11.3 Longest common subsequence and edit distance
4 min read•july 30, 2024
shines in solving sequence comparison problems like the and . These algorithms find patterns and measure similarities between strings, crucial in fields like bioinformatics, text analysis, and version control.
Both LCS and Edit Distance use clever table-filling techniques to break down complex problems into manageable subproblems. By storing and reusing intermediate results, they achieve efficient solutions, showcasing the power of dynamic programming in tackling real-world challenges.
Longest Common Subsequence Problem
Definition and Applications
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- (LCS) finds the longest common to two or more sequences
- Subsequence appears in the same relative order but not necessarily contiguous
- Applications in bioinformatics compare genetic sequences and identify similarities between DNA or protein sequences
- Version control systems use LCS to determine differences between file versions and generate efficient diff outputs
- Natural language processing employs LCS for plagiarism detection and text similarity analysis
- Multiple problem extends LCS to multiple sequences crucial in evolutionary biology and comparative genomics
- LCS serves as foundation for related problems (shortest common supersequence, longest increasing subsequence)
Problem Characteristics
- Exhibits allows problem to be broken down into smaller subproblems
- Contains solutions to subproblems can be reused
- Suitable for dynamic programming approach due to these characteristics
- Can be solved recursively but inefficient for large inputs due to redundant computations
- Efficient solution requires storing and reusing intermediate results
Dynamic Programming for LCS
Table Construction and Filling
- Construct 2D table to store LCS lengths for all prefixes of two input sequences
- Table dimensions (m+1) x (n+1) where m and n are lengths of sequences
- Fill table using recurrence relation based on character comparison:
- If characters match add 1 to LCS length of prefixes without these characters
- If characters don't match take maximum of LCS lengths excluding one character from either sequence
- Recurrence relation: 0 & \text{if } i = 0 \text{ or } j = 0 \\ LCS[i-1][j-1] + 1 & \text{if } X[i] = Y[j] \\ \max(LCS[i-1][j], LCS[i][j-1]) & \text{if } X[i] \neq Y[j] \end{cases}$$
- Example: For sequences "ABCDGH" and "AEDFHR" LCS is "ADH" with length 3
Algorithm Complexity and Optimization
- Time complexity O(mn) where m and n are lengths of input sequences
- Space complexity O(mn) can be optimized to O(min(m,n)) by using only two rows of table at a time
- Backtracking through filled table reconstructs actual LCS sequence not just its length
- Optimization techniques:
- Using bit vectors for binary alphabet
- Hirschberg's algorithm combines divide-and-conquer with dynamic programming to achieve O(min(m,n)) space complexity
Edit Distance Problem
Problem Definition and Significance
- Edit distance () measures minimum number of single-character edits to transform one string into another
- Single-character edits include insertions deletions and substitutions
- Fundamental in computational linguistics DNA sequence alignment and spell-checking algorithms
- Provides quantitative measure of string similarity crucial in fuzzy string matching and approximate string matching
- Natural language processing uses edit distance for automatic spelling error correction and speech recognition
- Bioinformatics applies edit distance to analyze mutations and evolutionary relationships between genetic sequences
- Generalizes to weighted edit distance where different have varying costs allowing nuanced comparisons
Applications and Variations
- Spell checkers suggest corrections based on words with low edit distance from misspelled word
- DNA sequence alignment uses edit distance to measure similarity between genetic sequences
- Plagiarism detection compares document similarity using edit distance on word or sentence level
- Fuzzy string search finds strings that approximately match a pattern within a specified edit distance
- Weighted edit distance assigns different costs to various edit operations (transposition might cost less than substitution)
- Example: Edit distance between "kitten" and "sitting" is 3 (substitute 'k' for 's' substitute 'e' for 'i' insert 'g' at the end)
Dynamic Programming for Edit Distance
Table Construction and Filling
- Construct (m+1) x (n+1) table where m and n are lengths of input strings
- Base cases represent edit distances between empty strings and prefixes of input strings
- Fill table using recurrence relation considering three operations at each step:
- Insertion
- Deletion
- Substitution (or match if characters are the same)
- Recurrence relation: D[i-1][j] + 1 & \text{(deletion)} \\ D[i][j-1] + 1 & \text{(insertion)} \\ D[i-1][j-1] + \text{cost} & \text{(substitution or match)} \end{cases}$$ Where cost = 0 if characters match 1 otherwise
- Example: Edit distance between "cat" and "cut" is 1 (substitute 'a' with 'u')
Algorithm Implementation and Extensions
- Time complexity O(mn) where m and n are lengths of input strings
- Space complexity O(mn) optimizable to O(min(m,n)) using only two rows of table
