Search results for "Theoretical Computer Science"
showing 10 items of 1151 documents
On computing the degree of convexity of polyominoes
2015
In this paper we present an algorithm which has as input a convex polyomino $P$ and computes its degree of convexity, defined as the smallest integer $k$ such that any two cells of $P$ can be joined by a monotone path inside $P$ with at most $k$ changes of direction. The algorithm uses space $O(m + n)$ to represent a polyomino $P$ with $n$ rows and $m$ columns, and has a running time $O(min(m; r k))$, where $r$ is the number of corners of $P$. Moreover, the algorithm leads naturally to a decomposition of $P$ into simpler polyominoes.
Loop-free Gray code algorithm for the e-restricted growth functions
2011
The subject of Gray codes algorithms for the set partitions of {1,2,...,n} had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of [email protected]?s algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of {1,2,...,n}, Ruskey and Savage (1994) [9] generalized [email protected]?s results and give two Gray codes for the set of partitions of {1,2,...,n}, and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques. In this paper, we introduce the set of e-restricte…
DEFECT THEOREMS FOR TREES
2000
We generalize different notions of a rank of a set of words to sets of trees. We prove that almost all of those ranks can be used to formulate a defect theorem. However, as we show, the prefix rank forms an exception.
A note on Sturmian words
2012
International audience; We describe an algorithm which, given a factor of a Sturmian word, computes the next factor of the same length in the lexicographic order in linear time. It is based on a combinatorial property of Sturmian words which is related with the Burrows-Wheeler transformation.
Highly irregular graphs with extreme numbers of edges
1997
Abstract A simple connected graph is highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper we find: (1) the greatest number of edges of a highly irregular graph with n vertices, where n is an odd integer (for n even this number is given in [1]), (2) the smallest number of edges of a highly irregular graph of given order.
On approximate-type systems generated by L-relations
2014
The aim of this work is to study approximate-type systems induced by L-relations in the framework of the general theory of M-approximate systems introduced in [42] and its generalizations. Special attention is payed to the structural properties of lattices of such systems and to the study of connections between categories of such systems and the corresponding categories of sets endowed with L-relations.
Improved constructions of quantum automata
2008
We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use \frac{4}{\epsilon} \log 2p + O(1) states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of \log p than the previously known construction of Ambainis and Freivalds (quant-ph/9802062). Similarly to Ambainis and Freivalds, our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction.
Spatial Search on Grids with Minimum Memory
2015
We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log N)$, where $N$ is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum walks.
Exceptional Quantum Walk Search on the Cycle
2016
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random walk's hitting time. In this context, however, the mixing time to prepare the initial uniform state…
Improved constructions of mixed state quantum automata
2009
Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language. There was an unpublished ''folk theorem'' proving that quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable. We prove that there is an infinite sequence of distinct int…