Search results for "Combinatorics"
showing 10 items of 1770 documents
Lower space bounds for randomized computation
1994
It is a fundamental problem in the randomized computation how to separate different randomized time or randomized space classes (c.f., e.g., [KV87, KV88]). We have separated randomized space classes below log n in [FK94]. Now we have succeeded to separate small randomized time classes for multi-tape 2-way Turing machines. Surprisingly, these “small” bounds are of type n+f(n) with f(n) not exceeding linear functions. This new approach to “sublinear” time complexity is a natural counterpart to sublinear space complexity. The latter was introduced by considering the input tape and the work tape as separate devices and distinguishing between the space used for processing information and the spa…
On the family ofr-regular graphs with Grundy numberr+1
2014
Abstract The Grundy number of a graph G , denoted by Γ ( G ) , is the largest k such that there exists a partition of V ( G ) , into k independent sets V 1 , … , V k and every vertex of V i is adjacent to at least one vertex in V j , for every j i . The objects which are studied in this article are families of r -regular graphs such that Γ ( G ) = r + 1 . Using the notion of independent module, a characterization of this family is given for r = 3 . Moreover, we determine classes of graphs in this family, in particular, the class of r -regular graphs without induced C 4 , for r ≤ 4 . Furthermore, our propositions imply results on the partial Grundy number.
An exact, complete and efficient implementation for computing planar maps of quadric intersection curves
2005
We present the first exact, complete and efficient implementation that computes for a given set P=p1,...,pn of quadric surfaces the planar map induced by all intersection curves p1∩ pi, 2 ≤ i ≤ n, running on the surface of p1. The vertices in this graph are the singular and x-extreme points of the curves as well as all intersection points of pairs of curves. Two vertices are connected by an edge if the underlying points are connected by a branch of one of the curves. Our work is based on and extends ideas developed in [20] and [9].Our implementation is complete in the sense that it can handle all kind of inputs including all degenerate ones where intersection curves have singularities or pa…
Enumerating the Walecki-Type Hamiltonian Cycle Systems
2017
Let Kv be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly HCS(v)) is a set of Hamiltonian cycles of Kv whose edges partition the edge set of Kv. By means of a slight modification of the famous HCS(4n+1) of Walecki, we obtain 2n pairwise distinct HCS(4n+1) and we enumerate them up to isomorphism proving that this is equivalent to count the number of binary bracelets of length n, i.e. the orbits of Dn, the dihedral group of order 2n, acting on binary n-tuples.
Padding and the expressive power of existential second-order logics
1998
Padding techniques are well-known from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential second-order logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ϕ in the presence of padding, if ϕ is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A.
Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams
2017
We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is \(2^{\varOmega (n/log n)}\) and present quantum OBD…
NP-completeness of the hamming salesman problem
1985
It is shown that the traveling salesman problem, where cities are bit strings with Hamming distances, is NP-complete.
Combinatorics of Finite Words and Suffix Automata
2009
The suffix automaton of a finite word is the minimal deterministic automaton accepting the language of its suffixes. The states of the suffix automaton are the classes of an equivalence relation defined on the set of factors. We explore the relationship between the combinatorial properties of a finite word and the structural properties of its suffix automaton. We give formulas for expressing the total number of states and the total number of edges of the suffix automaton in terms of special factors of the word.
Claws contained in all n-tournaments
1993
Abstract We prove that any claw of order n with degree d≤ 3 8 n is n-unavoidable, which means that any tournament of order n contains it as a subdigraph. A simple corollary is that any tournament has a directed Hamiltonian path.
Graph connectivity and monadic NP
2002
Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. As an application it is shown that graph connectivity cannot be expressed by existential second-order formulas, where the second-order quantification is restricted to unary relations (monadic NP), even, in the presence of a built-in linear order. As a second application it is stated, that, on the other hand, the presence of a linear order increases the power of monadi…