Łukasz Kuszner

Distributed Largest-First Algorithm for Graph Coloring

In the paper we present a distributed probabilistic algorithm for coloring the vertices of a graph. Since this algorithm resembles a largest-first strategy, we call it the distributed LF (DLF) algorithm. The coloring obtained by DLF is optimal or near optimal for numerous classes of graphs e.g. complete k-partite, caterpillars, crowns, bipartite wheels. We also show that DLF runs in O(Δ2 log n) rounds for an arbitrary graph, where n is the number of vertices and Δ denotes the largest vertex degree.

Self-stabilizing algorithms for graph coloring with improved performance guarantees

In the self-stabilizing model we consider a connected system of autonomous asynchronous nodes, each of which has only local information about the system. Regardless of the initial state, the system must achieve a desirable global state by executing a set of rules assigned to each node. The paper deals with the construction of a solution to graph coloring in this model, a problem motivated by code assignment in wireless networks. A new method based on spanning trees is applied to give the first (to our knowledge) self-stabilizing algorithms working in a polynomial number of moves, which color bipartite graphs with exactly two colors. The complexity and performance characteristics of the presented algorithms are discussed for different graph classes.

On greedy graph coloring in the distributed model

In the paper we consider distributed algorithms for greedy graph coloring. For the largest-first (LF) approach, we propose a new distributed algorithm which is shown to color a graph in an expected time of O(ΔlognlogΔ) rounds, and we prove that any distributed LF-coloring algorithm requires at least Ω(Δ) rounds. We discuss the quality of obtained colorings in the general case and for particular graph classes. Finally, we show that other greedy graph coloring approaches, such as smallest-last (SL) or dynamic-saturation (SLF), are not suitable for application in distributed computing, requiring Ω(n) rounds.

A self-stabilizing algorithm for finding a spanning tree in a polynomial number of moves

In the self-stabilizing model each node has only local information about the system. Regardless of the initial state, the system must achieve a desirable global state. We discuss the construction of a solution to the spanning tree problem in this model. To our knowledge we give the first self-stabilizing algorithm working in a polynomial number of moves, without any fairness assumptions. Additionally we show that this approach can be applied under a distributed daemon. We briefly discuss implementation aspects of the proposed algorithm and its application in broadcast routing and in distributed computing.

Deterministic Rendezvous in Restricted Graphs

In this paper we consider the problem of synchronous rendezvous in which two anonymous mobile entities (robots) A and B are expected to meet at the same time and point in a graph G = (V,E). Most of the work devoted to rendezvous in graphs assumes that robots have access to the same sets of nodes and edges, where the topology of connections may be initially known or unknown. In our work we assume the movement of robots is restricted by the topological properties of the graph space coupled with the intrinsic characteristics of robots preventing them from visiting certain edges in E.

Rendezvous of heterogeneous mobile agents in edge-weighted networks☆

We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time \(T_{\textup{OPT}}\) in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time \(O(n T_{\textup{OPT}})\) in a n-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to \(\Theta (T_{\textup{OPT}})\) when the agents are allowed to exchange Θ(n) bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time \(\Theta (T_{\textup{OPT}})\).

Distributed Evacuation in Graphs with Multiple Exits

We consider the problem of efficient evacuation using multiple exits. We formulate this problem as a discrete problem on graphs where mobile agents located in distinct nodes of a given graph must quickly reach one of multiple possible exit nodes, while avoiding congestion and bottlenecks. Each node of the graph has the capacity of holding at most one agent at each time step. Thus, the agents must choose their movements strategy based on locations of other agents in the graph, in order to minimize the total time needed for evacuation. We consider two scenarios: (i) the centralized (or offline) setting where the agents have full knowledge of initial positions of other agents, and (ii) the distributed (or online) setting where the agents do not have prior knowledge of the location of other agents but they can communicate locally with nearby agents and they must modify their strategy in an online fashion while they move and obtain more information. In the former case we present an offline polynomial time solution to compute the optimal strategy for evacuation of all agents. In the online case, we present a constant competitive algorithm when agents can communicate at distance two in the graph. We also show that when the agents are heterogeneous and each agent has access to only a subgraph of the original graph then computing the optimal strategy is NP-hard even with full global knowledge. This result holds even if there are only two types of agents.