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Approximating Edit Distance

Safarnejad Boroujeni, Mahdi | 2021

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 54309 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Ghodsi, Mohammd
  7. Abstract:
  8. Edit distance or Levenshtein distance is one of the most fundamental problems in computer science and engineering to measure the dissimilarity between strings.This problem is solvable in O(n^2) time using dynamic programming and in 2015 it was conditionally proven that no truly subquadratic time algorithm (\textit{i.e.}, O(n^{2-ε})) exists for edit distance.Since 2001, approximating edit distance within a constant factor was recognized as one of the biggest unsolved problems in the field of combinatorial pattern matching.This thesis designs seven approximation algorithms for edit distance and its related problems.Firstly, two quantum approximation algorithms are designed with constant approximation factors and truly subquadratic running times, one of them has an approximation factor of 3+ε and the approximation factor of the other is a large constant.The framework used for these two algorithms is used in 2018 by Chakraborty \textit{et al.} to approximate edit distance in classical computers.Another approach of this dissertation is approximating edit distance in parallel computers and in particular MapReduce model, for which two algorithms are designed.One of these two algorithms has an approximation factor of 1 + ε and truly subquadratic parallel running time, and the other one has an approximation factor of 3 + ε and in addition to its parallel running time, the total running time is also truly subquadratic.Another parallel algorithm is also proposed for edit distance between two permutations, namely Ulam distance, which is almost optimal, meaning its total running time is almost linear and its approximation factor is 1 + ε.Finally, the editing distance between tree structures is studied. The best-known exact algorithm for this problem runs in time O(n^3), and in this thesis, we present an approximation algorithm for this problem with a running time of \tildorder_ε(n^2) and an approximation of 1+ε. Moreover, an almost linear-time approximation algorithm is presented for tree edit distance within an approximation factor of O(\sqrt{n}).In addition to the direct results of these seven algorithms, this dissertation develops new techniques to design these algorithms, and these techniques are used by other researchers to approximate similar problems such as the largest common subsequence, the largest increasing subsequence, and the approximate solution of edit distance in other computational models such as the streaming model
  9. Keywords:
  10. Parallel Algorithm ; Quantum Algorithm ; Randomized Algorithm ; Approximate Algorithm ; Edit Distance ; Ulam Distance ; Tree Edit Distance

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