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

Performing work in broadcast networks

We consider the problem of how to schedule t similar and independent tasks to be performed in a synchronous distributed system of p stations communicating via multiple-access channels. Stations are prone to crashes whose patterns of occurrence are specified by adversarial models. Work, defined as the number of the available processor steps, is the complexity measure. We consider only reliable algorithms that perform all the tasks as long as at least one station remains operational. It is shown that every reliable algorithm has to perform work Omega(t + p root t) even when no failures occur. An optimal deterministic algorithm for the channel with collision detection is developed, which performs work O(t + p root t). Another algorithm, for the channel without collision detection, performs work O(t + p root t + p min {f, t}), where f < p is the number of failures. This algorithm is proved to be optimal, provided that the adversary is restricted in failing no more than f stations. Finally, we consider the question if randomization helps against weaker adversaries for the channel without collision detection. A randomized algorithm is developed which performs the expected minimum amount O(t + p root t) of work, provided that the adversary may fail a constant fraction of stations and it has to select failure-prone stations prior to the start of an execution of the algorithm

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