Fred S. Annexstein

On finding solutions for extended Horn formulas

Abstract We present a simple quadratic-time algorithm for solving the satisfiability problem for a special class of boolean formulas. This class properly contains the class of extended Horn formulas and balanced formulas. Previous algorithms for these classes require testing membership in the classes. However, the problem of recognizing balanced formulas is complex, and the problem of recognizing extended Horn formulas is not known to be solvable in polynomial time. Our algorithm requires no such test for membership.

A unified framework for off-line permutation routing in parallel networks

In this paper we present a general strategy for finding efficient permutation routes in parallel networks. Among the popular parallel networks to which the strategy applies are mesh networks, hypercube networks, hypercube-derivative networks, ring networks, and star networks. The routes produced are generally congestion-free and take a number of routing steps that is within a small constant factor of the diameter of the network. Our basic strategy is derived from an algorithm that finds (in polynomial time) efficient permutation routes for aproduct network, G×H, given efficient permutation routes forG andH. We investigate the use of this algorithm for routingmultiple permutations and extend its applicability to a wide class of graphs, including several families ofCayley graphs. Finally, we show that our approach can be used to find efficient permutation routes among the remaining live nodes infaulty networks.

On the diameter and bisector size of Cayley graphs

We present bounds on two combinatorial properties of Cayley graphs in terms relating to the structure of their underlying group, Included in this work is a presentation of lower bounds on the diameter of Cayley graphs of groups with nilpotent subgroups and upper bounds on the size of node bisectors of Cayley graphs of groups with solvable subgroups.Cayley graphs, being endowed with algebraic structure, have been increasingly recognized as a source of interconnection networks underlying parallel computers. Their structure has been shown to endow parallel architectures with advantages, for example, in terms of algorithmic efficiency. Our results demonstrate limits on the communication power of certain classes of well-structured interconnection networks.

Indexing techniques for file sharing in scalable peer-to-peer networks

File sharing is a very popular service provided by peer-to-peer (P2P) networks. In a P2P file-sharing network, users share files and issue queries to the network to find the locations of files residing at other peer nodes. Recently, proxy-enabled peers, or supernodes, have been incorporated to enhance scalability by providing indexing services to nodes on slower network connections. Typically, supernodes build a vector or multi-index of shared files stored on other (slower) peer nodes connected to them. We consider a new model whereby the index tables of individual nodes are merged into a single data structure stored by the supernode. We analyze this model in relation to the standard vectorized data structure. We compare the performance of these supernode indexing algorithms and provide a theoretical analysis that is asymptotic and probabilistic in nature. However, there are several significant constant factors that the theory does not account for, and which are important for designing an optimal system solution. We report on a series of simulation experiments which provide verification of the asymptotic analysis of the formal framework and tools to determine the magnitude of the constant factors. Our general conclusion is that when the query rate exceeds the rate of data updates, the new merged model is preferable to the vector model. However, the details of our analysis allow us to consider combinations of several parameters, and thereby enable the design of optimal indexing schemes via the incorporation of measurements of the parameters of particular applications.

Latency effects on reachability in large-scale peer-to-peer networks

In this paper we study the latency effects introduced in large scale internet applications. In particular, we study the effects of heterogeneous latency on reachability in decentralized, distributed networks operating under flooding protocols. We show that the standard protocol mechanisms of time-to-live (TTL) and unique message identification (UID), used to govern flooding message transmissions, can combine to potentially devastating effect on the reachability of message broadcast. We call this combined effect short-circuiting, and we investigate consequences of this phenomenon. We show that in the worstcase, short-circuiting resulting from heterogeneous latencies can near-completely eliminate the reach of broadcast messages, even with an ability to place k ⪈ 1 broadcast servers optimally. This dramatic negative effect on reachability shows that application protocol designs need to be sensitive to latency models. In addition, we show that short-circuiting can have a negative effect on well known approximation algorithms for maximizing reachability. We show that with respect to distance measures that account for short-circuiting, the standardk-center problem can not be approximated within n1-e, unless P = NP. Our theoretical results suggest that it may be quite difficult to find near optimal solutions to certain internet server selection problems. We consider the significance of our results on a large-scale peer-to-peer searching application, that relies on related flooding protocols. We report measurements through experimental studies with both simulated networks and a large network application known as Gnutella. Our empirical results, using statistics obtained from both the simulations and real applications, support the conclusion that, on average, the real effects of short-circuiting are significant, but not devastating to the performance of an overall large-scale system.

