125 research outputs found
Longest property-preserved common factor
In this paper we introduce a new family of string processing problems. We are given two or more strings and we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. Here we consider two fundamental string properties: square-free factors and periodic factors under two different settings, one per property. In the first setting, we are given a string x and we are asked to construct a data structure over x answering the following type of on-line queries: given string y, find a longest square-free factor common to x and y. In the second setting, we are given k strings and an integer 1 < k’ ≤ k and we are asked to find a longest periodic factor common to at least k’ strings. We present linear-time solutions for both settings. We anticipate that our paradigm can be extended to other string properties
Computing longest common square subsequences
A square is a non-empty string of form YY. The longest common square subsequence (LCSqS) problem is to compute a longest square occurring as a subsequence in two given strings A and B. We show that the problem can easily be solved in O(n^6) time or O(|M|n^4) time with O(n^4) space, where n is the length of the strings and M is the set of matching points between A and B. Then, we show that the problem can also be solved in O(sigma |M|^3 + n) time and O(|M|^2 + n) space, or in O(|M|^3 log^2 n log log n + n) time with O(|M|^3 + n) space, where sigma is the number of distinct characters occurring in A and B. We also study lower bounds for the LCSqS problem for two or more strings
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On the Classification of Universal Rotor-Routers
The combinatorial theory of rotor-routers has connections with problems of
statistical mechanics, graph theory, chaos theory, and computer science. A
rotor-router network defines a deterministic walk on a digraph G in which a
particle walks from a source vertex until it reaches one of several target
vertices. Motivated by recent results due to Giacaglia et al., we study
rotor-router networks in which all non-target vertices have the same type. A
rotor type r is universal if every hitting sequence can be achieved by a
homogeneous rotor-router network consisting entirely of rotors of type r. We
give a conjecture that completely classifies universal rotor types. Then, this
problem is simplified by a theorem we call the Reduction Theorem that allows us
to consider only two-state rotors. A rotor-router network called the
compressor, because it tends to shorten rotor periods, is introduced along with
an associated algorithm that determines the universality of almost all rotors.
New rotor classes, including boppy rotors, balanced rotors, and BURD rotors,
are defined to study this algorithm rigorously. Using the compressor the
universality of new rotor classes is proved, and empirical computer results are
presented to support our conclusions. Prior to these results, less than 100 of
the roughly 260,000 possible two-state rotor types of length up to 17 were
known to be universal, while the compressor algorithm proves the universality
of all but 272 of these rotor types
Longest Property-Preserved Common Factor
International audienceIn this paper we introduce a new family of string processing problems. We are given two or more strings and we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. Here we consider three fundamental string properties: square-free factors, periodic factors, and palindromic factors under three different settings, one per property. In the first setting, we are given a string x and we are asked to construct a data structure over x answering the following type of on-line queries: given string y, find a longest square-free factor common to x and y. In the second setting, we are given k strings and an integer 1 < k ≤ k and we are asked to find a longest periodic factor common to at least k strings. In the third setting, we are given two strings and we are asked to find a longest palindromic factor common to the two strings. We present linear-time solutions for all settings. We anticipate that our paradigm can be extended to other string properties or settings
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