On Strings Having the Same Length-k Substrings

Giulia Bernardini*, Alessio Conte, Esteban Gabory, Roberto Grossi, Grigorios Loukides, Solon P. Pissis, Giulia Punzi, Michelle Sweering

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review


Let Substrk(X) denote the set of length-k substrings of a given string X for a given integer k > 0. We study the following basic string problem, called z-Shortest Sk-Equivalent Strings: Given a set Sk of n length-k strings and an integer z > 0, list z shortest distinct strings T1,..., Tz such that Substrk(Ti) = Sk, for all i ∈ [1, z]. The z-Shortest Sk-Equivalent Strings problem arises naturally as an encoding problem in many real-world applications; e.g., in data privacy, in data compression, and in bioinformatics. The 1-Shortest Sk-Equivalent Strings, referred to as Shortest Sk-Equivalent String, asks for a shortest string X such that Substrk(X) = Sk. Our main contributions are summarized below: Given a directed graph G(V, E), the Directed Chinese Postman (DCP) problem asks for a shortest closed walk that visits every edge of G at least once. DCP can be solved in Õ(|E||V |) time using an algorithm for min-cost flow. We show, via a non-trivial reduction, that if Shortest Sk-Equivalent String over a binary alphabet has a near-linear-time solution then so does DCP. We show that the length of a shortest string output by Shortest Sk-Equivalent String is in O(k + n2). We generalize this bound by showing that the total length of z shortest strings is in O(zk + zn2 + z2n). We derive these upper bounds by showing (asymptotically tight) bounds on the total length of z shortest Eulerian walks in general directed graphs. We present an algorithm for solving z-Shortest Sk-Equivalent Strings in O(nk + n2 log2 n + zn2 log n + |output|) time. If z = 1, the time becomes O(nk + n2 log2 n) by the fact that the size of the input is Θ(nk) and the size of the output is O(k + n2).

Original languageEnglish
Title of host publication33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022
EditorsHideo Bannai, Jan Holub
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772341
Publication statusPublished - 1 Jun 2022
Event33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022 - Prague, Czech Republic
Duration: 27 Jun 202229 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022
Country/TerritoryCzech Republic


  • Chinese Postman
  • combinatorics on words
  • de Bruijn graph
  • string algorithms


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