Przemysław Uznański

Bounds on the cover time of parallel rotor walks

We consider the cover time of multi-agent rotor-router.Rotor-router is a deterministic process in which the environment determines moves of the agents.Rotor-router is a form of derandomization of the random walk.We show that k agents explore any graph between ? ( log ? k ) and ? ( k ) faster than single agent.It is conjectured that the speedup for multiple random walks is also between ? ( log ? k ) and ? ( k ) . The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node successively propagates walkers visiting it along its outgoing arcs in round-robin fashion, according to a fixed ordering. We consider the cover time of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the initialization of the walks. We show that for any graph with m edges and diameter D, this cover time is at most ? ( m D / log ? k ) and at least ? ( m D / k ) , which corresponds to a speedup of between ? ( log ? k ) and ? ( k ) with respect to the cover time of a single walk.

Order-Preserving Pattern Matching with k Mismatches

We study a generalization of the order-preserving pattern matching recently introduced by Kubica et al. (Inf. Process. Let., 2013) and Kim et al. (submitted to Theor. Comp. Sci.), where instead of looking for an exact copy of the pattern, we only require that the relative order between the elements is the same. In our variant, we additionally allow up to k mismatches between the pattern of length m and the text of length n, and the goal is to construct an efficient algorithm for small values of k. Our solution detects an order-preserving occurrence with up to k mismatches in \(\mathcal{O}(n(\log\log m+k\log\log k))\) time.

Fast collaborative graph exploration

We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k=Dn1+e 0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D. We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large k. For any constant c>1, we show that in the global communication model, a team of k=Dnc agents can always complete exploration in

Rendezvous of Distance-Aware Mobile Agents in Unknown Graphs

D(1+ \frac{1}{c-1} +o(1))

Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams.

time steps, whereas at least

Bounds on the Cover Time of Parallel Rotor Walks

D(1+ \frac{1}{c} -o(1))

Sublinear-Space Distance Labeling Using Hubs

steps are sometimes required. In the local communication model,

Approximation Strategies for Generalized Binary Search in Weighted Trees

D(1+ \frac{2}{c-1} +o(1))

Randomized algorithms for finding a majority element

steps always suffice to complete exploration, and at least

Prime Factorization of the Kirchhoff Polynomial: Compact Enumeration of Arborescences

D(1+ \frac{2}{c} -o(1))