Computing Majority with Triple Queries

Consider a bin containing $n$ balls colored with two colors. In a $k$-query, $k$ balls are selected by a questioner and the oracle's reply is related (depending on the computation model being considered) to the distribution of colors of the balls in this $k$-tuple; however, the oracle never reveals the colors of the individual balls. Following a number of queries the questioner is said to determine the majority color if it can output a ball of the majority color if it exists, and can prove that there is no majority if it does not exist. We investigate two computation models (depending on the type of replies being allowed). We give algorithms to compute the minimum number of 3-queries which are needed so that the questioner can determine the majority color and provide tight and almost tight upper and lower bounds on the number of queries needed in each case.Comment: 22 pages, 1 figure, conference version to appear in proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011

Deterministic broadcasting time with partial knowledge of the network

We consider the time of deterministic broadcasting in networks whose nodes have limited knowledge of network topology. Each node u knows only the part of the network within knowledge radius r from it, i.e., it knows the graph induced by all nodes at distance at most r from u. Apart from that, each node knows the maximum degree Delta of the network. One node of the network, called the source, has a message which has to reach all other nodes. We adopt the widely studied communication model called the one-way model in which, in every round, each node can communicate with at most one neighbor, and in each pair of nodes communicating in a given round, one can only send a message while the other can only receive it. This is the weakest of all store-and-forward models for point-to-point networks, and hence our algorithms work for other models as well, in at most the same time.We show trade-offs between knowledge radius and time of deterministic broadcasting, when the knowledge radius is small, i.e., when nodes are only aware of their close vicinity. While for knowledge radius 0, minimum broadcasting time is theta(e), where e is the number of edges in the network, broadcasting can be usually completed faster for positive knowledge radius. Our main results concern knowledge radius 1. We develop fast broadcasting algorithms and analyze their execution time. We also prove lower bounds on broadcasting time, showing that our algorithms are close to optimal

Randomized Algorithms for Determining the Majority on Graphs

Every node of an undirected connected graph is colored white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same color) or 1 (different colors). The aim is to discover a node of the majority color, 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

Introduction to morphological and functional evaluation of the heart and coronary arteries

In the last years, the number of clinical indications for the evaluation of the heart – with both computed tomography (CT) and magnetic resonance (MR) – exponentially grew. This evidence reflects the remarkable technological developments of both techniques allowing unprecedented spatial, temporal and contrast resolution levels and to comprehensively evaluate cardiac pathology, combining anatomical information with functional assessment and tissue characterization of myocardial diseases

Faster deterministic wakeup in multiple access

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^3 log^3 n), which improves on the currently best upper bound of O(n^4 log^5 n)

Anonymous Processors with Synchronous Shared Memory: Monte Carlo Algorithms

We consider synchronous distributed systems in which processors communicate by shared read- write variables. Processors are anonymous and do not know their number n. The goal is to assign individual names by all the processors to themselves. We develop algorithms that accomplish this for each of the four cases determined by the following independent properties of the model: concurrently attempting to write distinct values into the same shared memory register either is allowed or not, and the number of shared variables either is a constant or it is unbounded. For each such a case, we give a Monte Carlo algorithm that runs in the optimum expected time and uses the expected number of O(n log n) random bits. All our algorithms produce correct output upon termination with probabilities that are 1?n^{??(1)}, which is best possible when terminating almost surely and using O(n log n) random bits