>
using namespace std;

#define MAXN 10000
#define SQRSIZE 100

int arr[MAXN]; // original array
int block[SQRSIZE]; // decomposed array
int blk_sz; // block size

// Time Complexity : O(1)
void update(int idx, int val)
{
    int blockNumber = idx / blk_sz;
    block[blockNumber] += val - arr[idx];
    arr[idx] = val;
}

// Time Complexity : O(sqrt(n))
int query(int l, int r)
{
    int sum = 0;
    while (l < r and l % blk_sz != 0 and l != 0) {
        // traversing first block in range
        sum += arr[l];
        l++;
    }
    while (l + blk_sz - 1 <= r) {
        // traversing completely overlapped blocks in range
        sum += block[l / blk_sz];
        l += blk_sz;
    }
    while (l <= r) {
        // traversing last block in range
        sum += arr[l];
        l++;
    }
    return sum;
}

// Fills values in input[]
void preprocess(int input[], int n)
{
    // initiating block pointer
    int blk_idx = -1;

    // calculating size of block
    blk_sz = sqrt(n);

    // building the decomposed array
    for (int i = 0; i < n; i++) {
        arr[i] = input[i];
        if (i % blk_sz == 0) {
            // entering next block
            // incrementing block pointer
            blk_idx++;
        }
        block[blk_idx] += arr[i];
    }
}

// Driver code
int main()
{
    // We have used separate array for input because
    // the purpose of this code is to explain SQRT
    // decomposition in competitive programming where
    // we have multiple inputs.
    int input[] = { 1, 5, 2, 4, 6, 1, 3, 5, 7, 10 };
    int n = sizeof(input) / sizeof(input[0]);

    preprocess(input, n);

    cout << "query(3,8) : " << query(3, 8) << endl;
    cout << "query(1,6) : " << query(1, 6) << endl;
    update(8, 0);
    cout << "query(8,8) : " << query(8, 8) << endl;
    return 0;
}

// Java program to demonstrate working of
// Square Root Decomposition.
import java.util.*;

class GFG {

    static int MAXN = 10000;
    static int SQRSIZE = 100;

    static int[] arr = new int[MAXN]; // original array
    static int[] block
        = new int[SQRSIZE]; // decomposed array
    static int blk_sz; // block size

    // Time Complexity : O(1)
    static void update(int idx, int val)
    {
        int blockNumber = idx / blk_sz;
        block[blockNumber] += val - arr[idx];
        arr[idx] = val;
    }

    // Time Complexity : O(sqrt(n))
    static int query(int l, int r)
    {
        int sum = 0;
        while (l < r && l % blk_sz != 0 && l != 0) {
            // traversing first block in range
            sum += arr[l];
            l++;
        }
        while (l + blk_sz - 1 <= r) {
            // traversing completely
            // overlapped blocks in range
            sum += block[l / blk_sz];
            l += blk_sz;
        }
        while (l <= r) {
            // traversing last block in range
            sum += arr[l];
            l++;
        }
        return sum;
    }

    // Fills values in input[]
    static void preprocess(int input[], int n)
    {
        // initiating block pointer
        int blk_idx = -1;

        // calculating size of block
        blk_sz = (int)Math.sqrt(n);

        // building the decomposed array
        for (int i = 0; i < n; i++) {
            arr[i] = input[i];
            if (i % blk_sz == 0) {
                // entering next block
                // incrementing block pointer
                blk_idx++;
            }
            block[blk_idx] += arr[i];
        }
    }

    // Driver code
    public static void main(String[] args)
    {

        // We have used separate array for input because
        // the purpose of this code is to explain SQRT
        // decomposition in competitive programming where
        // we have multiple inputs.
        int input[] = { 1, 5, 2, 4, 6, 1, 3, 5, 7, 10 };
        int n = input.length;

        preprocess(input, n);

        System.out.println("query(3, 8) : " + query(3, 8));
        System.out.println("query(1, 6) : " + query(1, 6));
        update(8, 0);
        System.out.println("query(8, 8) : " + query(8, 8));
    }
}

// This code is contributed by PrinciRaj1992

# Python 3 program to demonstrate working of Square Root
# Decomposition.
from math import sqrt

MAXN = 10000
SQRSIZE = 100

arr = [0]*(MAXN)         # original array
block = [0]*(SQRSIZE)     # decomposed array
blk_sz = 0                 # block size

# Time Complexity : O(1)


def update(idx, val):
    blockNumber = idx // blk_sz
    block[blockNumber] += val - arr[idx]
    arr[idx] = val

# Time Complexity : O(sqrt(n))


def query(l, r):
    sum = 0
    while (l < r and l % blk_sz != 0 and l != 0):

        # traversing first block in range
        sum += arr[l]
        l += 1

    while (l + blk_sz - 1 <= r):

        # traversing completely overlapped blocks in range
        sum += block[l//blk_sz]
        l += blk_sz

    while (l <= r):

        # traversing last block in range
        sum += arr[l]
        l += 1

    return sum

# Fills values in input[]


def preprocess(input, n):

    # initiating block pointer
    blk_idx = -1

    # calculating size of block
    global blk_sz
    blk_sz = int(sqrt(n))

    # building the decomposed array
    for i in range(n):
        arr[i] = input[i]
        if (i % blk_sz == 0):

            # entering next block
            # incrementing block pointer
            blk_idx += 1

        block[blk_idx] += arr[i]

