Solving the BRCKTS Problem of SPOJ using Dynamic Programming and Catalan Numbers

The BRCKTS problem on SPOJ involves determining the number of valid ways to arrange brackets in a string, aligning this with the challenge of generating valid bracket sequences. This article explores this problem through dynamic programming and also delves into how Catalan numbers can provide a solution.

Problem Understanding

The goal is to count the number of valid bracket sequences that can be formed with n pairs of brackets. A valid sequence is defined as one where every opening bracket has a corresponding closing bracket, and at no point in the sequence do closing brackets outnumber opening brackets.

Dynamic Programming Approach

Define the DP Array

Let dp[i] be the number of valid bracket sequences that can be formed with i pairs of brackets.

Base Case

dp[0] 1 There is one way to arrange zero pairs of brackets: the empty sequence.

Recurrence Relation

For each i from 1 to n, calculate dp[i] using the formula:

dp[i] sum_{j0}^{i-1} dp[j] * dp[i-1-j]

This formula states that for every valid configuration of j pairs of brackets, you can form a valid configuration of i pairs by placing a pair of brackets around the valid configuration of j pairs and the remaining i-1-j pairs.

Implementation in Python

def count_bracket_sequences(n):    # Create a dp array to store the number of valid sequences for each count of pairs    dp  [0] * (n   1)    dp[0]  1  # Base case: one way to arrange zero pairs    # Fill the dp array using the recurrence relation    for i in range(1, n   1):        for j in range(i):            dp[i]   dp[j] * dp[i - 1 - j]    return dp[n]

Example Usage

The above function can be used as follows:

n  int(input())print(count_bracket_sequences(n))

Complexity

Time Complexity: O(n^2) as there are two nested loops iterating up to n.

O(n) for storing the dp array.

Alternative Approach: Catalan Numbers

The number of valid bracket sequences can also be computed using Catalan numbers, where the n-th Catalan number is given by:

C_n frac{1}{n 1} binom{2n}{n}

This can be computed using factorials, but the dynamic programming approach is often more convenient for programming competitions.

Conclusion

Using either the dynamic programming approach or the Catalan number formula will lead you to the correct solution for the BRCKTS problem on SPOJ. The DP method is generally easier to implement in a competitive programming setting, allowing for efficient computation within time constraints.