On the complexity of computation in the presence of link failures: the case of a ring

Summary We investigate the message complexity of distributed computations on rings of asynchronous processors. In such computations, each processor has an initial local value and the task is to compute some predetermined function of all local values. Our work deviates from previous works concerning the complexity of ring computations in that we consider the effect oflink failures. A link is said to fail if some message sent through it never reaches its destination. We show that the message complexity of any function, which is sensitive to all its inputs, is (n logn) whenn, the number of processors, is a-priori known; and is (n ² ) whenn is not known. Interestingly, these tight bounds do not depend on whether the identity of a leader is a-priori known before the computation starts. These results stand in sharp contrast to the situation in asynchronous rings with no link failures, where the message complexity is affected by the a-priori knowledge of a leader but is not affected by the knowledge ofn. Oded Goldreich was born in Tel-Aviv, Israel, on February 4th 1957. Received B.A., M.Sc., and D.Sc. in Computer Science from the Technion — Israel Institute of Technology, Haifa, Israel, in 1980, 1982, and 1983, respectively. He is currently an Associate Professor of Computer Science in the Technion. From 1983 to 1986, he was a postdoctoral fellow in MIT's Laboratory for Computer Science. His research interests include cryptography and related areas, relation between randomness and algorithms, and distributed computation. Luiba Shrira was born in Vilnius, Lithuania. Received B.A., M.Sc., and D.Sc. in Computer Science from the Technion—Israel Institute of Technology, Haifa, Israel in 1977, 1980, and 1985, respectively. from 1986 to 1989 she was a postdoctoral fellow at Laboratory for Computer science at MIT, where she is currently a Research Associate. Her research interests include highly-available and reliable distributed algorithms and systems, persistent object systems, and programming methodology.Part of the work has been done while the first author was in the Laboratory for Computer Science of MIT and the second author was in the Computer Science Department of the Technion. First author was partially supported by a Weizmann Postdoctoral Fellowship, an IBM Postdoctoral Fellowship, and Albert Einstein Research Fund (through Technion's V.P.R. Fund)     

Counting networks with arbitrary fan-out

Summary It is shown that an acyclic smoothing network (and hence counting network) with fan-outn cannot be constructed from balancers of fan-outb ₁,...,b _k , if there exists a prime factorp ofn, such thatp does not divideb _i , for alli, 1ik. This holds regardless of the depth, fan-in or size of the network, as long as they are finite. On the positive side, a simple construction ofcyclic counting networks with fan-outn, for arbitraryn, is presented. An acyclic counting network with fan-in and fan-outp2 ^k , for any integerk0, is constructed out of 2-balancers andp-balancers. Eran Aharonson received the B.A. and M.Sc. degrees in Computer Science from the Technion, Israel Institute of Technology (Haifa, Israel) in 1989 and 1992, respectively. He is currently vice president for research and development at ART-Advanced Recognition Technolgies Ltd., a company dedicated to handwriting and voice recognition. His general research interests are distributed computation, theoretical computer science and pattern recognition. Hagit Attiya received the B.Sc. degree in Mathematics and Computer Science from the Hebrew University of Jerusalem, in 1981, the M.Sc. and Ph.D. degrees in Computer Science from the Hebrew University of Jerusalem, in 1983 and 1987, respectively. She is presently a senior lecturer at the department of Computer Science at the Technion, Israel Institute of Technology. Prior to this, she has been a post-doctoral research associate at the Laboratory for Computer Science at M.I.T. Her general research interests are distributed computation and theoretical computer science. More specific interests include fault-tolerance, timing-based and asynchronous algorithms.A preliminary version of this paper appears in proceedings of the3rd Annual ACM-SIAM Symposium on Discrete Algorithms, January 1992, pp. 104–113. This research was supported by Technion V.P.R.-B. and G. Greenberg Research Fund (Ottawa)Supported by Rashi Enterprise graduate fellowship     

On the complexity of global computation in the presence of link failures: the general case

