Shmuel Zaks

Tight lower and upper bounds for some distributed algorithms for a complete network of processors

Distributed algorithms for complete asynchronous networks of processors (i.e., networks where each pair of processors is connected by a communication line) are discussed. The main result is <italic>O</italic>(<italic>nlogn</italic>) lower and upper bounds on the number of messages required by any algorithm in a given class of distributed algorithms for such networks. This class includes algorithms for problems like finding a leader or constructing a spanning tree (as far as we know, all known algorithms for those problems may require <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>) messages when applied to complete networks). <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>) bounds for other problems, like constructing a maximal matching or a Hamiltonian circuit are also given. In proving the lower bound we are counting the edges which carry messages during the executions of the algorithms (ignoring the actually number of messages carried by each edge). Interestingly, this number is shown to be of the same order of magnitude of the total number of messages needed by these algorithms. In the upper bounds, the length of any message is at most log<subscrpt>2</subscrpt>[4<italic>mlog</italic><subscrpt>2</subscrpt><italic>n</italic>] bits, where <italic>m</italic> is the maximum identity of a node in the network. One implication of our results is that finding a spanning tree in a complete network is easier than finding a minimum weight spanning tree in such a network, which may require <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>) messages.

A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor

Fischer, Lynch and Paterson showed in a fundamental paper that achieving a distributed agreement for N > I processors is impossible in the presence of one faulty processor. This result was later extended by Moran and Wolfstahl who showed that it holds for any task with a connected input graph and a disconnected decision graph (whcrc a vcrtcx in the input [decision] graph is an N-tuple of input [decision] values of the processors, and there is an edge connecting two vertices if and only if they differ in exactly one component), In this paper we extend that latter result, and in fact we set the exact bordedine between solvable and unsolvable tasks, by giving a necessary and sufficient condition for a task to be solvable in the presence of a faulty processor. We present a universal protocol which solves any task which is found to be solvable by our condition. Using our characterization, we derive a novel technique to prove lower bounds on the number of messages that must be sent due to processor failure; specifically, we show that for each fixed JV > 2 there exist distributed tasks for Iv processors that can be solved in the presence of a faulty processor, but any protocol that solves them must send arbitrarily many messages in the worst case.

The layout of virtual paths in ATM networks

We study the problem of designing a layout of virtual paths (VPs) on a given ATM network. We first define a mathematical model that captures the characteristics of virtual paths. In this model, we define the general VP layout problem, and a more restricted case; while the general case layout should cater connections between any pair of nodes in the network, the restricted case layout should only cater connections between a specific node to the other nodes. For the latter case, we present an algorithm that finds a layout by decomposing the network into subnetworks and operating on each subnetwork, recursively; we prove an upper bound on the optimality of the resulting layout and a matching lower bound for the problem, that are tight under certain realistic assumptions. Finally, we show how the solution for the restricted case is used as a building block in various solutions to more general cases (trees, meshes, K-separable networks, and general topology networks) and prove a lower bound for some of our results. The results exhibit a tradeoff between the efficiency of the call setup and both the utilization of the VP routing tables and the overhead during recovery from link disconnections.

The Cycle Lemma and Some Applications

Two proofs of a frequently rediscovered combinatorial lemma are presented. Using the lemma, a combinatorial proof is given that the average height of an ordered (plane-planted) tree is approximately twice the average node (vertex) level.

Generating Trees and Other Combinatorial Objects Lexicographically

We show a one-to-one correspondence between all the ordered trees that have

On sets of Boolean n-vectors with all k-projections surjective

n_0 + 1

On the complexity of the regenerator placement problem in optical networks

leaves and

A 10/7 + /spl epsi/ approximation for minimizing the number of ADMs in SONET rings

n_i

Optimal lower bounds for some distributed algorithms for a complete network of processors

internal nodes with

Efficient elections in chordal ring networks

k_i