Dimacs/005 DIMACS Series in Discrete Mathematic S and Theoretical Computer Scienc E

https://doi.org/10.1090/dimacs/005 DIMACS Series in Discrete Mathematic s and Theoretical Computer Scienc e

Volume 5

Reliability o f Compute r and Communicatio n Network s

Proceedings o f a DIMAC S Worksho p December 2-4 , 198 9

Fred Robert s Frank Hwan g Clyde Monm a Editors

NSF Scienc e an d Technolog y Cente r in Discret e Mathematic s an d Theoretica l Compute r Scienc e A consortiu m o f Rutger s University , Princeto n University , AT&T Bel l Labs , Bellcor e Th e DIMACS Workshop on Reliability of Compute r and Communicatio n Net- work s was held at Rutger s University , New Brunswick , New Jersey , on Decembe r 2-4, 1989.

198 0 Mathematics Subject Classification (1985 Revision). Primary 05-06, 05C99 , 68-06, 68M10 , 68M15 , 68R10 , 90-06 , 90B25 , 94-06 , 94C15 .

Librar y of Congres s Cataloging-in-Publicatio n Data Reliabilit y of compute r and communicatio n networks : proceeding s of a DIMAC S Workshop , Decembe r 2-4, 1989/Fre d Roberts , Fran k Hwang , Clyd e Monma , editors . p. cm.—(DIMACS serie s in discret e mathematic s and theoretica l compute r science ; ISS N 1052-1798 ; v. 5) Include s bibliographica l references . AMS : ISBN 0-8218-6592- 7 ACM ; ISBN 0-89791-387- 6 1. Compute r networks-Reliability-Congresses . I. Roberts , Fre d S. II. Hwang , Frank . III. Monma , Clyd e L. IV. America n Mathematica l Society . V. Associatio n for Computin g Machinery . VI. Series . TK5105.5.R44 5 1991 91-9953 004.6-dc2 0 CIP

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Preface v Program i x List o f Participant s x v Robust Desig n o f Dynamic Routin g Network s GERALD R . AS H 1 Graph Searching , Path-Width , Tree-Widt h an d Relate d Problem s (A Survey) DANIEL BIENSTOC K 3 3 On Residua l Connectednes s Networ k Reliabilit y F. BOESCH , A . SATYANARAYANA , AN D C . SUFFE L 5 1 Survivable Fibe r Network Desig n R. H . CARDWEL L 6 1 Decomposable Probabilisti c Influenc e Diagram s C. C . CHY U 7 9 Bounding Networ k Parameter s b y Approximating Graph s CHARLES J . COLBOUR N AN D EUGEN E I . LITVA K 9 1 The Optima l Multitermina l Cu t Proble m WILLIAM H . CUNNINGHA M 10 5 Polyhedral Approache s t o Network Survivabilit y M. GROTSCHEL , C . MONMA , AN D M . STOE R 12 1 A Surve y O n Doubl e Loo p Network s F. K . HWAN G 14 3 Quantitative Reliabilit y Analysi s o f Redundan t Multistag e Interconnection Network s NITA M . KINI , ANU P KUMAR , AN D DHARM A P . AGRAWA L 15 3 An Axiomatic Characterizatio n o f th e Reliabilit y Polynomia l RICHARD P . MCLEA N AN D DOUGLA S H . BLAI R 17 1