Summary This paper presents (m logn) and (mn) messages lower bounds on the problem of computing a gobal sensitive function in biderectional networks with link failures (i.e., dynamically changing topology), wheren andm are the total number of nodes and links in the network. The (m logn) lower bound is under the assumption thatn is a-priori known to the nodes, while the second bound is for the case in which such knowledge is not available. A global sensitive function ofn variables is a function that may not be computed without the knowledge of the values of all then variables (e.g. maximum, sum, etc). Thus, computing such a function at one node of a distributed network requires this node to communicate with every other node in the network. Though lower bounds higher than (m) messages are known for this problem in the context of link failures, none holds for dense bidirectional networks. Moreover, we are not aware of any other nontrivial lower bound higher than (m) for dense bidirectional networks. Yehuda Afek received a B.Sc. in Electrical Engineering from the Technion and an M.Sc. and Ph.D. in Computer Science from the University of California, Los-Angeles. In 1985 he joined the Distributed Systems Research Department in AT&T Bell Laboratories as a Member of Technical Staff. In 1988 he joined the Computer Science Department in Tel-Aviv University, where he now holds a permanent position. From 1989 to 1994 he was also a consultant for AT&T Bell Laboratories. His interests include communication protocols, distributed computing and asynchronous shared memory systems. Danny Hendler was born in Kiryat-Haim near Haifa, Israel, on April 17th 1961. He received his B.Sc. and M.Sc. in Computer Science from Tel-Aviv University, Israel, in 1986 and 1993, respectively. In the past 8 years he has worked as a free lance software-consultant, specializing mainly in communication, telephony and voice-mail applications.     

Using mappings to prove timing properties

Summary A new technique for proving timing properties for timing-based algorithms is described; it is an extension of the mapping techniques previously used in proofs of safety properties for asynchronous concurrent systems. The key to the method is a way of representing a system with timing constraints as an automaton whose state includes predictive timing information. Timing assumptions and timing requirements for the system are both represented in this way. A multi-valued mapping from the assumptions automaton to the requirements automaton is then used to show that the given system satisfies the requirements. One type of mapping is based on a collection of progress functions providing measures of progress toward timing goals. The technique is illustrated with two examples, a simple resource manager and a two-process race system. Nancy A. Lynch received the B.S. degree in mathematics from Brooklyn College, Brooklyn, NY, in 1968, and the Ph.D. degree in mathematics from the Massachusetts Institute of Technology, Cambridge, MA, in 1972. She is presently a professor of computer science and electrical engineering at Massachusetts Institute of Technology. She has also been on the computer science faculty at Georgia Institute of Technology and on the mathematics faculty at Tufts University and the University of Southern California. Her research interests are in distributed and real-time computing and theoretical computer science. In particular, she has worked on formal models and verification methods, on algorithm design and analysis, and on impossibility results. She also likes to hike and ski. Hagit Attiya received the B.Sc. degree in Mathematics and Computer Science from the Hebrew University of Jerusalem, in 1981, the M.Sc. and Ph.D. degrees in Computer Science from the Hebrew University of Jerusalem, in 1983 and 1987, respectively. She is presently a senior lecturer at the department of Computer Science at the Technion, Israel Institute of Technology. Prior to this, she has been a post-doctoral research associate at the Laboratory for Computer Science at M.I.T. Her general research interests are distributed computation and theoretical computer science. More specific interests include fault-tolerance, timing-based and asynchronous algorithms.This work was supported by ONR contracts N00014-85-K-0168 and N00014-91-J-1046, by NSF grants CCR-8611442 and CCR-8915206, and by DARPA contracts N00014-87-K-0825 and N00014-89-J-1988     

On deterministic approximation of DNF

We develop several quasi-polynomial-time deterministic algorithms for approximating the fraction of truth assignments that satisfy a disjunctive normal form formula. The most efficient algorithm computes for a given DNF formulaF onn variables withm clauses and > 0 an estimateY such that ¦Pr[F] –Y¦ in time which is , for any constant. Although the algorithms themselves are deterministic, their analysis is probabilistic and uses the notion of limited independence between random variables.Research supported in part by National Science Foundation Operating Grant CCR-9016468, National Science Foundation Operating Grant CCR-9304722, United States-Israel Binational Science Foundation Grant No. 89-00312, United States-Israel Binational Science Foundation Grant No. 92-00226, and ESPRIT Basic Research Grant EC-US 030.Research partially done while visiting the International Computer Science Institute and while at Carnegie Mellon University.     