Directional Routing via Generalized st -Numberings

We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph st-numberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in different directions from the same source are node-disjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3-connected graphs have 3-directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to k-connected graphs. We also show that a directed acyclic graph (dag) that is k-connected to a set of sinks has a k-directional embedding in (k-1)-space with the sink set as the extreme nodes.

Depth-Latency Tradeoffs in Multicast Tree Algorithms

The construction of multicast trees is complicated by the need to balance a number of important objectives, including: minimizing latencies, minimizing depth/hops, and bounding the degree. In this paper, we study the problem of determining a degree-bounded directed spanning tree of minimum average-latency in a complete graph where the inter-node latencies are used to determine a metric. In particular, we focus on measuring the effects on average latency when imposing depth constraints (i.e., bounds on hop count) on degree-bounded spanning trees. The general problem is a well known NP-hard problem, and several works have proposed approximate solutions which aim at minimizing either depth or latency. In this work, we present a new heuristic algorithm which improves upon previous solutions by considering both depth and latency and the tradeoffs between them. Our algorithms are shown to improve the theoretical worst-case approximation factors, and we demonstrate improvements under empirical evaluation. Our experiments examine and analyze several different topologies, including, low-dimensional random geometric networks, random transit-stub networks, and high- dimensional hypercube networks. We show how our solutions can be applied in the context of enabling multicasting support in locality aware peer-to-peer overlay networks.

Compact encodings for all local path information in web taxonomies with application to wordnet

We consider the problem of finding a compact labelling for large, rooted web taxonomies that can be used to encode all local path information for each taxonomy element. This research is motivated by the problem of developing standards for taxonomic data, and addresses the data intensive problem of evaluating semantic similarities between taxonomic elements. Evaluating such similarities often requires the processing of large common ancestor sets between elements. We propose a new class of compact labelling schemes, designed for directed acyclic graphs, and tailored for applications to large web taxonomies. Our labelling schemes significantly reduce the complexity of evaluating similarities among taxonomy elements by enabling the gleaning of inferences from the labels alone, without searching the data structure. We provide an analysis of the label lengths for the proposed schemes based on structural properties of the taxonomy. Finally, we provide supporting empirical evidence for the quality of these schemes by evaluating the performance on the WordNet taxonomy.

A multi-tree routing scheme using acyclic orientations

Abstract We propose a mathematical model for fault-tolerant routing based on acyclic orientations, or acorns, of the underlying network G=(V,E). The acorn routing model applies routing tables that store the set of parent pointers associated with each out-neighborhood defined by the acorn. Unlike the standard single-parent sink-tree model, which is vulnerable to faults, the acorn model affords a full representation of the entire network and is able to dynamically route around faults. This fault tolerance is achieved when using the acorn model as a multi-tree generator for gathering data at a destination node, as well as an independent tree generator for global point-to-point communication. A fundamental fault-tolerant measure of the model is the capacity of an acorn, i.e., the largest integer k such that each vertex outside the neighborhood N(v) of the destination v has at least k parent pointers. A capacity-k acorn A to destination v is k-vertex fault-tolerant to v. More strongly, we show A supports a k independent sink-tree generator, i.e., the parent pointers of each vertex w ∈ V−N(v) can be partitioned into k nonempty classes labeled 1,2,…,k such that any set of sink trees T1,T2,…,Tk are pairwise independent, where tree Ti is a sink tree generated by parent pointers labeled i together with the parent pointers into v. We present an linear time optimization algorithm for finding an acorn A of maximum capacity in graphs, based upon a minimax theorem. We also present efficient algorithms that label the parent pointers of capacity-k acorn A, yielding a k-independent sink tree generating scheme.

Embedding hypercubes and related networks into mesh-connected processor arrays

Abstract In this paper we study the problem of how computations programmed for hypercubes, and their bounded-degree relatives, the shuffle-exchange and cube-connected-cycles, can be efficiently emulated by mesh-connected arrays of processing elements. The emulations we present are implemented via graph embeddings. The graph embeddings we present are all optimal with respect to congestion, and are also noteworthy for they yield efficient emulation programs that are written in a SIMD style without the use of indirect addressing. The paper includes the three following emulation results, where each emulation algorithm requires no indirection. For any n ≥ 1 and fixed r ≥ 1, an n -sided r -dimensional mesh can emulate an n 2 n -node cube-connected-cycles network with slowdown Θ(2 n / n r −1 ), a 2 n -node shuffle-exchange network with slowdown Θ(2 n / n r ), and a 2 n -node hypercube network operating in a weak model, with slowdown Θ(2 n log n / n r ).