in IV CONTENTS

Fault-Tolerant VLS I Architectures Base d on d e Bruijn Graph s (Galileo in the Mid Nineties ) DHIRAJ K . PRADHA N 18 3 The Use of Binomial Moments fo r Boundin g Network Reliabilit y ANDRAS PREKOPA , ENDR E BOROS , AN D KEH-WE I LI H 19 7 Boolean Decomposition Scheme s and the Complexity o f Reliabilit y Computations J. SCOT T PR O VAN 21 3 On Dynamic Full Access in Multistage Interconnection Network s C. S . RAGHAVENDR A AN D M . A . SRIDHA R 22 9 Algebraic Methods fo r Boundin g Network Reliabilit y DOUGLAS R . SHIE R 24 5 Preface This volume contains original papers prepared followin g talks given at the DIMACS Worksho p o n Reliabilit y o f Compute r an d Communicatio n Net - works, held Decembe r 2-4 , 1989 , at Rutger s University i n New Brunswick , New Jersey . Al l o f th e paper s (excep t fo r Chyu' s whic h wil l als o appea r elsewhere) ar e in final form. Reliability problem s aris e wit h increasin g frequenc y a s ou r moder n sys - tems o f telecommunications, informatio n transmission , transportation , an d distribution becom e mor e an d mor e complex . Thi s workshop wa s designe d to analyz e th e discret e mathematica l method s whic h ar e relevan t t o thes e problems, to identify th e latest trends and important ope n problems, and to survey potentia l practica l applications , wit h a n emphasi s o n compute r an d communication networks . The worksho p analyze d bot h question s o f computatio n o f reliabilit y o f existing systems and questions dealing with the design of highly reliable systems to begin with . (Th e two questions ar e closel y related, sinc e one need s to know how to compute reliability in order to design a reliable system.) Th e workshop als o analyzed the closel y related notio n o f survivability. Networ k survivability mean s th e abilit y t o restor e "service " i n th e even t o f a catas - trophic failure o f a network component. W e also discussed ho w to build redundancy int o a network, ho w to define an d measur e redundancy , an d ho w redundancy relate s to reliability . Th e workshop deal t wit h both singl e stag e and multistage, interconnected networks, which have been studied in connec- tion with computer networks, and in particular with fault tolerance and other reliability issue s i n mind . Th e worksho p emphasize d practica l application s through invite d speaker s fro m a variety o f companie s tha t ar e dealin g wit h practical networ k reliabilit y problems . In all , ther e wer e 8 9 attendee s a t th e workshop , wit h th e averag e dail y attendance abou t 50 . Th e attendee s include d theoretica l mathematicians , computer scientists , and electrical engineers from academi a and industry, a s well a s networ k practitioners , engineers , an d reliabilit y planner s fro m suc h companies as AT&T Bell Labs, Bellcore, Pacific Bell, GTE, Bolt, Beranek and Newman, MITRE, and IBM. The worksho p starte d wit h a sessio n o n networ k reliability . Th e speak - ers i n tha t sessio n wer e Charle s Colbour n (Waterloo) , Scot t Pr o van (Nort h Carolina), an d Dougla s Shie r (Willia m an d Mary) . Thes e thre e speaker s summarized the traditional problem o f computing the reliability polynomia l of a grap h whe n ther e ca n b e edg e failures . Colbour n discusse d th e us e o f graph transformations t o obtain bounds o n the reliability. Prova n discusse d

