Showing papers by "Andrzej Pelc published in 2000"

Strategies for Hotlink Assignments

Abstract: Consider a DAG (directed acyclic graph) G = (V, E) representing a collection V of web pages connected via links E All web pages can be reached from a designated source page, represented by a source node s of G Each web page carries a weight representative of the frequency with which it is visited By adding hotlinks, at most one per page, we are interested in minimizing the expected number of steps needed to visit a selected set of web pages from the source page For arbitrary DAGs we show that the problem is NP-complete We also give algorithms for assigning hotlinks, as well as upper and lower bounds on the expected number of steps to reach the leaves from the source page s located at the root of a complete binary tree Depending on the probability distribution (arbitrary, uniform, Zipf) the expected number of steps is at most c ċ n, where c is a constant less than 1 For the geometric distribution we show how to obtain a constant average number of steps

Better adaptive diagnosis of hypercubes

Abstract: We consider the problem of adaptive fault diagnosis in hypercube multiprocessor systems. Processors perform tests on one another and later tests can be scheduled on the basis of previous test results. Fault-free testers correctly identify the fault status of tested processors, while faulty testers can give arbitrary test results. The goal is to identify correctly the status of all processors, assuming that the number of faults does not exceed the hypercube dimension. We propose an adaptive diagnosis algorithm whose efficiency is drastically better than that of any previously known strategies. While the worst-case number of tests for any of them exceeds 2/sup n/ log n for an n-dimensional hypercube, our method uses at most 2/sup n/+3n/2 tests in the worst case. We can also modify our algorithm to improve the number of testing rounds. By slightly increasing the number of tests to 2/sup n/+(n+1)/sup 2/ (still a much better performance than 2/sup n/ log n), we can carry out diagnosis in at most 11 rounds in the worst case (as opposed to over n rounds in the best previously known strategy).

Impact of topographic information on graph exploration efficiency

Abstract: A robot has to explore an undirected connected graph by visiting all its nodes and traversing all edges. It may either have a complete a priori knowledge of the graph or only have an unoriented map of it, or, finally, lack any knowledge of the graph. We study the impact of this varying amount of knowledge on exploration performance. It is shown that the best exploration algorithm lacking any knowledge of the graph uses twice as many edge traversals in the worst case as does the best algorithm which has an unoriented map of the graph. On the other hand, the latter uses twice as many edge traversals in the worst case as does the best algorithm having a complete knowledge of the graph. Similar results for the restricted case of exploration algorithms working only for trees are also established. © 2000 John Wiley & Sons, Inc.

The wakeup problem in synchronous broadcast systems (extended abstract)

Abstract: This paper studies the differences between two levels of synchronization in a distributed broadcast system (or a multiple access channel). In the globally synchronous model, all processors have access to a global clock. In the locally synchronous model, processors have local clocks ticking at the same rate, but each clock starts individually, when the processor wakes up.We consider the fundamental problem of waking up all of n processors of a completely connected broadcast system. Some processors wake up spontaneously, while others have to be woken up. Only wake processors can send messages; a sleeping processor is woken up upon hearing a message. The processors hear a message in a given round if and only if exactly one processor sends a message in that round. Our goal is to wake up all processors as fast as possible in the worst case, assuming an adversary controls which processors wake up and when. We analyze the problem in both the globally synchronous and locally synchronous models, with or without the assumption that n is known to the processors. We propose randomized and deterministic algorithms for the problem, as well as lower bounds in some of the cases. These bounds establish a gap between the globally synchronous and locally synchronous models.

Assigning Labels in Unknown Anonymous Networks

Abstract: We consider the task of distributedly assigning distinct labels to nodes of an unknown anonymous network. A priori, nodes do not have any identities (anonymous network) and do not know the topology or the size of the network (unknown network). They execute identical algorithms, apart from a distinguished node, called the source, which starts the labeling process. Our goal is to assign short labels, as fast as possible. The quality of a labeling algorithm is measured by the range from which the algorithm picks the labels, or alternatively the length of the assigned labels. Natural efficiency measures are the time, i.e., the number of rounds required for the label assignment, and the message and bit complexities of the label assignment protocol, i.e., the total number of messages (resp., bits) circulating in the network. We present label assignment algorithms whose time and message complexity are asymptotically optimal and which assign short labels. On the other hand, we establish inherent trade-offs between quality and efficiency for labeling algorithms.