Algebraic and functional specification of an interactive serializable database interface

Summary A formal functional specification of a serializable interface for an interactive database is given and refined into two different versions with distinct strategies for solving read/write conflicts. The formalization is based on techniques of algebraic specification for defining the basic data structures and functional system specification by streams and stream processing functions for defining the properties concerning interaction. It is especially demonstrated how different specification techniques can be used side by side. Manfred Broy finished his studies with the Diplom in Mathematics and Computer Science at the Technical University of Munich. Till 1983 he was research and teaching assistant at the Institut für Informatik and the Sonderforschungsbereich 49 Programmiertechnik. At the Technical University of Munich he also did his Ph.D. (in February 1980 with the subject: Transformation parallel ablaufender Programme) and qualified as an university lecturer (in 1982 with the subject: A Theory for Nondeterminism, Parallelism, Communication and Concurrency). In April 1983 he became a Full Professor for Computer Science at the Faculty of Mathematics and Computer Science at the University of Passau. Since October 1989 he has been Full Professor for Computer Science at the Technical University of Munich. His fields of interests are: Programming languages, program development, programming methodology and distributed systems.This work was supported by the DFG Project Transformation paralleler Programme and by the Sonderforschungsbereich 342 Werkzeuge und Methoden für die Nutzung paralleler Architekturen     

Extremal Relations between Additive Loss Functions and the Kolmogorov Complexity

Conditions are presented under which the maximum of the Kolmogorov complexity (algorithmic entropy) K(₁... _N ) is attained, given the cost f( _i ) of a message ₁... _N . Various extremal relations between the message cost and the Kolmogorov complexity are also considered; in particular, the minimization problem for the function f( _i ) – K(₁... _N ) is studied. Here, is a parameter, called the temperature by analogy with thermodynamics. We also study domains of small variation of this function.     

Synchronous condition-based consensus

The condition-based approach studies restrictions on the inputs to a distributed problem, called conditions, that facilitate its solution. Previous work considered mostly the asynchronous model of computation. This paper studies conditions for consensus in a synchronous system where processes can fail by crashing. It describes a full classification of conditions for consensus, establishing a continuum between the asynchronous and synchronous models, with the following hierarchy where includes all conditions (and in particular the trivial one made up of all possible input vectors). For a condition , we have:

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

In statistical analysis of measurement results it is often necessary to compute the range of the population variance when we only know the intervals of possible values of the x _i . While can be computed efficiently, the problem of computing is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute in time O(n · log(n)). In this paper we provide new algorithms that compute (in all cases) and (for the above case) in linear time O(n). Similar linear-time algorithms are described for computing the range of the entropy when we only know the intervals of possible values of probabilities p _i . In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute in n calls to ƒ.     

Distributed algorithms for constructing a minimum-weight spanning tree in a broadcast network