V VI PREFACE two types of approximation scheme s for reliability, one using the property of shellability and the other using series-parallel and delta-wye transformations . Shier discussed algebrai c methods for bounding reliability . A second session on network reliability featured talk s by Richard McLea n (Rutgers) and Andras Prekopa (Rutgers) . McLean' s talk brought to the reliability problem the point of view of the economist, and in particular the game theorist, an d presente d axiom s whic h completel y characterize d th e reliabil - ity polynomial . Thi s poin t o f vie w wa s quit e nove l an d wa s receive d wit h great interest by the more traditional reliability theorists. Prekopa' s talk also presented a nontraditional approach, making use of some recently develope d lower and upper bounds fo r th e probability o f logical functions o f events to obtain lowe r bounds fo r the reliability o f a network . The first day concluded with a session on applications which featured talk s by A.G . (Sandy ) Frase r (AT& T Bel l Labs ) an d Pete r Kuba t (GT E Labora - tory). A third tal k i n thi s session , b y Mariann e Lep p (Bolt , Berane k an d Newman), wa s cancelled du e to the speaker' s illness . Thi s first of three applied session s a t th e worksho p wa s intende d t o hav e practitioner s presen t both important use s of theoretical reliability methods and new practical net - work reliability problems. Fraser' s talk dealt with congestio n an d instabilit y in data networks, and emphasized the new reliability problems posed by the large capacity o f fiber optic connections. Ther e wa s a great dea l o f interac - tion between the speaker and the audience, with many suggestion s made fo r possible approaches to the problems posed, and with many ideas for potential future reliabilit y researc h bein g discussed . Afte r th e session , i t wa s agree d that the talk was such a success that i t should be followed u p by a more de- tailed presentatio n t o a DIMACS group by a Bell Labs scientist. On e of th e positive aspects of Marianne Lepp's cancellation was that we were able to al- low this part of the session to continue for one hour instead o f the scheduled half hour. Kubat' s tal k emphasized a variety o f approaches to the desig n o f reliable an d qualit y telecommunication s system s fro m th e poin t o f vie w o f future cost s and als o resulted i n a lively discussion . The second day began with a session on reliability o f multistage network s and faul t tolerance . Th e sessio n wa s introduce d wit h a brie f summar y o f the topi c b y Anuja n Varm a (IBM) . Th e speaker s i n thi s sessio n wer e D.K . Pradhan (Massachusetts) , D.P . Agrawa l (Nort h Carolin a State) , an d C.S . Raghavendra (Souther n California) . Thi s sessio n feature d talk s b y electri - cal engineers/computer scientists , an d wa s a very nic e complemen t t o bot h the mathematically oriente d talks of the first day and the applied talks. On e of th e thing s whic h man y o f th e participant s pointe d t o a s a highligh t o f the meetin g wa s tha t w e go t togethe r s o man y divers e researcher s wh o d o not usuall y atten d th e sam e meetings . Pradha n talke d abou t a particula r multiprocessor networ k whic h i s o f grea t interes t today , th e d e Bruij n net - work. ( A later speaker, Jean-Claude Bermond, also talked about the de Bruijn network.) Agrawa l talke d abou t redundan t multistag e interconnectio n net - works (MIN's ) an d presente d a n algorith m fo r computin g thei r reliability . Raghavendra als o talked abou t MIN's and presented a variety o f technique s for achievin g fault-tolerance . PREFACE vn The next sessio n deal t wit h network survivability , th e problem o f how to design a network which had prescribed survivability characteristics, given the desire to build suc h a network a t minimu m cost . Th e speaker s i n thi s ses - sion wer e Marti n Grotsche l (Augsburg ) an d Danie l Bienstoc k (Columbia) . Grotschel's talk was a beautiful mi x of theory and practice. I t described how he and Clyde Monma had worked with Bellcore scientists (including Richard Cardwell, a networ k practitioner ) t o convinc e th e regiona l Bel l Telephon e companies tha t survivabilit y coul d b e built int o a network wit h smal l addi - tional cost . I t discusse d ho w they interacted wit h the practitioners t o defin e the precise mathematical problem , whic h the y then solve d usin g polyhedra l methods. Whe n one of the attendees questioned the seeming oversimplifica - tion in the network survivabilit y mode l used by Grotschel, he was reassured by Cardwell that thi s was exactly the appropriate mode l to use. Bienstock' s talk dealt with problems of searching a graph, seeking out an intruder by the use of guards. I t covered a variety o f important recen t developments havin g to do with such graphical parameters a s path-width an d tree-width . A second sessio n on network survivability featured talk s by William Cun - ningham (Carleto n University , Ottawa ) an d Fran k Boesc h (Stevens) . Cun - ningham discussed a variety of network attack problems, which are optimiza- tion problems in which one tries to minimize the difference between the effor t required by an attacker to destroy the edges in a subset o f a network and the resulting damag e t o th e network . Boesc h gav e a very nic e summar y o f th e node reliability problem, the reliability problem where nodes fail, rather than edges. The second da y ended wit h our secon d sessio n o n applications. Thi s session agai n featured practitioners , i n this cas e Gerald As h (AT& T Bell Labs) and Richar d Cardwel l (Bellcore) . A third talk , b y Neal Crysta l o f Bellcore , had to be cancelled because the speaker was unable to attend. As h described the 198 7 AT& T switc h fro m hierarchica l routin g o f call s throug h it s net - work to dynamic nonhierarchical routing . H e emphasized ho w much mone y AT&T had save d as a result and how much more reliable things had becom e as a result. I n answe r t o a question fro m th e audience , h e emphasize d th e important rol e tha t theoretica l development s ha d playe d i n th e decisio n t o make the switch to dynamic nonhierarchical routing, and in the design of the new system . H e cite d th e importan t contributio n o f a foundationa l pape r by Fan Chung , Ron Graham , an d Fran k Hwang , and th e significanc e o f th e development o f th e linea r programmin g algorith m o f Karmarkar . As h als o discussed futur e routin g methodologies an d the technical problems involve d in implementing them. Cardwel l talked about alternative network topologie s for building survivable fiber networks. H e emphasized a ring topology which many of the participants found ver y interesting. Ther e was considerable discussion abou t ho w t o balance of f th e initia l cos t saving s o f suc h a networ k with th e fac t tha t onc e the networ k reache s capacity, i t i s very difficul t an d expensive t o modify . Man y interestin g an d importan t theoretica l question s arose from thi s discussion . The third da y began wit h a sessio n o n redundancy an d reliability o f spe - cial networks . Th e fou r speaker s wer e Joel Cohe n (Rockefelle r University) , Vlll PREFACE

