Lev B. Levitin

On a new class of codes for identifying vertices in graphs

We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex /spl upsi/ to be the set of vertices in G that are at distance at most t from /spl upsi/. The vertex /spl upsi/ is then said to cover itself and every other vertex in the ball with center /spl upsi/. Our formal problem statement is as follows: given an undirected graph G and an integer t/spl ges/1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C. The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C. We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

The maximum speed of dynamical evolution

Abstract We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time — its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of ΔE, the standard deviation of the energy of the system; here we give a strict bound that depends only on E − E0, the systems average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding 1 J of energy to a given computer can never increase its processing rate by more than about 3 × 1033 operations per second.

Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight

How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.

On robust and dynamic identifying codes

A subset C of vertices in an undirected graph G=(V,E) is called a 1-identifying code if the sets I(v)={u/spl isin/C:d(u,v)/spl les/1}, v/spl isin/V, are nonempty and no two of them are the same set. It is natural to consider classes of codes that retain the identification property under various conditions, e.g., when the sets I(v) are possibly slightly corrupted. We consider two such classes of robust codes. We also consider dynamic identifying codes, i.e., walks in G whose vertices form an identifying code in G.

Scalable cycle-breaking algorithms for gigabit Ethernet backbones

Ethernet networks rely on the so-called spanning tree protocol (IEEE 802.1d) in order to break cycles, thereby avoiding the possibility of infinitely circulating packets and deadlocks. This protocol imposes a severe penalty on the performance and scalability of large gigabit Ethernet backbones, since it makes inefficient use of expensive fibers and may lead to bottlenecks. We propose a significantly more scalable cycle-breaking approach, based on the novel theory of turn-prohibition. Specifically, we introduce, analyze and evaluate a new algorithm, called tree-based turn-prohibition (TBTP). We show that this polynomial-time algorithm maintains backward-compatibility with the IEEE 802.1d standard and never prohibits more than 1/2 of the turns in the network, for any given graph and any given spanning tree. Through extensive simulations on a variety of graph topologies, we show that it can lead to an order of magnitude improvement over the spanning tree protocol with respect to throughput and end-of-end delay metrics. In addition, we propose and evaluate heuristics to determine the replacement order of legacy switches that results in the fastest performance improvement.

A new approach to the general minimum distance decoding problem: The zero-neighbors algorithm

Minimum distance decoding (MDD) for a general error-correcting linear code is a hard computational problem that recently has been shown to be NP -hard. The complexity of known decoding algorithms is determined by \min (2^{k},2^{n-k}) , where n is the code length and k is the number of information digits. Two new algorithms are suggested that reduce substantially the complexity of MDD. The algorithms use a new concept of zero neighbors--a special set of codewords. Only these codewords (which can be computed in advance) should be stored and used in the decoding procedure. The number of zero neighbors is shown to be very small compared with \min (2^{k},2^{n-k}) for n \gg 1 and a wide range of code rates R = k/n . For example, for R \approx 0.5 this number grows approximately as a square root of the number of codewords.

On the covering of vertices for fault diagnosis in hypercubes

Abstract We investigate the optimal covering of vertices by Hamming balls of radius t in a hypercube Z2n such that any vertex in Z2n can be uniquely identified by examining the vertices that cover it. Given Z2n and an integer t ⩾ 1, we find a (minimal) set C of vertices such that every vertex in Z2n belongs to a unique set of balls of radius t centered at the vertices in C . This is useful in diagnosing processor faults in hypercube-based multiprocessor systems.

Data verification and reconciliation with generalized error-control codes

We consider the problem of data reconciliation, which we model as two separate multisets of data that must be reconciled with minimum communication. Under this model, we show that the problem of reconciliation is equivalent to a variant of the graph coloring problem and provide consequent upper and lower bounds on the communication complexity of reconciliation. Further, we show by means of an explicit construction that the problem of reconciliation is, under certain general conditions, equivalent to the problem of finding error-correcting codes for a general class of errors. Under this equivalence, reconciling with little communication is linked to codes with large size, and vice versa. We show analogous results for the problem of multiset verification, in which we wish to determine whether two multisets are equal using minimum communication. As a result, a wide body of literature in coding theory may be applied to the problems of reconciliation and verification.

Information Theory for Quantum Systems

Basic concepts and results of physical information theory are presented. The entropy defect and Shannon’s measure of information are introduced and the entropy defect principle is formulated for both quasiclassical and consistently quantum description of a physical system. Results related to ideal physical information channels are discussed. The entropy defect and the amount of information coincide in the quasiclassical case, but the latter quantity is, in general, smaller than the former in quantum case due to the quantum-mechanical irreversibility of measurement. The physical meaning of both quantities is analyzed in connection with Gibbs paradox and the maximum work obtainable from a non-equilibrium system. Indirect (generalized) vs. direct (von Neumann’s) quantum measurements are considered. It is shown that in any separable infinite-dimensional Hilbert space direct and indirect quantum measurements yield equal maximum information.

Minimal Sets of Turns for Breaking Cycles in Graphs Modeling Networks

The problem of preventing deadlocks and livelocks in computer communication networks, in particular, those with wormhole routing, is considered. The method to prevent deadlocks is to prohibit certain turns (i.e., the use of certain pairs of connected edges) in the routing process, in such a way that eliminates all cycles in the graph. We propose a new algorithm that constructs a minimal (irreducible) set of turns that breaks all cycles and preserves connectivity of the graph. The algorithm is tree-free and is considerably simpler than earlier cycle-breaking algorithms. We prove its properties and present lower and upper bounds for minimum cardinalities of cycle-breaking connectivity preserving sets for graphs of general topology as well as for planar graphs. In particular, the algorithm guarantees that not more than 1/3 of all turns in the network become prohibited. We also present experimental results on the fraction of prohibited turns, the distance dilation, as well as on the message delivery times and saturation loads for the proposed algorithm in comparison with known tree-based algorithms. The proposed algorithm outperforms substantially the tree-based algorithms in all characteristics considered.