Optimally edge-colouring outerplanar graphs is in NC

We prove that every outerplanar graph can be optimally edge-coloured in polylogarithmic time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM)

Faster Longest Common Extension Queries in Strings over General Alphabets

Longest common extension queries (often called longest common prefix queries) constitute a fundamental building block in multiple string algorithms, for example computing runs and approximate pattern matching. We show that a sequence of $q$ LCE queries for a string of size $n$ over a general ordered alphabet can be realized in $O(q \log \log n+n\log^*n)$ time making only $O(q+n)$ symbol comparisons. Consequently, all runs in a string over a general ordered alphabet can be computed in $O(n \log \log n)$ time making $O(n)$ symbol comparisons. Our results improve upon a solution by Kosolobov (Information Processing Letters, 2016), who gave an algorithm with $O(n \log^{2/3} n)$ running time and conjectured that $O(n)$ time is possible. We make a significant progress towards resolving this conjecture. Our techniques extend to the case of general unordered alphabets, when the time increases to $O(q\log n + n\log^*n)$. The main tools are difference covers and the disjoint-sets data structure.Comment: Accepted to CPM 201

Internal Pattern Matching Queries in a Text and Applications

We consider several types of internal queries: questions about subwords of a text. As the main tool we develop an optimal data structure for the problem called here internal pattern matching. This data structure provides constant-time answers to queries about occurrences of one subword $x$ in another subword $y$ of a given text, assuming that $|y|=\mathcal{O}(|x|)$, which allows for a constant-space representation of all occurrences. This problem can be viewed as a natural extension of the well-studied pattern matching problem. The data structure has linear size and admits a linear-time construction algorithm. Using the solution to the internal pattern matching problem, we obtain very efficient data structures answering queries about: primitivity of subwords, periods of subwords, general substring compression, and cyclic equivalence of two subwords. All these results improve upon the best previously known counterparts. The linear construction time of our data structure also allows to improve the algorithm for finding $\delta$-subrepetitions in a text (a more general version of maximal repetitions, also called runs). For any fixed $\delta$ we obtain the first linear-time algorithm, which matches the linear time complexity of the algorithm computing runs. Our data structure has already been used as a part of the efficient solutions for subword suffix rank & selection, as well as substring compression using Burrows-Wheeler transform composed with run-length encoding.Comment: 31 pages, 9 figures; accepted to SODA 201

Optimally edge-colouring outerplanar graphs is in NC

AbstractWe prove that every outerplanar graph can be optimally edge-coloured in polylogarithmic time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM)

Parallel O(log(n)) time edge-colouring of trees and Halin graphs

We present parallel O(log(n))-time algorithms for optimal edge colouring of trees and Halin graphs with n processors on a a parallel random access machine without write conflicts (P-RAM). In the case of Halin graphs with a maximum degree of three, the colouring algorithm automatically finds every Hamiltonian cycle of the graph

Prime normal form and equivalence of simple grammars

AbstractA prefix-free language is prime if it cannot be decomposed into a concatenation of two prefix-free languages. We show that we can check in polynomial time if a language generated by a simple context-free grammar is prime. Our algorithm computes a canonical representation of a simple language, converting its arbitrary simple grammar into prime normal form (PNF); a simple grammar is in PNF if all its nonterminals define primes. We also improve the complexity of testing the equivalence of simple grammars. The best previously known algorithm for this problem worked in O(n13) time. We improve it to O(n7log2n) and O(n5polylogv) time, where n is the total size of the grammars involved, and v is the length of a shortest string derivable from a nonterminal, maximized over all nonterminals

On the Greedy Algorithm for the Shortest Common Superstring Problem with Reversals

We study a variation of the classical Shortest Common Superstring (SCS) problem in which a shortest superstring of a finite set of strings $S$ is sought containing as a factor every string of $S$ or its reversal. We call this problem Shortest Common Superstring with Reversals (SCS-R). This problem has been introduced by Jiang et al., who designed a greedy-like algorithm with length approximation ratio $4$. In this paper, we show that a natural adaptation of the classical greedy algorithm for SCS has (optimal) compression ratio $\frac12$, i.e., the sum of the overlaps in the output string is at least half the sum of the overlaps in an optimal solution. We also provide a linear-time implementation of our algorithm.Comment: Published in Information Processing Letter