Introduction

In computer science, dealing with strings and subsequences is a common task, often essential for various algorithms and data processing techniques. One intriguing problem in this domain is identifying the longest subsequence within a string where all occurrences of a specific character are consolidated or brought together in one contiguous block. This problem merges concepts of sequence processing and character manipulation, making it highly applicable in fields like data compression, DNA sequence analysis, and text processing.

Problem Definition

Given a string `S` and a character `c`, the goal is to find the longest subsequence where all occurrences of `c` are contiguous. This problem can also be viewed as rearranging characters such that the length of the subsequence is maximized by placing all occurrences of `c` in one place.

Example

Consider the string `S = "aabcada"` and the character `c = "a"`. The input string can be rearranged or processed into subsequences. One such subsequence could be `"abcad"`, where all `a`'s are consolidated.

Technical Explanation

Approach

To solve the problem effectively:

1. Identify Positions of `c`: First, identify all the positions of the character `c` within the string.
2. Build Subsequence: Construct a new string that keeps every character once and places all occurrences of `c` together.
3. Calculate Length: Compute the length of this subsequence.

Algorithm

1. Traverse the String: Initialize pointers or use a loop to scan through the string and mark positions of `c`.
2. Formulate the Sequence:
   • Start with an empty result string.
   • Append all characters except `c`, keeping only one occurrence of each.
   • Append all occurrences of `c` together either at the beginning or end.
3. Evaluate: The length of this formed subsequence is the desired result.

Complexity Analysis

The algorithm generally runs in $O(n)$ time complexity, where $n$ is the length of the string. Space complexity largely depends on the storage of the result string and auxiliary structures, typically $O(n)$.

Sample Code

• Empty String: Return an empty result.
• No Occurrence of `c`: The longest subsequence is simply the unique characters from the string since `c` does not exist.
• For extremely large datasets, consider parallel processing to count and rearrange character subsequences more efficiently.