Nicholas Maxemchuk (AT& T Bell Labs), Philip Boland (Universit y College , Dublin), and C.Y . Chy u (Berkeley) . Cohe n talked about calculatin g the reliability o f a random grap h o r digraph, an d als o about calculatin g the redun - dancy a s measured b y the expecte d numbe r o f spannin g trees. H e explore d the relationship s betwee n th e two . Maxemchu k discusse d a particular net - work topology , th e Manhatta n stree t network , an d it s reliabilit y properties . Boland discusse d reliabilit y problem s i n whic h component s o f a system ar e arranged linearl y an d th e syste m fail s i f an d onl y i f k consecutiv e compo - nents fail. H e introduced positive dependence between adjacent component s and argued that the system reliability i s a decreasing function o f this dependence for k large . Chy u talked about influence diagrams, structured digraphs which are used in applications to medical decisionmaking, reliability analysis, etc. A secon d sessio n o n redundanc y an d reliabilit y o f specia l network s fea - tured tw o speakers , Jean-Claud e Bermon d (CNRS , Valbonne ) an d Fran k Hwang (AT& T Bell Labs). Bermon d continued the discussion o f special network topologie s b y discussin g reliabilit y propertie s o f d e Bruij n an d Kaut z graphs, and b y introducing various generalizations o f thes e graphs. H e sur - veyed both recen t result s and ope n problems. Hwan g discussed doubl e loo p networks, whic h ar e directe d circulants , an d whic h als o hav e a potentiall y useful ne w networ k topology . H e describe d a nove l metho d fo r obtainin g the most reliable double loop networks. H e also gave a routing algorithm fo r such networks. The final sessio n o f th e worksho p wa s the thir d sessio n o n applications . This feature d a tal k b y Yin g Chen g (AT& T Bel l Labs) . A secon d talk , b y Ralph Evan s (privat e consultant) , wa s cancelle d du e t o th e illnes s o f th e speaker. Cheng' s tal k deal t wit h AT M (asynchronou s transfe r mode ) net - works. H e use d algebrai c an d combinatoria l method s t o develo p a n erro r correction/error detectio n schem e in suc h networks. The workshop was a success because it fostered s o many new interactions between researcher s an d between researcher s an d practitioners . I t ha s als o already ha d a very visibl e an d dramati c impact . A number o f participant s from suc h companies as Pacific Bel l were so excited by the potential value of the methods fo r designin g highly survivable fiber optic networks which were discussed at the meeting, that they suggested that Bellcore arrange a follow-up meeting fo r Bellcor e employee s an d employee s o f th e Bel l system operatin g companies. A s a direc t resul t o f ou r DIMAC S activity , suc h a workshop , entitled "Telecommunication s Networ k Survivabilit y Symposium, " wa s or - ganized fo r Bellcor e an d th e Bellcor e clien t companies . On e o f us , Clyd e Monma, wa s a co-organizer. Th e response to the workshop wa s very enthu - siastic: ther e were over 20 0 participants an d the feedback fro m participant s was that the workshop wa s timely and important . The editor s hop e tha t th e foundation s lai d b y th e workshop , it s follow - up workshop , an d thi s volume wil l lead to continuin g development s i n thi s exciting and important field. Frank Hwan g Clyde Monm a Fred S . Roberts Program

Friday, December 1 , 1989

6:30-8:30 pm Registratio n

Saturday, December 2, 1989

7:30-8:30 Breakfas t 8:30-9:00 Registratio n and Coffe e 9:15-9:30 Openin g Chair Fre d S. Roberts, Rutgers Universit y Daniel Gorenstein , Director, DIMAC S Clyde Monma, Bellcore, for the Organizin g Committee

SESSION O N NETWORK RELIABILITY , PAR T 1 Chair Michael Ball, University o f Marylan d 9:30-10:00 Charles Colbourn, University o f Waterlo o Bounding Network Reliability by Graph Trans- formations 10:00-10:10 Discussion 10:10-10:40 J. Scot t Provan, University o f North Carolin a Approximating Two-Terminal Reliability in Graphs 10:40-10:50 Discussion 10:50-11:20 Coffee 11:20-11:50 Douglas Shier, College of William and Mar y Algebraic Methods for Bounding Reliability 11:50-12:00 Discussion 12:00-1:15 Lunch

SESSION O N NETWORK RELIABILITY , PAR T 2 Chair Endre Boros, Rutgers Universit y x PROGRA M