- Backtracking through filled table reconstructs actual sequence of edit operations
- Algorithm extensions:
- Handle more complex edit operations (transpositions swapping adjacent characters)
- Custom costs for different types of edits (making vowel substitutions cheaper than consonant substitutions)
- Damerau-Levenshtein distance includes transposition as a single edit operation
- Practical implementations often use optimizations:
- Early termination when edit distance exceeds a threshold
- Bit-parallel algorithms for faster computation on modern processors
Key Terms to Review (21)
Big O Notation: Big O notation is a mathematical concept used to describe the upper limit of an algorithm's running time or space requirement in relation to the size of the input. It provides a high-level understanding of the performance and efficiency of algorithms by characterizing their growth rates, which is essential for comparing different algorithms and determining their scalability as the problem size increases.
Bounded edit distance: Bounded edit distance refers to the concept of measuring how closely two sequences (like strings) can be transformed into one another through a limited number of specific operations, such as insertions, deletions, or substitutions. This concept is particularly useful in the context of string similarity and comparison algorithms, allowing for efficient comparisons by limiting the number of allowable edits. Understanding bounded edit distance helps in applications like spell checking, DNA sequence alignment, and natural language processing.
Divide and Conquer: Divide and conquer is a powerful algorithmic technique that breaks a problem down into smaller, more manageable subproblems, solves each subproblem individually, and then combines their solutions to solve the original problem. This method is particularly useful for designing efficient algorithms by tackling complex problems in a structured manner, leading to improved performance and simpler implementations.
Dynamic Programming: Dynamic programming is a problem-solving technique used in computer science and mathematics to simplify complex problems by breaking them down into simpler subproblems and solving each subproblem just once, storing the results for future use. This method is particularly useful for optimization problems where decisions need to be made sequentially, allowing for more efficient computation compared to naive approaches.
Edit distance: Edit distance is a measure of how dissimilar two strings are, defined as the minimum number of edit operations required to transform one string into the other. The main operations considered are insertion, deletion, and substitution of characters. This concept is crucial in understanding string comparison, as it helps to quantify the similarity or difference between sequences and is widely used in applications like spell checking, DNA sequence analysis, and natural language processing.
Edit operations: Edit operations refer to the basic actions that can be performed to transform one string into another. These operations include insertion, deletion, and substitution of characters, and they are fundamental in determining the edit distance, which quantifies how dissimilar two strings are. Understanding these operations is crucial for algorithms that solve problems like finding the longest common subsequence and calculating the minimum edit distance between sequences.
Greedy Algorithms: Greedy algorithms are a class of algorithms that build up a solution piece by piece, always choosing the next piece that offers the most immediate benefit. This approach is often used in optimization problems, where the goal is to find the best solution among many possible options. Greedy algorithms do not always yield the optimal solution but can be efficient and effective for a range of problems.
Levenshtein Distance: Levenshtein distance is a metric for measuring the difference between two sequences by counting the minimum number of single-character edits required to transform one sequence into the other. This includes operations such as insertions, deletions, and substitutions. It is a vital concept in understanding edit distances, particularly in relation to algorithms that find similarities between strings or sequences.
Longest common subsequence: The longest common subsequence (LCS) is a classic problem in computer science that identifies the longest sequence that can appear in the same order within two or more sequences without rearranging them. This concept is crucial for applications in areas like bioinformatics, text comparison, and version control, where finding similarities between data is essential.
Longest Common Subsequence (LCS): The Longest Common Subsequence (LCS) is a classic problem in computer science that focuses on finding the longest sequence that can appear in the same relative order in two given sequences, but not necessarily consecutively. This concept is crucial in various applications, including bioinformatics for DNA sequencing, data comparison, and version control systems. Understanding LCS allows for the assessment of similarity between sequences, which is essential for problems involving alignment and editing distance.
