Grunnleggende konsepter
The authors present optimal static and dynamic algorithms for computing the bounded edit distance between two strings, where the edit weights are small integers. They achieve e
O(n + k2) time for the static problem and e
O(k) time per update for the dynamic problem, matching the conditional lower bounds for the unweighted case.
Sammendrag
The content discusses the edit distance problem, which is the minimum number of character insertions, deletions, and substitutions needed to transform one string into another. The authors focus on the bounded edit distance problem, where the complexity is parameterized by the value k of the edit distance.
The key insights are:
For the static problem with small integer weights, the authors develop an e
O(n + k2 min{W,
√
k log n} log5 n)-time algorithm, which matches the conditional lower bound for the unweighted case when W = O(1).
For the dynamic problem with small integer weights, the authors present a deterministic algorithm that maintains the edit distance and an optimal sequence of edits, supporting updates in e
O(W 2k log6 n) time after e
O(n logo(1) n + Wk2 log6 n)-time initialization.
The authors leverage tools from the context of static weighted edit distance algorithms, including a combinatorial lemma that allows stitching optimal alignments between two pairs of strings into an optimal alignment between their concatenations.
The dynamic algorithm uses balanced straight-line programs to represent the hierarchical decomposition of the input strings, enabling efficient updates that exploit the compressibility of the strings.
The authors also provide a divide-and-conquer approach that reduces the general problem to instances with small self-edit distance, which can be handled more efficiently.
Statistikk
The content does not provide any specific numerical data or metrics to support the key logics. It focuses on presenting the algorithmic techniques and their theoretical time complexities.
Sitater
The content does not contain any striking quotes that support the key logics.