Igor Potapov

Gossiping with Unit Messages in Known Radio Networks

A gossiping is a communication primitive in which each node of the network possesses a unique message that is to be communicated to all other nodes in the network. We study the gossiping problem in known ad hoc radio networks, where during each transmission only unit messages originated at any node of the network can be transmitted successfully. We survey a number of radio network topologies. Assuming that the size (a number of nodes) of the network is n we show that the exact complexity of radio gossiping in stars is 2n–1, in rings is 2n±O(1), and on a line of processors is 3n ± O(1). We later prove that radio gossiping in free trees is harder and it requires at least 31/6n – 16 time steps to be completed. For free trees we also show a gossiping algorithm with time complexity 5n + 8. In conclusion we prove that in general graphs radio gossiping requires Ω(n log n) time, and we propose radio gossiping algorithm that works in time O(n log2 n).

Temporal logic with predicate /spl lambda/-abstraction

A predicate linear temporal logic LTL/sub /spl lambda/=/ without quantifiers but with predicate /spl lambda/-abstraction mechanism and equality is considered. The models of LTL/sub /spl lambda/=/ can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LTL/sub /spl lambda/=/ for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LTL/sub /spl lambda/=/ is not recursively axiomatizable and, therefore, fully automated verification of LTL/sub /spl lambda/=/ specifications via validity checking is not, in general, possible. The result is based on computational universality of the above abstract computational model of pebble systems, which is of independent interest due to the range of possible interpretations of such systems.

Real-time traversal in grammar-based compressed files

Summary form only given. In text compression applications, it is important to be able to process compressed data without requiring (complete) decompression. In this context it is crucial to study compression methods that allow time/space efficient access to any fragment of a compressed file without being forced to perform complete decompression. We study here the real-time recovery of consecutive symbols from compressed files, in the context of grammar-based compression. In this setting, a compressed text is represented as a small (a few Kb) dictionary D (containing a set of code words), and a very long (a few Mb) string based on symbols drawn from the dictionary D. The space efficiency of this kind of compression is comparable with standard compression methods based on the Lempel-Ziv approach. We show, that one can visit consecutive symbols of the original text, moving from one symbol to another in constant time and extra O(|D|) space. This algorithm is an improvement of the on-line linear (amortised) time algorithm presented in (L. Gasieniec et al, Proc. 13th Int. Symp. on Fund. of Comp. Theo., LNCS, vol.2138, p.138-152, 2001).

Time Efficient Gossiping in Known Radio Networks

We study here the gossiping problem (all-to-all communication) in known radio networks, i.e., when all nodes are aware of the network topology. We start our presentation with a deterministic algorithm for the gossiping problem that works in at most n units of time in any radio network of size n. This is an optimal algorithm in the sense that there exist radio network topologies, such as: a line, a star and a complete graph in which the radio gossiping cannot be completed in less then n units of time. Furthermore, we show that there isn’t any radio network topology in which the gossiping task can be solved in time \(<\lfloor\log(n-1)\rfloor+2.\) We show also that this lower bound can be matched from above for a fraction of all possible integer values of n; and for all other values of n we propose a solution admitting gossiping in time ⌈log(n − 1)⌉ + 2. Finally we study asymptotically optimal O(D)-time gossiping (where D is a diameter of the network) in graphs with max-degree \(\Delta=O(\frac{D^{1-1/(i+1)}}{\log^{i} n}),\) for any integer constant i≥ 0 and D large enough.

Reachability problems in quaternion matrix and rotation semigroups

We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2- and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.

On the Computational Complexity of Matrix Semigroup Problems

Most computational problems for matrix semigroups and groups are inherently difficult to solve and even undecidable starting from dimension three. The questions about the decidability and complexity of problems for two-dimensional matrix semigroups remain open and are directly linked with other challenging problems in the field. In this paper we study the computational complexity of the problem of determining whether the identity matrix belongs to a matrix semigroup (the Identity Problem) generated by a finite set of 2 × 2 integral unimodular matrices. The Identity Problem for matrix semigroups is a well-known challenging problem, which has remained open in any dimension until recently. It is currently known that the problem is decidable in dimension two and undecidable starting from dimension four. In particular, we show that the Identity Problem for 2 × 2 integral unimodular matrices is NP-hard by a reduction of the Subset Sum Problem and several new encoding techniques. An upper bound for the nontrivial decidability result by C. Choffrut and J. Karhumaki is unknown. However, we derive a lower bound on the minimum length solution to the Identity Problem for a constructible set of instances, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. This shows that the most obvious candidate for an NP algorithm, which is to guess the shortest sequence of matrices which multiply to give the identity matrix, does not work correctly since the certificate would have a length which is exponential in the size of the instance. Both results lead to a number of corollaries confirming the same bounds for vector reachability, scalar reachability and zero in the right upper corner problems.

Membership and Reachability Problems for Row-Monomial Transformations

In this paper we study the membership and vector reachability problems for labelled transition systems with row-monomial transformations. We show the decidability of these problems for row-monomial martix semigroups over rationals and extend these results to the wider class of matrix semigroups. After that we apply our methods to reachability problems for a class of transition systems which turn out to be equivalent to specific counter machines.

REACHABILITY PROBLEMS IN LOW-DIMENSIONAL ITERATIVE MAPS

In this paper we analyze the dynamics of one-dimensional piecewise maps. We show that one-dimensional piecewise affine maps are equivalent to pseudo-billiard or so called “strange billiard” systems. We also show that use of more general classes of functions lead to undecidability of reachability problem for one-dimensional piecewise maps.

Computation in one-dimensional piecewise maps and planar pseudo-billiard systems

The computation in low-dimensional system is related to many long standing open problems. In this paper we show the universality of a one-dimensional iterative map defined by elementary functions. The computation in iterative maps have a number of connections with other unconventional models of computations. In particular, one-dimensional iterative maps can be simulated by a planar pseudo-billiard system. As a consequence of our main result we show that a planar pseudo-billiard system is not only can demonstrate a chaotic behaviour, but also has ability of universal computation.

Deterministic Communication in Radio Networks with Large Labels

We study deterministic gossiping in ad-hoc radio networks, where labels of the nodes are large, i.e., they are polynomially large in the size n of the network. A label-free model was introduced in the context of randomized broadcasting in ad-hoc radio networks, see [2]. Most of the work on deterministic communication in ad-hoc networks was done for the model with labels of size O(n), with few exceptions; Peleg [19] raised the problem of deterministic communication in ad-hoc radio networks with large labels and proposed the first deterministic O(n2 log n)- time broadcasting algorithm. In [11] Chrobak et al. proved that deterministic radio broadcasting can be performed in time O(n log2 n); their result holds for large labels.Here we propose two new deterministic gossiping algorithms for ad-hoc radio networks with large labels. In particular: - a communication procedure giving an O(n5/3 log3 n)-time deterministic gossiping algorithm for directed networks and an O(n4/3 log3 n)- time algorithm for undirected networks; - a gossiping procedure designed particularly for undirected networks resulting in an almost linear O(n log3 n)-time algorithm.