MO’s Algorithm aka Square Root Decomposition, a very efficient and easy technique to solve Range Query Problems (RQP). For MO’s Algorithm to work, the RQP has to be offline. In this post, we will understand about RQP, Offline RPQ, Naive Approach to solve RQP and an Efficient Approach using MO’s Algorithm.
What is a Range Query Problem?
You are given a sequence A of N values A₁, A₂, A₃, …, Aₙ. You also are given Q queries. In each query, you will be given two values l and r. Your task is to perform a function f(l, r) on the elements in the subsequence Aₗ, Aₗ₊₁, …, Aᵣ₋₁, Aᵣ
What is an Offline Query Problem?
An RQP is offline if it satisfies the following conditions
All Queries are known beforehand.
There are no update operations on the given sequence.
You are given a sequence A of N values A₁, A₂, A₃, …, Aₙ. You also are given Q queries. Each query will have two values l and r. Your task is to find the number of vowels in the range Aₗ, Aₗ₊₁, …, Aᵣ₋₁, Aᵣ
A Simple approach is to iterate from l to r for each query (l, r) and find the number of vowels in the range. A sample code snippet is given below.
Java Code Snippet for Naive Approach
Time Complexity The time complexity of the above code would be O(NQ)* where N = Number of elements in the Sequence and Q = Number of Queries.
MO’s algorithm follows two simple steps to make the solution more efficient
Divide the sequence into *√N blocks.*
Rearrange all the queries in such a way that a query (l₁, r₁) comes before (l₂, r₂) if
MO’s Order Conditions
By rearranging the queries in the above manner, we tend to process all the queries of one block before processing the queries of another block, thus minimizing the pointer movement of i and j. Now, we apply MO’s technique in our problem
Now we have a sequence of letters of size 12. Now we apply the first step, calculate block size in our sequence
BLOCK_SIZE = √12 ≈ 3, So, we divide our sequence to 3 blocks each of size 4.
Sequence divided into √N blocks
Now consider that the following queries are given to us (4, 9), (3, 7), (2, 8), (0, 5), (7, 10), (1, 11)
Out next step is to arrange the above queries based on MO’s order. The java code snippet for sorting logic is given below
Code Snippet for Soring Logic
Now the queries will be rearranged as (0, 5), (2, 8), (1, 11), (3, 7), (4, 9), (7, 10)
Now we process the queries in the new order
Initially, i, j, and vowelsCount will be set to 0.
The whole idea is based on 2 observations
a. While **incrementing** ***i ***we are excluding ***A[i]*** from our range and while **decrementing *i*** we are including ***A[i]*** in our range. b. While **incrementing *j ***we are including ***A[j]*** from our range and while **decrementing *j*** we are including ***A[j]*** in our range.
- The first query is (0, 5). i is in the right position. So, we increment j till it reaches 5. While incrementing j if the letter present at A[j] is vowel we increment vowelsCount.
Process query (0, 5)
- The next query is (2, 8). Currently, i and j are in positions 0 and 5 respectively. So we increment j till it reaches 8 and increment i till it reaches index 2. While incrementing j, if the letter at A[j] is a vowel we increment vowelsCount, and while incrementing i, if the letter at A[i] is a vowel we decrement vowelsCount.
Process query (2, 8)
- The next query is (1, 11). Currently, i and j are in positions 2 and 8 respectively. So we increment j till it reaches 11 and decrement i till it reaches index 1. While decrementing i we increment vowelsCount and while incrementing j, we increment vowelsCount.
- Similarly, we can adjust the pointer i and j based on the remaining queries. The remaining steps are pictured below
Process Query (3, 7)
Process Query (4, 9)
Process Query (7, 10)
Once all the queries are processed, we can print the result in the given order of queries.
Time Complexity For each query, there are two movements one for pointer i and another for pointer j
Movement of pointer i All queries are sorted in such a way that l values are grouped by blocks. So, we won’t move to the next block until we process all the queries in the current block. So, i ‘s movement will be O(√N) for each query. So for Q queries, the complexity will be O(Q √N)*
Movement of pointer j All queries are sorted in increasing order of r. So, for a given block j will be always increasing. So, *j can travel all the way up to **N *for each block. So the complexity will be *O(N √N)**
So, the overall complexity will be O(Q √N) + O(N √N) = O((N + Q) √N)*
MO’s algorithm will be very useful in competitive programming. You can find practice problems for MO’s Algorithms on codechef, SPOJ, codeforces, etc. One of the famous problems is DQUERY from SPOJ SPOJ.com - Problem DQUERY Đọc đề đẹp hơn ở: codeforces.com/group/FLVn1Sc504/contest/274.. Given a sequence of n numbers a 1, a…spoj.com
You can find the solution for the above problem in below Github URL ganeshkumarm1/DSAlgo Contribute to ganeshkumarm1/DSAlgo development by creating an account on GitHub.github.com
Thank you 🤘
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