Random walks on randomly evolving graphs

A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time; on the other hand, many distributed networks are inherently dynamic, and their topology is subjected to potentially drastic changes. In this work we study the mixing (i.e., convergence) properties of random walks on graphs subjected to random changes over time. Specifically, we consider the edge-Markovian random graph model: for each edge slot, there is a two-state Markov chain with transition probabilities p (add a non-existing edge) and q (remove an existing edge). We derive several positive and negative results that depend on both the density of the graph and the speed by which the graph changes

Information Gathering in Ad-Hoc Radio Networks with Tree Topology

We study the problem of information gathering in ad-hoc radio networks without collision detection, focussing on the case when the network forms a tree, with edges directed towards the root. Initially, each node has a piece of information that we refer to as a rumor. Our goal is to design protocols that deliver all rumors to the root of the tree as quickly as possible. The protocol must complete this task within its allotted time even though the actual tree topology is unknown when the computation starts. In the deterministic case, assuming that the nodes are labeled with small integers, we give an O(n)-time protocol that uses unbounded messages, and an O(n log n)-time protocol using bounded messages, where any message can include only one rumor. We also consider fire-and-forward protocols, in which a node can only transmit its own rumor or the rumor received in the previous step. We give a deterministic fire-and- forward protocol with running time O(n^1.5), and we show that it is asymptotically optimal. We then study randomized algorithms where the nodes are not labelled. In this model, we give an O(n log n)-time protocol and we prove that this bound is asymptotically optimal

Probabilistic Properties of Highly Connected Random Geometric Graphs

In this paper we study the probabilistic properties of reliable networks of minimal total edge lengths. We study reliability in terms of k-edge-connectivity in graphs in d-dimensional space. We show this problem fits into Yukich’s framework for Euclidean functionals for arbitrary k, dimension d and distant-power gradient p, with p < d. With this framework several theorems on the convergence of optimal solutions follow. We apply Yukich’s framework for functionals so that we can use partitioning algorithms that rapidly compute near-optimal solutions on typical examples. These results are then extended to optimal k-edge-connected power assignment graphs, where we assign power to vertices and charge per vertex. The network can be modelled as a wireless network

The temporal explorer who returns to the base.

In this paper we study the problem of exploring a temporal graph (i.e. a graph that changes over time), in the fundamental case where the underlying static graph is a star on n vertices. The aim of the exploration problem in a temporal star is to find a temporal walk which starts at the center of the star, visits all leaves, and eventually returns back to the center. We present here a systematic study of the computational complexity of this problem, depending on the number k of time-labels that every edge is allowed to have; that is, on the number k of time points where each edge can be present in the graph. To do so, we distinguish between the decision version STAREXP(k) , asking whether a complete exploration of the instance exists, and the maximization version MAXSTAREXP(k) of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We fully characterize MAXSTAREXP(k) and show a dichotomy in terms of its complexity: on one hand, we show that for both k=2 and k=3 , it can be efficiently solved in O(nlogn) time; on the other hand, we show that it is APX-complete, for every k≥4 (does not admit a PTAS, unless P = NP, but admits a polynomial-time 1.582-approximation algorithm). We also partially characterize STAREXP(k) in terms of complexity: we show that it can be efficiently solved in O(nlogn) time for k∈{2,3} (as a corollary of the solution to MAXSTAREXP(k) , for k∈{2,3} ), but is NP-complete, for every k≥6

Roto-translazione di figure bi-tridimensionali con la stazione grafica WHIZZARD 7200 della MEGATEX. Applicazione: programma per il display e l'analisi interattiva dei dati del MICROVERTEX di UA1

SIGLEITItal

On temporally connected graphs of small cost

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u,v, u≠v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n, and at most the optimal cost plus 2. To show this, we prove a lower bound on the cost for any undirected graph. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P=NP. On the positive side, we show that in dense graphs with random edge availabilities, all but Θ(n) labels are redundant whp. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels

Subquadratic non-adaptive threshold group testing

We consider threshold group testing â a generalization of a well known and thoroughly examined problem of combinatorial group testing. In the classical setting, the goal is to identify a set of positive individuals in a population, by performing tests on pools of elements. The output of each test is an answer to the question: is there at least one positive element inside a query set Q? The threshold group testing is a natural generalization of this classical setting which arises when the answer to a test is positive if at least t > 0 elements under test are positive. We show that there exists a testing strategy for the threshold group testing consisting of (formula presented ) tests, for d positive items in a population of size N. For any value of the threshold t, we also provide a lower bound of order (formula presented). Our subquadratic bound shows a complexity separation with the classical group testing (which corresponds to t = 1) where Ω(d2logdN) tests are needed [25]. Next, we introduce a further generalization, the multi-threshold group testing problem. In this setting, we have a set of s > 0 thresholds, t1, t2, â¦, ts. The output of each test is an integer between 0 and s which corresponds to which thresholds get passed by the number of positives in the queried pool. Here, one may be interested in minimizing not only the number of tests, but also the number of thresholds which is related to the accuracy of the tests. We show the existence of two strategies for this problem. The first one of size (formula presented ) is an extension of the above-mentioned result. The second strategy is more general and works for a range of parameters. As a consequence, we show that (formula presented ) tests are sufficient for t ⤠d/2. Both strategies use respectively O(âd) and O(ât) thresholds