Two distributed algorithms are presented for a network using a common communication channel (e.g. radio) in which all nodes are within signal range and in line of sight of each other: (a) an algorithm to compute all internode distances (in terms of propagation delays) in the network. the algorithm requires only 2 messages per node, and provides each node with the distances to all other nodes. (b) An algorithm for constructing a minimum-weight spanning tree (MST) in such a network. This algorithm starts out with the information provided by (a) and ends with each node possessing the explicit knowledge of the full MST. The algorithm requires at most log₂ N messages per node. The internal processing in each node needsO(N logN) time andO(N) space. All messages required by (a) and (b) contain at most one edge weight plus 2 log₂ N bits. Some possible applications of the algorithms are: position-location, tuning acknowledgement time-out mechanisms, tuning the scheduling functions of access protocols that are sensitive to individual internode propagation delays, and selecting performance effective transmission sequences for round robin access protocols.Yaron I. Gold received the B.Sc. (Cum Laude, 1970) and M.Sc. (1975), both in Electrical Engineering, from the Technion, Israel institute of Technology, and the Ph.D. (1981) in Computer Science, from the University of Minnesota, Minneapolis.From 1970 to 1975 Yaron Gold served as Research and Development Officer in the Israeli Defense Forces, leading a group of several scientists, engineers and technicians. From 1982 to 1984 he was on the faculty of the Department of Electrical Engineering the Department of Electrical Engineering and Computer Science at the University of Connecticut. During that period he also served as consultant for United Technologies Corporation and for Battelle Laboratories. Presently, Dr. Gold is on the faculty of the Computer Science Department at the Technion.His research interests include Computer Networks and Communications, Simulation and Intelligent Systems.Shlomo Moran received the B.Sc. and D.Sc. degrees in mathematics from Technion, Israel Institute of Technology, Haifa, in 1975 and 1979, respectively.From 1979 to 1981 he was assistant professor and a visiting research specialist at the University of Minnesota, Minneapolis. From 1981 to 1985 he was a senior lecturer at the Department of Computer Science, Technion, and from 1985 to 1986 he was a World Trade visitor at IBM Thomas J. Watson Research Center, Yorktown Heights. Currently he is associate professor at the Department of Computer Science, Technion.His research interests include distributed algorithms, computational complexity, combinatorics, graph theory and geometric aspects of communication networks.This work was supported in part by NSF grant ECS-8307186Part of this research was done while this author was with the University of ConnecticutPart of this research was done while this author was with IBM, Thomas J. Watson Research Center     

On scheduling tasks with exponential service times and in-tree precedence constraints

Summary In this paper we extend the work of Chandy and Reynolds in which they considered the problem of scheduling tasks on two identical processors. The processing times of the tasks are independent, identically distributed exponential random variables. The tasks are subject to in-tree precedence constraints. Chandy and Reynolds have shown that the expected value of the makespan is minimized if and only if the scheduling policy is Highest-Level-First (HLF). We extend their result by showing that a policy maximizes the probability that all tasks finish by some time t0 if and only if the policy is HLF. Additionally, we show that a policy maximizes the probability that the sum of the finishing times of all the tasks is less than some value s0 if and only if the policy is HLF.This work has been partially supported by a Lady Davis Fellowship at the Technion, Haifa Israel; by the Research Institute for Advanced Computer Science at the NASA Ames Research Center, Moffett Field CA; and by NSF Grant MCS 80-04257.     

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients a _j of on are proportional to where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the Nth term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f(x)=x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence     

Thek-server dual and loose competitiveness for paging

Weighted caching is a generalization ofpaging in which the cost to evict an item depends on the item. We study both of these problems as restrictions of the well-knownk-server problem, which involves moving servers in a graph in response to requests so as to minimize the distance traveled.We give a deterministic on-line strategy for weighted caching that, on any sequence of requests, given a cache holdingk items, incurs a cost within a factor ofk/(k–h+1) of the minimum cost possible given a cache holdingh items. The strategy generalizes least recently used, one of the best paging strategies in practice. The analysis is a primal-dual analysis, the first for an on-line problem, exploiting the linear programming structure of thek-server problem.We introduceloose competitiveness, motivated by Sleator and Tarjan's complaint [ST] that the standard competitive ratios for paging strategies are too high. Ak-server strategy isloosely c(k)-competitive if, for any sequence, foralmost all k, the cost incurred by the strategy withk serverseither is no more thanc(k) times the minimum costor is insignificant.We show that certain paging strategies (including least recently used, and first in first out) that arek-competitive in the standard model are looselyc(k)-competitive providedc(k)/Ink and bothk/c(k) andc(k) are nondecreasing. We show that the marking algorithm, a randomized paging strategy that is (Ink)-competitive in the standard model, is looselyc(k)-competitive providedk–2 In Ink and both 2 Ink–c(k) andc(k) are nondecreasing.This paper is the journal version of On-line Caching as Cache Size Varies, which appeared in theProceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms (1991). Details beyond those in this paper may be found in Competitive Paging and Dual-Guided Algorithms for Weighted Caching and Matching, which is the author's thesis and is available as Technical Report CS-TR-348-91 from the Computer Science Department at Princeton University.This research was performed while the author was at the Computer Science Department, Princeton University, Princeton, NJ 08544, USA, and was supported by the Hertz Foundation.     

