Square Root Decomposition

Unlocking the Power of Square Root Decomposition: Simplifying Complex Mathematical Problems

Square root decomposition is a data structure technique that allows us to perform range queries on an array in an efficient way. The basic idea is to divide the array into blocks of equal size, and precompute some information about each block that can be used to answer queries on that block. This allows us to answer queries on a range of elements by combining the pre-computed information for each block in that range.

For example, let’s say we have an array A of size N and we want to be able to perform range sum queries on it. We can use square root decomposition to build a data structure that allows us to answer these queries in O(sqrt(N)) time.

First, we divide the array into blocks of size B = sqrt(N). If N is not a perfect square, we add one more block to the end of the array to make it a perfect square.

Next, for each block i, we precompute the sum of all elements in that block and store it in an array S. We also keep track of the starting and ending indices of each block in another array L.

For example, if we have the following array A:

A = [3, 1, 4, 2, 5, 6, 7, 8]

We can divide it into blocks of size B = sqrt(8) = 2:

A = [[3, 1], [4, 2], [5, 6], [7, 8]]

And precompute the sums for each block and store them in the array S:

S = [4, 6, 11, 15]

And store the starting and ending indices of each block in the array L:

L = [(0, 1), (2, 3), (4, 5), (6, 7)]

Now, to answer a range sum query for a range [l, r], we first find the blocks that contain the starting and ending indices of the range. We can do this by finding the block indices for l and r using integer division:

start_block = l / B
end_block = r / B

Next, we sum up the elements from the blocks that are fully contained in the range:

sum = 0
if start_block == end_block:
    # The range is contained within a single block
    for i in range(l, r+1):
        sum += A[i]
else:
    # The range spans multiple blocks
    for i in range(l, (start_block+1)*B):
        sum += A[i]
    for i in range(start_block+1, end_block):
        sum += S[i]
    for i in range(end_block*B, r+1):
        sum += A[i]

The first loop handles the case where the range is contained within a single block. In this case, we simply sum up the elements in the range.

The second loop handles the case where the range spans multiple blocks. We sum up the elements from the starting block up to the end of the block. We then sum up the precomputed sums for all the blocks fully contained within the range. Finally, we sum up the elements from the start of the end block up to the end of the range.

This approach allows us to answer range sum queries on an array in O(sqrt(N)) time, which is much faster than a naive approach that would take O(N) time to sum up all the elements in the range.