1:15-1:45 Richar d McLean, Rutgers Universit y A Characterization of the Reliability Polyno- mial for Polyhedral Network Problems 1:45-1:55 Discussio n 1:55-2:25 Andra s Prekopa, Rutgers Universit y The Use of Binomial Moments for Approxi- mate Network Reliability Calculation 2:25-2:35 Discussio n 2:35-3:05 Coffe e

SESSION O N APPLICATIONS , PAR T 1 Chair Michae l Rothkopf , Rutger s Universit y 3:05-3:35 A . G. Fraser, AT& T Bell Laboratories Congestion and Instability in Data Networks 3:35-3:45 Discussio n 3:45-4:15 Mariann e Lepp, Bolt, Beranek and Newma n Reliability in Internet Routing 4:15-4:25 Discussio n 4:25-4:55 Pete r Kubat, GTE Laborator y Network Reliability and Cost Effective Telecommunication System Design 4:55-5:05 Discussio n 5:15-6:15 Win e and chees e party 6:15-7:15 Dinne r

Sunday, December 3, 1989

SESSION O N RELIABILIT Y O F MULTISTAG E NETWORK S AND FAUL T TOLERANC E Chair Anuja n Varma, IBM 7:30-8:30 Breakfas t 8:45-9:15 D . K . Pradhan, University o f Massachusett s The de Bruijn Multiprocessor Network: A Fault- Tolerant Versatile Parallel Processing Network 9:15-9:25 Discussio n 9:25-9:55 D . P. Agrawal, North Carolin a Stat e Univer - sity Reliability Evaluation of Redundant Multistage Interconnection Networks PROGRAM

9:55-10:05 Discussion 10:05-10:35 Coffee 10:35-11:05 C. S. Raghavendra, University o f Souther n California Reliability and Fault-Tolerance in Multi-Stage Interconnection Networks 11:05-11:15 Discussion

SESSION O N NETWOR K SURVIVABILITY , PART 1 Chair Danie l Kleitman, MI T 11:15-11:45 Marti n Grotschel , University o f Augsburg Polyhedral Approaches to Network Survivability 11:45-l 1:55 Discussio n 11:55-12:25 Danie l Bienstock, Columbi a Universit y Graph Searching, Tree-Width and Path-Width 12:25-12:35 Discussio n 12:35-2:00 Lunc h

SESSION O N NETWOR K SURVIVABILITY , PAR T 2 Chair Richar d Va n Slyke, Polytechnic Institute o f New York 2:00-2:30 Willia m Cunningham, Carleto n Universit y Network Attack Problems 2:30-2:40 Discussio n 2:40-3:10 Fran k Boesch , Stevens Institute o f Technolog y The Residual Node Reliability Problem 3:10-3:20 Discussio n 3:20-3:50 Coffe e

SESSION O N APPLICATIONS , PAR T 2 Chair Dvor a Tzvieli, AT&T Bell Laboratories 3:50-4:20 Geral d Ash, AT&T Bell Laboratorie s Dynamic Routing in Telecommunications Net- works: Improved Robustness at Lower Cost 4:20-4:30 Discussio n 4:30-5:00 Richar d Cardwell , Bellcor e Survivable Fiber Network Design Xll PROGRAM

5:00-5:10 Discussio n 6:00-7:00 Dinne r

Monday, December 4, 1989

SESSION O N REDUNDANC Y AN D RELIABILIT Y OF SPECIA L NETWORKS, PAR T 1 Chair Chin-Lin I, AT&T Bell Laboratories 7:30-8:30 Breakfast 9:00-9:30 Joel Cohen, Rockefeller Universit y Reliability and Redundancy of Anisotropic Random Graphs and Random Directed Graphs 9:30-9:40 Discussion 9:40-10:10 Nicholas F. Maxemchuk, AT& T Bell Laborato- ries Failure Mechanisms in the Manhattan Street Network 10:10-10:20 Discussion 10:20-10:50 Coffee 10:50-11:20 Philip Boland, University College , Dublin Linear Dependence in Consecutive k out ofn : F Systems 11:20-11:30 Discussion 11:30-12:00 C. C. Chyu, University o f California, Berkele y Decomposable Probabilistic Influence Diagrams 12:00-12:10 Discussion 12:10-1:15 Lunch

