Frantisek Franek

A simple fast hybrid pattern-matching algorithm

The Knuth-Morris-Pratt (KMP) pattern-matching algorithm guarantees both independence from alphabet size and worst-case execution time linear in the pattern length; on the other hand, the Boyer-Moore (BM) algorithm provides near-optimal average-case and best-case behaviour, as well as executing very fast in practice. We describe a simple algorithm that employs the main ideas of KMP and BM (with a little help from Sunday) in an effort to combine these desirable features. Experiments indicate that in practice the new algorithm is among the fastest exact pattern-matching algorithms discovered to date, apparently dominant for alphabet size above 15-20.

AN ASYMPTOTIC LOWER BOUND FOR THE MAXIMAL NUMBER OF RUNS IN A STRING

An asymptotic lower bound for the maxrun function ρ(n) = max {number of runs in string x | all strings x of length n} is presented. More precisely, it is shown that for any e > 0, (α−e)n is an asymptotic lower bound, where . A recent construction of an increasing sequence of binary strings “rich in runs” is modified and extended to prove the result.

2-colorings of complete graphs with a small number of monochromatic K 4 subgraphs

Denote by kt(G) the number of cliques of order t in the graph G. Let kt(n)=min {kt(G) + kt(G): |G| = n}, where Ḡ denotes the complement of G, and |G| denotes the order of G. Let ct(n) = k t(n)/(nt, and let ct = limn→∞ct(n). An old conjecture of Erdos (1962), related to Ramseys theorem, states that ct = 21−(t2) . It was shown false by Thomason (1989) for all t⩾4. We present a class of simply describable Cayley graphs which also show the falsity of Erdos conjecture for t = 4. These graphs were found by a computer search and, although of large orders (210-214), they are rather simple and highly regular. The smallest upper bound for c4 obtained by us is 0.976501 x 132, and is given by the graph on the power set of a 10-element set (and, hence, of order 210) determined by the configuration {1,3,4,7,8,10}. and by the graph on the power set of 11 elements (and, hence, of order 211) determined by the configuration {1,3,4,7,8,10,11}. It is also shown that the ratio of edges to nonedges in a sequence contradicting the conjecture for t = 4 may approach 1, unlike in the sequences of graphs Thomason used in 1989.

How many double squares can a string contain

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson?showed in 1998 that a string of length n contains at most 2 n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2 n - ? ( log n ) . We show that a string of length n contains at most ? 11 n / 6 ? distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most ? 5 n / 6 ? particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpsons result.

Two-factorizations of small complete graphs

Abstract We enumerate 2-factorizations of K9 of all types, as well as those of several types for K11.

COMPARISON OF VARIOUS ROUTINES FOR UNKNOWN ATTRIBUTE VALUE PROCESSING: THE COVERING PARADIGM

Simple inductive learning algorithms assume that all attribute values are available. The well-known Quinlans paper1 discusses quite a few routines for the processing of unknown attribute values in the TDIDT family and analyzes seven of them. This paper introduces five routines for the processing of unknown attribute values that have been designed for the CN4 learning algorithm, a large extension of the well-known CN2. Both algorithms CN2 and CN4 induce lists of decision rules from examples applying the covering paradigm. CN2 offers two ways for the processing of unknown attribute values. The CN4s five routines differ in style of matching complexes with examples (objects) that involve unknown attribute values. The definition of matching is discussed in detail in the paper. The strategy of unknown value processing is described both for learning and classification phases in individual routines. The results of experiments with various percentages of unknown attribute values on real-world (mostly medical) data are presented and performances of all five routines are compared.

Completing the spectrum of 2-chromatic S(2,4,ν)

We construct 2-chromatic S(2,4,v) for v=37,40, and 73. This completes the proof of the existence of 2-chromatic Steiner systems S(2,4,v) [equivalently, of Steiner systems S(2,4,v) with a blocking set] for all v ≡ 1 or 4 (mod 12).

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup S there exists a uniquely de- termined complete Boolean algebra B(S) - the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical system for S and it is an extremally disconnected compact Hausdor space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that B(S) is either atomic or atomless; that B(S) is weakly homogenous provided S has a minimal left ideal; and that for countable semigroups B(S) is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reection G(S) of a group-like semigroup S can be constructed via universal minimal dynamical system for S and, moreover, B(S) and B(G(S)) are the same.

More results on overlapping squares

Three recent papers (Fan et al., 2006; Simpson, 2007; Kopylova and Smyth, 2012) [5,11,8] have considered in complementary ways the combinatorial consequences of assuming that three squares overlap in a string. In this paper we provide a unifying framework for these results: we show that in 12 of 14 subcases that arise the postulated occurrence of three neighboring squares forces a breakdown into highly periodic behavior, thus essentially trivial and easily recognizable. In particular, we provide a proof of Subcase 4 for the first time, and we simplify and refine the previously established results for Subcases 11-14.

Triangles in 2-factorizations

The triangle-spectrum for 2-factorizations of the complete graph Kv is the set of all numbers δ such that there exists a 2-factorization of Kv in which the total number of triangles equals δ. By applying mainly design-theoretic methods, we determine the triangle spectrum for all v ≡ 1 or 3 (mod 6), v ≥ 43, as well as for v = 7, 9, 13, 15, 21, and 27. For orders v = 19, 25, 31, 33, 37, 39, we leave only a total of 11 values undecided. To determine the triangle-spectrum for v ≡ 5 (mod 6) remains an open problem.