Gianluca De Marco

Asynchronous deterministic rendezvous in graphs

Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|Lmin|2) when D is known and O((D + |Lmax|)3) if D is unknown, where |Lmin| and |Lmax| are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|Lmin|), if the size n of the ring is known, and of cost O(n|Lmax|), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|Lmin|) if the topology of the graph and the initial positions are known to agents.

Broadcasting in hypercubes and star graphs with dynamic faults

Abstract We consider the problem of broadcasting in the n -dimensional hypercube under the hypothesis that each node can inform in one unit of time all of its n neighbors and that any n − 1 message transmissions can fail during each time unit. Under these assumptions we prove that broadcasting can be accomplished in only 7 time units more than that it would be necessary to broadcast in absence of transmission failures. This improves on previously published results. We also prove an analogous result for the n -dimensional star interconnection network.

The plurality problem with three colors and more

The plurality problem is a game between two participants: Paul and Carole. We are given n balls, each of them is colored with one out of c colors. At any step of the game, Paul chooses two balls and asks whether they are of the same color, whereupon Carole answers yes or no. The game ends when Paul either produces a ball a of the plurality color (meaning that the number of balls colored like a exceeds those of the other colors), or when Paul states that there is no plurality. How many questions Lc(n) does Paul have to ask in the worst case?For c = 2, the problem is equivalent to the well-known majority problem which has already been solved (Combinatorica 11 (1991) 383-387). In this paper we show that 3 ⌊n/2⌋-2 ≤ L3(n) ≤ ⌊5n/3⌋ - 2. Moreover, for any c ≤ n, we show that surprisingly the naive algorithm for the plurality problem is asymptotically optimal.

Randomized Algorithms for Determining the Majority on Graphs

Every node of an undirected connected graph is coloured white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same colour) or 1 (different colours). The aim is to discover a node of the majority colour, or to conclude that there is the same number of black and white nodes. We consider randomized algorithms for this task and establish upper and lower bounds on their expected running time. Our main contribution are lower bounds showing that some simple and natural algorithms for this problem cannot be improved in general.

Approximation algorithms for a hierarchically structured bin packing problem

In this paper we study a variant of the bin packing problem in which the items to be packed are structured as the leaves of a tree. The problem is motivated by document organization and retrieval. We show that the problem is NP-hard and we give approximation algorithms for the general case and for the particular case in which all the items have the same size.

Faster deterministic wakeup in multiple access channels

We consider the fundamental problem of waking up n processors sharing a multiple access channel. We assume the weakest model of synchronization, the locally synchronous model, in which no global clock is available: processors have local clocks ticking at the same rate, but each clock starts counting the rounds in the round in which the correspondent processor wakes up. Moreover, the number n of processors is not known to the processors. We propose a new deterministic algorithm for this problem in time O(n^3log^3n), which improves on the currently best upper bound of O(n^4log^5n).

The Plurality Problem with Three Colors

The plurality problem with three colors is a game between two participants: Paul and Carol. Suppose we are given n balls colored with three colors. At any step of the game, Paul chooses two balls and asks whether they are of the same color, whereupon Carol answers yes or no. The game ends when Paul either produces a ball a of the plurality color (meaning that the number of balls colored like a exceeds those of the other colors), or when Paul states that there is no plurality. How many questions L(n) does Paul have to ask in the worst case? We show that \(3\lfloor n/2 \rfloor - 2 \leq L(n) \leq \lfloor 5n/3 \rfloor - 2\)

Contention Resolution in a Non-synchronized Multiple Access Channel

Multiple access channel is a well-known communication model that deploys properties of many network systems, such as Aloha multi-access systems, local area Ethernet networks, satellite communication systems, packet radio networks. The fundamental aspect of this model is to provide efficient communication and computation in the presence of restricted access to the communication resource: at most one station can successfully transmit at a time, and a wasted round occurs when more than one station attempts to transmit at the same time. In this work we consider the problem of contention resolution in a multiple access channel in a realistic scenario when up to k stations out of n join the channel at different times. The goal is to let at least one station to transmit alone, which results in successful delivery of the message through the channel. We present three deterministic algorithms: two of them working under some constrained scenarios, and achieving asymptotically optimal time complexity Θ(k log(n/k)), while the third general algorithm accomplishes the goal in time O(k logn log log n).

FAST NONADAPTIVE DETERMINISTIC ALGORITHM FOR CONFLICT RESOLUTION IN A DYNAMIC MULTIPLE-ACCESS CHANNEL ∗

A classical problem in addressing a decentralized multiple-access channel is resolving conflicts when a set of stations attempt to transmit at the same time on a shared communication channel. In a static scenario, i.e., when all stations are activated simultaneously, Komlos and Greenberg [IEEE Trans. Inform. Theory, 31 (1985), pp. 302--306] in their seminal work showed that it is possible to resolve the conflict among

Searching for majority with k-tuple queries

k