Longest Common Subsequence
Given two strings, return the length of their longest common subsequence (LCS). A subsequence maintains relative order but needn't be contiguous.
Example:
abcde ace
3
- The two input strings are
abcdeandace, and we need to find their longest common subsequence (LCS). - We compare the characters of both strings and find the common characters in the same relative order:
a,c, ande. - The length of this LCS is 3, since it contains three characters.
- The final output is the length of the LCS, which is 3.
Constraints:
- 1 <= len(text1), len(text2) <= 1000
- Strings consist of lowercase English letters
Background Knowledge
The Longest Common Subsequence (LCS) problem is a classic example of a dynamic programming problem. To understand this problem, you need to grasp the concept of a subsequence, which is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, given the string "ABC", some possible subsequences are "A", "B", "C", "AB", "BC", and "ABC". The LCS of two strings is the longest subsequence that is common to both strings.
In dynamic programming, we break down complex problems into smaller subproblems, solve each subproblem only once, and store the solutions to subproblems to avoid redundant computation. This approach is particularly useful for problems that have overlapping subproblems, meaning that the problem can be broken down into subproblems that may have some overlap. The LCS problem has overlapping subproblems because the LCS of two strings can be derived from the LCS of their prefixes.
To solve the LCS problem, you need to understand the concept of a 2D array or matrix, where each cell represents the LCS of two prefixes of the input strings. The value of each cell can be computed based on the values of its neighboring cells. This is a key insight in dynamic programming, where we build a solution to a complex problem by combining solutions to smaller subproblems.
Algorithm/Approach
The general approach to solve the LCS problem is to use dynamic programming with a 2D array or matrix. The algorithm pattern involves:
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