Sum of Scores of Built Strings

This problem requires calculating the sum of scores for a string by checking the longest common prefix between each substring and the full string. Efficient algorithms like binary search, rolling hash, and suffix arrays can be employed to solve the problem. Focus on optimizing the process to handle large strings within time constraints.

Problem Statement

You are given a string s of length n, built by prepending characters one at a time. Each prefix of the string is labeled from s1 to sn, where s1 is just the first character and sn is the complete string. For each prefix si, its score is determined by the length of the longest common prefix between si and sn.

The task is to return the sum of the scores of all prefixes. For example, given the string "babab", the score of s1 is 1 (since the longest common prefix with s5 is "b"), the score of s2 is 0, and so on. The final result is the sum of all these scores.

Examples

Example 1

Input: s = "babab"

Output: 9

For s1 == "b", the longest common prefix is "b" which has a score of 1. For s2 == "ab", there is no common prefix so the score is 0. For s3 == "bab", the longest common prefix is "bab" which has a score of 3. For s4 == "abab", there is no common prefix so the score is 0. For s5 == "babab", the longest common prefix is "babab" which has a score of 5. The sum of the scores is 1 + 0 + 3 + 0 + 5 = 9, so we return 9.

Example 2

Input: s = "azbazbzaz"

Output: 14

For s2 == "az", the longest common prefix is "az" which has a score of 2. For s6 == "azbzaz", the longest common prefix is "azb" which has a score of 3. For s9 == "azbazbzaz", the longest common prefix is "azbazbzaz" which has a score of 9. For all other si, the score is 0. The sum of the scores is 2 + 3 + 9 = 14, so we return 14.

Constraints

• 1 <= s.length <= 105
• s consists of lowercase English letters.

Solution Approach

Binary Search Over Valid Answer Space

The problem can be optimized by performing binary search on the valid prefix lengths. By checking each potential prefix length in a logarithmic manner, we can efficiently calculate the longest common prefix for each substring and its corresponding suffix.

Rolling Hash

To speed up the comparison of prefixes, use a rolling hash technique. This allows you to compute the hash of a substring in constant time, making the comparison between prefixes more efficient.

Suffix Array and String Matching

A suffix array is a sorted list of all suffixes of the string. This data structure helps in efficiently finding the longest common prefix between any substring and the full string. Combined with binary search, it provides an optimal solution.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the chosen algorithm. With binary search and rolling hash, the solution can achieve O(n log n) time complexity, while using suffix arrays may improve the process, resulting in O(n log n) or better depending on implementation specifics. Space complexity also varies based on the approach but typically requires O(n) space for storing intermediate results.

What Interviewers Usually Probe

• Can the candidate identify the binary search pattern over the valid answer space?
• Does the candidate recognize the need for rolling hashes in optimizing string comparisons?
• Is the candidate familiar with advanced string algorithms like suffix arrays or string matching?

Common Pitfalls or Variants

Common pitfalls

• Misunderstanding the problem by calculating scores directly without considering efficient string matching techniques.
• Focusing too much on brute force solutions without optimizing with binary search or hashing.
• Incorrectly calculating the longest common prefix, especially when dealing with large strings or multiple substrings.

Follow-up variants

• What if the string s contains only one character? How does that simplify the problem?
• What if the string contains repeated substrings? How does that impact the approach?
• How would the problem change if instead of summing the scores, we had to find the maximum score?

FAQ

What is the binary search pattern for the Sum of Scores of Built Strings?

The binary search pattern focuses on efficiently determining the valid prefix length for each substring by narrowing down the range of possible lengths, minimizing comparisons.

How does the rolling hash improve performance in this problem?

Rolling hash allows for quick recalculation of substring hashes, making it easier to compare prefixes without recomputing them from scratch each time.

What is the role of suffix arrays in solving this problem?

Suffix arrays help by providing a sorted list of suffixes, which allows for efficient determination of the longest common prefix between substrings and the full string.

What is the expected time complexity for this problem?

The expected time complexity can be O(n log n) if using binary search along with efficient algorithms like rolling hashes and suffix arrays.

How can I optimize my approach for large inputs?

By using binary search combined with rolling hashes and suffix arrays, you can optimize both time and space complexity to handle large inputs efficiently.