Optimizing Palindromic Subsquare Identification on SPOJ

Introduction

Welcome to this comprehensive guide on optimizing the solution to a problem on SPOJ related to identifying palindromic subsquares in a 2D array. This article will delve into the intricacies of the algorithm, the logic behind the code, and the implementation details. By following this guide, you will be able to solve similar problems efficiently.

Understanding the Problem

The challenge involves finding the largest palindromic sub square for each position ij in a 2D array. The solution requires identifying the largest palindromic sub squares with ij as the bottom right corner. The final answer will be the maximum value among all such palindromic sub squares.

Algorithm Overview

The key steps in the algorithm are:

Compute palindromes for each position horizontally and vertically. Store the information in arrays to facilitate quick lookup. Determine the largest palindromic sub square at each position.

Step-by-Step Explanation

Step 1: Computing Palindromes

The first step involves identifying palindromic sub squares of varying lengths for the given position ij. We use a 3D boolean array b[n][n][n] where b[i][j][k]1 indicates the presence of a palindrome of length k ending at position ij.

To determine these palindromes, we check the following conditions:

For a palindrome of length k at position ij, the value of the array at the position ij-k 1 horizontally (or ij-k 1 vertically) must be the same as the value at position ij. Additionally, there should be a palindrome of length k-2 ending at ij-1 vertically (or ij-1 horizontally).

Step 2: Storing Information

Once the palindromes are identified, we store the information in arrays to facilitate quick lookup and manipulation. The arrays h[n][n][n] and v[n][n][n] are used to store the count of consecutive rows and columns with palindromes of a particular length, respectively.

The arrays are defined as follows:

h[i][j][k]: Stores the number of consecutive rows containing a palindrome of length k ending at position ij. v[i][j][k]: Stores the number of consecutive columns containing a palindrome of length k ending at position ij.

Each element in these arrays is computed based on the presence of palindromes and their subsequent lengths.

Step 3: Determining the Largest Palindromic Subsquare

Finally, we determine the largest palindromic sub square that can be formed at each position ij. If for a given position ij and a given length k, h[i][j][k] k and v[i][j][k] k, it means that there is a palindromic sub square of length k ending at position ij.

Time Complexity

The overall time complexity of this solution is O(N^3), making it efficient for large 2D arrays.

In conclusion, by following the steps outlined in this guide, you can effectively solve the SPOJ problem related to identifying palindromic subsquares in a 2D array. The algorithm leverages dynamic programming principles to optimize the solution, providing a robust and efficient approach to the problem.