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Square Root Decomposition – A Lesser-Known Yet Powerful CP Technique

Introduction

In competitive programming, efficiently handling queries and updates on an array is a common challenge. Many beginners rely on brute force, which becomes impractical for large constraints. While segment trees and Fenwick trees (BIT) are popular alternatives, they can be complex to implement.

Enter Square Root Decomposition, a lesser-known yet highly effective technique that balances simplicity and efficiency. This method processes queries and updates in O(√N) time, making it a valuable tool when more advanced data structures are not necessary.

When to Use Square Root Decomposition

Square Root Decomposition is particularly useful when: - The array is mostly static or changes infrequently. - The constraints do not justify the complexity of segment trees. - You need to handle range queries efficiently (such as sum, minimum, maximum, or frequency counting).

This approach offers a simpler alternative to segment trees, especially for problems involving: - Range Sum Queries (RSQ) - Range Minimum/Maximum Queries - Frequency counting within a range (as used in Mo’s Algorithm)

Understanding Square Root Decomposition

1. The Core Idea

Instead of processing queries on the entire array, the array is divided into blocks of size approximately √N. Precomputed answers for each block allow for quicker query resolution.

Why use √N? - For an array with N elements, dividing it into blocks of size √N ensures that each query only needs to process at most 2√N elements, significantly reducing time complexity.

2. Dividing the Array into Blocks

Given an array arr[] of size N: 1. Split the array into approximately √N blocks. 2. Precompute the result for each block (such as the sum, minimum, or maximum). 3. Answer range queries by combining results from entire blocks and the remaining individual elements.

Implementation – Range Sum Queries

Below is an example implementation of Range Sum Queries (RSQ) using Square Root Decomposition.

Step 1: Preprocessing (Building Blocks)

• Divide arr[] into blocks of size √N.
• Compute and store the sum of each block in a separate array.

Step 2: Answering Range Sum Queries

• Sum the results from full blocks that lie completely within the query range.
• Process the remaining elements in the partially covered blocks separately.

Step 3: Updating an Element

• Identify the block containing the element to be updated.
• Update the block's precomputed sum to reflect the new value.

C++ Implementation

#include <bits/stdc++.h>
using namespace std;

const int MAXN = 100000; // Maximum array size
int arr[MAXN], block[MAXN]; // Original array and block sum storage
int blockSize, numBlocks;

// Function to preprocess the array into √N blocks
void preprocess(int n) {
    blockSize = sqrt(n);
    numBlocks = (n + blockSize - 1) / blockSize; // Calculate number of blocks

    // Initialize block sums to 0
    for (int i = 0; i < numBlocks; i++)
        block[i] = 0;

    // Compute block sums
    for (int i = 0; i < n; i++)
        block[i / blockSize] += arr[i];
}

// Function to get sum of elements in range [l, r]
int query(int l, int r) {
    int sum = 0;

    // Process first partial block
    while (l <= r && l % blockSize != 0)
        sum += arr[l++];

    // Process full blocks
    while (l + blockSize - 1 <= r) {
        sum += block[l / blockSize];
        l += blockSize;
    }

    // Process last partial block
    while (l <= r)
        sum += arr[l++];

    return sum;
}

// Function to update an element at index idx
void update(int idx, int val) {
    int blockIdx = idx / blockSize; // Find block index
    block[blockIdx] += (val - arr[idx]); // Update block sum
    arr[idx] = val; // Update the original array
}

int main() {
    int n = 9;
    int arrInput[] = {1, 5, 2, 4, 6, 3, 7, 8, 9};
    memcpy(arr, arrInput, sizeof(arrInput));

    preprocess(n);

    cout << "Sum of range [1, 6]: " << query(1, 6) << endl; // Expected output: 27
    update(4, 10);
    cout << "Sum of range [1, 6] after update: " << query(1, 6) << endl; // Expected output: 31

    return 0;
}

Complexity Analysis

• Preprocessing: O(N)
• Query Processing: O(√N)
• Update Operation: O(1) (for the block update)

Use Cases in Competitive Programming

1. Range Sum Queries (RSQ): As illustrated in the example.
2. Range Minimum/Maximum Queries: Modify the block array to store the minimum or maximum value of each block.
3. Frequency Counting in a Range: Store frequency counts in blocks and combine counts for efficient range queries.
4. Mo’s Algorithm: An advanced application that leverages the same principle to answer offline range queries in O((N + Q)√N) time.

Comparison with Other Techniques

Square Root Decomposition is best suited for cases where the input size is moderate (e.g., N ≤ 100,000) and where the problem constraints allow for a simpler implementation over more complex data structures.

Conclusion

Square Root Decomposition is an underrated yet powerful technique in competitive programming. Although it might not match the theoretical performance of segment trees or BITs in every scenario, its simplicity and ease of implementation make it a valuable tool for many problems.

Key takeaways: - Divide the array into blocks of size √N. - Precompute results for each block. - Process queries by combining the answers from full blocks and remaining elements. - Use this technique for range queries, minimum/maximum queries, and frequency counting, especially when the problem constraints are moderate.

Next time you encounter a problem that requires efficient range queries without the overhead of more complex structures, consider using Square Root Decomposition.