Use of potential functions in 3D rendering of fractal images from complex functions

Computer graphics is important in developing fractal images visualizing the Mandelbrot and Julia sets from a complex function. Computer rendering is a central tool for obtaining nice fractal images. We render 3D objects with the height of each complex point of a fractal image considering the diverging speed of its orbit. A potential function helps approximate this speed. We propose a new method for estimating the normal vector at the surface points given by a potential function. We consider two families of functions that exhibit interesting fractal images in a bounded region: a power function,f _{, c}(z)=z +c, where is a real number, and the Newton form of an equation, where ¦¦=1 and >0.     

Fast allocation and deallocation with an improved buddy system

We propose several modifications to the binary buddy system for managing dynamic allocation of memory blocks whose sizes are powers of two. The standard buddy system allocates and deallocates blocks in time in the worst case (and on an amortized basis), where n is the size of the memory. We present three schemes that improve the running time to O(1) time, where the time bound for deallocation is amortized for the first two schemes. The first scheme uses just one more word of memory than the standard buddy system, but may result in greater fragmentation than necessary. The second and third schemes have essentially the same fragmentation as the standard buddy system, and use bits of auxiliary storage, which is but for all and . Finally, we present simulation results estimating the effect of the excess fragmentation in the first scheme.Received: 4 May 2003, Published online: 22 December 2004Gerth Stølting Brodal: Supported by the Carlsberg Foundation (contract number ANS-0257/20). Partially supported by the Future and Emerging Technologies Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). Basic Research in Computer Science, www.brics.dk, funded by the Danish National Research Foundation.Erik D. Demaine: Partially supported by the Natural Science and Engineering Research Council of Canada (NSERC).J. Ian Munro: Supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the Canada Research Chair in Algorithm Design.This paper includes several results that appeared in preliminary form in the Proceedings of the 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FST & TCS99) [8].     

Exact Linear Thermodynamic Evolution Equations from the Robertson Formalism

A linear evolution equation for a thermodynamic variable F, odd under time-reversal, is obtained from the exact equation derived by Robertson from the Liouville equation for the information-theoretic phase-space distribution. One obtains an exact expression for , the relaxation time for F. For very short , is time-independent for t > if C(t) F{exp(-i t)}Fo, the equilibrium time correlation, decays exponentially for t > . is the Liouville operator. So long as C(t) is such that decays rapidly to a steady-state value, the t limit of agrees with the fluctuation-dissipation theorem in applications to fluid transport.     

A short note on simultaneous splitting

We show that for any alphabet there is a setL ^* such that ifC is any infinite co-infinite context-free language over , thenL splitsC (i.e., each ofL C,L , C, and is infinite).Preparation of this paper was supported in part by the National Science Foundation under Grant No. MCS77-11360.     

Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

Quantum Orlicz Spaces in Information Geometry

Let H₀ be a selfadjoint operator such that Tr is of trace class for some , and let denote the set of ε-bounded forms, i.e., for some 0 $$" align="middle" border="0"> . Let χ := Span . Let denote the underlying set of the quantum information manifold of states of the form . We show that if Tr ,

Biologically Inspired Autonomous Rover Control

Robotic missions beyond 2013 will likely be precursors to a manned habitat deployment on Mars. Such missions require robust control systems for long duration activities. Current single rover missions will evolve into deployment of multiple, heterogeneous cooperating robotic colonies. This paper describes the map-making memory and action selection mechanism of BISMARC ( iologically nspired ystem for ap-based utonomous over ontrol) that is currently under development at the Jet Propulsion Laboratory in Pasadena, CA (Huntsberger and Rose, Neutral Networks, 11(7/8):1497–1510). BISMARC is an integrated control system for long duration missions involving robots performing cooperative tasks.