Optimal adaptive broadcasting with a bounded fraction of faulty nodes

Abstract: We consider broadcasting among n processors, f of which can be faulty. A fault-free processor, called the source, holds a piece of information which has to be transmitted to all other fault-free processors. We assume that the fraction f/n of faulty processors is bounded by a constant γ<1 . Transmissions are fault free. Faults are assumed to be of the crash type: faulty processors do not send or receive messages. We use the whispering model: pairs of processors communicating in one round must form a matching. A fault-free processor sending a message to another processor becomes aware of whether this processor is faulty or fault free and can adapt future transmissions accordingly. The main result of the paper is a broadcasting algorithm working in O( log n) rounds and using O(n) messages of logarithmic size, in the worst case. This is an improvement of the result from [17] where O ((log n) 2 ) rounds were used. Our method also gives the first algorithm for adaptive distributed fault diagnosis in O( log n) rounds.

Assigning labels in unknown anonymous networks (extended abstract)

Abstract: We consider the task of distributedly assigning distinct labels to nodes of an unknown anonymous network. A priori, nodes do not have any identities (anonymous network) and do not know the topology or the size of the network (unknown network). They execute identical algorithms, apart from a distinguished node, called the source, which starts the labeling process. Our goal is to assign short labels, as fast as possible. The quality of a labeling algorithm is measured by the range from which the algorithm picks the labels, or alternatively, the length of the assigned labels. Natural efficiency measures are the time, i.e., the number of rounds required for the label assignment, and the message and bit complexities of the label assignment protocol, i.e., the total number of messages (resp., bits) circulating in the network. We present label assignment algorithms whose time and message complexity are asymptotically optimal and which assign short labels. On the other hand, we establish inherent trade-offs between quality and efficiency for labeling algorithms.

Efficient Communication in Unknown Networks

Abstract: We consider the problem of disseminating messages in networks. We are interested in information dissemination algorithms in which machines operate independently without any knowledge of the network topology or size. Three communication tasks of increasing difficulty are studied. In blind broadcasting (BB) the goal is to communicate the source message to all nodes. In acknowledged blind broadcasting (ABB) the goal is to achieve BB and inform the source about it. Finally, in full synchronization (FS) all nodes must simultaneously enter the state terminated after receiving the source message. The algorithms should be efficient both in terms of the time required and the communication overhead they put on the network. We limit the latter by allowing every node to send a message to at most one neighbor in each round. We show that BB is achieved in time at most 2n in any n-node network and show networks in which time 2n-o(n) is needed. For ABB we show algorithms working in time (2+Ɛ)n, for any fixed positive constant Ɛ and sufficiently large n. Thus for both BB and ABB our algorithms are close to optimal. Finally, we show a simple algorithm for FS working in time 3n.The optimal time of full synchronization remains an open problem.

Deterministic Broadcasting Time with Partial Knowledge of the Network

Abstract: We consider the time of deterministic broadcasting in networks whose nodes have limited knowledge of network topology. Each node v 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 v. Apart from that, each node knows only the maximum degree Δ of the network and the number n of nodes. 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 tradeoffs between knowledge radius and time of deterministic broadcasting, when knowledge radius is small, i.e., when nodes are only aware of their close vicinity. While for knowledge radius 0, minimum broadcasting time is Θ(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 radii 1 and 2. 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, for a given knowledge radius. For knowledge radius 1 we develop a broadcasting algorithm working in time O(min(n, D2Δ)), where n is the number of nodes, D is the diameter of the network, and Δ is the maximum degree. We show that for bounded maximum degree Δ this algorithm is asymptotically optimal. For knowledge radius 2 we show how to broadcast in time O(DΔ log n)) and prove a lower bound Ω(DΔ) on broadcasting time, when DΔ ∈ O(n). This lower bound is valid for any constant knowledge radius. For knowledge radius log* n + 3 we show how to broadcast in time O(DΔ). Finally, for any knowledge radius r, we show a broadcasting algorithm working in time O(D2Δ/r).

Reliable minimum finding comparator networks

Abstract: We consider the problem of constructing reliable comparator networks built from unreliable comparators. In case of a faulty comparator inputs are directly output without comparison. A trivial lower bound of O(logn + k) on the depth of n-input k-fault tolerant sorting network is well known. We are interested in establishing exact lower bounds on the depth of such networks. To this end we consider fairly simple minimum-finding networks. Our main result is the first nontrivial lower bound on depths of networks computing minimum among n > 2 items in the presence of k > 0 faulty comparators. We prove that the depth of any such network is at least max([logn] + 2k, logn + klog logn/k+1). We also describe a network whose depth nearly matches the lower bound.