Manning's Algorithm: Manning's Algorithm is a dynamic programming technique used to solve problems related to finding the longest common subsequence between two sequences and calculating edit distances. It provides a systematic way to build solutions by breaking them down into simpler subproblems, allowing for efficient computation of optimal solutions. The algorithm is particularly useful in applications such as bioinformatics, text comparison, and version control systems.
Np-complete: NP-complete refers to a class of problems in computational complexity theory that are both in NP (nondeterministic polynomial time) and as hard as any problem in NP. This means that if one can find a polynomial-time solution for any NP-complete problem, then every problem in NP can also be solved in polynomial time, essentially proving that P = NP. Understanding NP-complete problems is crucial for recognizing the limits of efficient computation and the challenges involved in solving certain types of problems.
Optimal Substructure: Optimal substructure is a property of a problem that states an optimal solution to the problem contains optimal solutions to its subproblems. This concept is crucial in designing algorithms as it allows complex problems to be broken down into simpler, manageable parts, facilitating efficient solution strategies such as dynamic programming and greedy algorithms.
Overlapping subproblems: Overlapping subproblems refer to a situation where a problem can be broken down into smaller, simpler subproblems that are reused multiple times throughout the solution process. This concept highlights the inefficiency of solving the same subproblem repeatedly, which can lead to an exponential increase in computational time. Recognizing overlapping subproblems is crucial for designing more efficient algorithms, particularly those that employ dynamic programming to optimize performance.
P: In computational theory, 'p' represents the class of decision problems that can be solved by a deterministic Turing machine in polynomial time. This concept is central to understanding the efficiency of algorithms and the complexity of problems, particularly in relation to how quickly they can be solved as the size of the input grows.
Recursive definition: A recursive definition is a way of defining an object or concept in terms of itself, allowing for the specification of elements based on previously defined elements. This method is commonly used in mathematics and computer science to define sequences or structures, where the definition builds upon smaller instances of the same type. In the context of algorithms, recursive definitions are essential for solving problems like finding the longest common subsequence and calculating edit distance, as they allow for breaking down complex problems into simpler subproblems.
Sequence alignment: Sequence alignment is a method used to arrange the sequences of DNA, RNA, or protein to identify regions of similarity that may indicate functional, structural, or evolutionary relationships between the sequences. This concept is crucial in bioinformatics and computational biology, where it serves as a foundational technique to compare biological sequences and analyze their similarities and differences. By utilizing dynamic programming principles, it allows for the efficient computation of the best possible alignment, which can also be extended to finding the longest common subsequence or calculating edit distance.
String transformation: String transformation refers to the process of converting one string into another through a series of operations such as insertion, deletion, or substitution. This concept is crucial for understanding how algorithms determine similarities and differences between strings, which is particularly useful in applications like spell checking, DNA sequence alignment, and data comparison.
Subsequence: A subsequence is a sequence derived from another sequence where certain elements may be removed without changing the order of the remaining elements. This concept is vital in understanding patterns and relationships between sequences, particularly when comparing strings or sequences to find similarities and differences. Identifying subsequences helps in solving problems like finding the longest common subsequence, which has applications in fields such as bioinformatics and data comparison.
Wagner-Fischer Algorithm: The Wagner-Fischer algorithm is a dynamic programming method used to compute the edit distance between two strings. This algorithm determines the minimum number of operations required to transform one string into another, which is essential for various applications such as spell checking, DNA sequencing, and natural language processing. It uses a matrix to store intermediate values, allowing for efficient calculations that consider insertions, deletions, and substitutions.
Weighted longest common subsequence: The weighted longest common subsequence (WLCS) is a variation of the longest common subsequence problem where each character in the sequences has an associated weight, and the goal is to find a common subsequence that maximizes the total weight instead of just the length. This concept is particularly useful in scenarios where certain characters or operations are more significant than others, enabling more nuanced comparisons of sequences. By incorporating weights, WLCS not only finds matches but also emphasizes the importance of those matches based on their assigned values.