# Driver code


# We have used separate array for input because
# the purpose of this code is to explain SQRT
# decomposition in competitive programming where
# we have multiple inputs.
input = [1, 5, 2, 4, 6, 1, 3, 5, 7, 10]
n = len(input)

preprocess(input, n)

print("query(3,8) : ", query(3, 8))
print("query(1,6) : ", query(1, 6))
update(8, 0)
print("query(8,8) : ", query(8, 8))

# This code is contributed by Sanjit_Prasad

// C# program to demonstrate working of
// Square Root Decomposition.
using System;

class GFG {
    static int MAXN = 10000;
    static int SQRSIZE = 100;

    static int[] arr = new int[MAXN]; // original array
    static int[] block
        = new int[SQRSIZE]; // decomposed array
    static int blk_sz; // block size

    // Time Complexity : O(1)
    static void update(int idx, int val)
    {
        int blockNumber = idx / blk_sz;
        block[blockNumber] += val - arr[idx];
        arr[idx] = val;
    }

    // Time Complexity : O(sqrt(n))
    static int query(int l, int r)
    {
        int sum = 0;
        while (l < r && l % blk_sz != 0 && l != 0) {
            // traversing first block in range
            sum += arr[l];
            l++;
        }
        while (l + blk_sz - 1 <= r) {
            // traversing completely
            // overlapped blocks in range
            sum += block[l / blk_sz];
            l += blk_sz;
        }
        while (l <= r) {
            // traversing last block in range
            sum += arr[l];
            l++;
        }
        return sum;
    }

    // Fills values in input[]
    static void preprocess(int[] input, int n)
    {
        // initiating block pointer
        int blk_idx = -1;

        // calculating size of block
        blk_sz = (int)Math.Sqrt(n);

        // building the decomposed array
        for (int i = 0; i < n; i++) {
            arr[i] = input[i];
            if (i % blk_sz == 0) {
                // entering next block
                // incrementing block pointer
                blk_idx++;
            }
            block[blk_idx] += arr[i];
        }
    }

    // Driver code
    public static void Main(String[] args)
    {

        // We have used separate array for input because
        // the purpose of this code is to explain SQRT
        // decomposition in competitive programming where
        // we have multiple inputs.
        int[] input = { 1, 5, 2, 4, 6, 1, 3, 5, 7, 10 };
        int n = input.Length;

        preprocess(input, n);

        Console.WriteLine("query(3, 8) : " + query(3, 8));
        Console.WriteLine("query(1, 6) : " + query(1, 6));
        update(8, 0);
        Console.WriteLine("query(8, 8) : " + query(8, 8));
    }
}

// This code is contributed by 29AjayKumar

<script>
// Javascript program to demonstrate working of 
// Square Root Decomposition.

let MAXN = 10000;
let SQRSIZE = 100;

let arr = new Array(MAXN);
for(let i = 0; i < MAXN; i++)
{
    arr[i] = 0;
}

let block = new Array(SQRSIZE);
for(let i = 0; i < SQRSIZE; i++)
{
    block[i] = 0;
}

let blk_sz;     

// Time Complexity : O(1)
function update(idx,val)
{
    let blockNumber = idx / blk_sz;
    block[blockNumber] += val - arr[idx];
    arr[idx] = val;
}

// Time Complexity : O(sqrt(n))
function query(l, r)
{
    let sum = 0;
    while (l < r && l % blk_sz != 0 && l != 0)
    {
        // traversing first block in range
        sum += arr[l];
        l++;
    }
    while (l+blk_sz-1 <= r)
    {
        // traversing completely
        // overlapped blocks in range
        sum += block[l / blk_sz];
        l += blk_sz;
    }
    while (l <= r)
    {
        // traversing last block in range
        sum += arr[l];
        l++;
    }
    return sum;
}

// Fills values in input[]
function preprocess(input, n)
{
    // initiating block pointer
    let blk_idx = -1;
  
    // calculating size of block
    blk_sz = Math.floor( Math.sqrt(n));
  
    // building the decomposed array
    for (let i = 0; i < n; i++)
    {
        arr[i] = input[i];
        if (i % blk_sz == 0)
        {
        
            // entering next block
            // incrementing block pointer
            blk_idx++;
        }
        block[blk_idx] += arr[i];
    }
}

// Driver code
// We have used separate array for input because
    // the purpose of this code is to explain SQRT
    // decomposition in competitive programming where
    // we have multiple inputs.
    let input = [1, 5, 2, 4, 6, 1, 3, 5, 7, 10];
    let n = input.length;
  
    preprocess(input, n);
  
    document.write("query(3, 8) : " +
                        query(3, 8)+"<br>");
    document.write("query(1, 6) : " +
                        query(1, 6)+"<br>");
    update(8, 0);
    document.write("query(8, 8) : " + 
                        query(8, 8)+"<br>");

// This code is contributed by rag2127
</script>

query(3,8) : 26
query(1,6) : 21
query(8,8) : 0

Time Complexity: O(N)
Auxiliary Space: O(MAXN), since MAXN extra space has been taken, where MAXN is the maximum value of N

Note: The above code works even if N is not a perfect square. In this case, the last block will contain even less number of elements than sqrt(N), thus reducing the number of iterations. 
Let's say n = 10. In this case, we will have 4 blocks first three blocks of size 3, and the last block of size 1.