SESSION O N REDUNDANC Y AN D RELIABILIT Y OF SPECIA L NETWORKS, PAR T 2 Chair Wen-Ching Winni e Li, Pennsylvania Stat e University 1:15-1:45 Jean-Claude Bermond , CNRS, Valbonne Reliability ofde Bruijn and Similar Networks 1:45-1:55 Discussion 1:55-2:25 Frank Hwang, AT&T Bell Laboratories Reliabilities of and Fault Tolerant Routing for Double Loop Networks 2:25-2:35 Discussion 2:35-2:55 Coffee PROGRAM xm

SESSION O N APPLICATIONS , PAR T 3 Chair Joh n Healey, Bellcor e 2:55-3:25 Yin g Cheng, AT&T Bell Laboratories An Error Correction/Detection Scheme in ATM Network 3:25-3:35 Discussio n 3:35-3:40 Closin g This page intentionally left blank List of Participants

AGRAWAL, Dharma P. Department o f Electrical and Computer Engineering , North Carolina Stat e University, Raleigh , NC ASH, Gerald AT&T Bell Laboratories, Holmdel, NJ AYANOGLU, Ende r AT&T Bell Laboratories, Holmdel, NJ BAGCHI, Ansuma n RUTCOR, Rutger s University, Ne w Brunswick, NJ BALL, Michael College of Business Management , University o f Maryland, Colleg e Park, M D BAUER, Dougla s Department o f Mathematics, Steven s Institute of Technology, Hoboken , NJ BENZAKEN, Claud e University o f Scienc e and Medicine , Grenoble, France BERMOND, Jean-Claud e CNRS, Valbonne, Franc e BIENSTOCK, Danie l Department o f Operations Research , Columbia University, New York, NY BITAR, Youaki n Department o f Computer Science , Princeton University, Princeton, NJ BLAIR, Dougla s Department o f Economics an d RUTCOR, Rutger s University, New Brunswick, N J BLOOM, Gar y Department o f Mathematics, Cit y University o f New York, New York, NY BOESCH, Francis T. Stevens Institute o f Technology , Hoboken, N J BOLAND, Phili p Department o f Mathematics, University College , Belfield, Dublin , Ireland

xviii LIS T O F PARTICIPANTS

PREKOPA, Andra s RUTCOR, Rutger s University, Ne w Brunswick, NJ PROVAN, J. Scot t Department o f Operations Research , University o f North Carolina , Chapel Hill, NC RAGHAVENDRA, C . S. Department o f Electrical Engineerin g Systems, University o f Souther n California, Lo s Angeles, CA REIBMAN, Andre w AT&T Bell Laboratories, Holmdel, NJ ROBERTS, Fred S . DIMACS, Rutgers University, Ne w Brunswick, N J ROTHBLUM, Urie l RUTCOR, Rutger s University, Ne w Brunswick, N J ROTHKOPF, Michae l RUTCOR, Rutger s University, Ne w Brunswick, NJ SACCOMAN, Joh n Stevens Institute o f Technology , Hoboken, NJ SAKAI, Denise RUTCOR, Rutger s University, Ne w Brunswick, N J SALIZKIY, Olg a Hofstra University , Hempstead, N Y SERPANOS, D. Department o f Computer Science , Princeton University , Princeton, N J SHA, Hsing-Mea n Department o f Computer Science , Princeton University , Princeton, N J SHAMIR, Ro n DIMACS, Rutgers University, Ne w Brunswick, N J SHAPIRO, John Fordham University , Bronx , NY SHULMAN, A . GTE Laboratories Inc., Waltham, M A SHIER, Dougla s Department o f Mathematics, Colleg e of William and Mary, Williamsburg , VA SQUIER, Richar d Department o f Computer Science , Princeton University , Princeton , N J SUFFEL, Charle s Stevens Institute o f Technology , Hoboken, N J TEWARI, Raji v Department o f Computer Science , Rutgers University, New Brunswick , NJ TONG, Yun g L. School of Mathematics, Georgi a Institute o f Technology, Atlanta , GA LIST O F PARTICIPANTS xi x

TONG, X . Department o f Computer Science , Rutgers University, Ne w Brunswick , NJ TRALDI, Lorenz o Department o f Mathematics, Lafayett e College, Easton, PA TZVIELI, Dvor a AT&T Bell Laboratories, Holmdel, NJ VAN SLYKE, Richard Polytechnic Institute o f New York, Brooklyn, NY VARMA, Anuja n IBM, T. J. Watson Research Center , Yorktown Heights, NY WANG, Ch i RUTCOR, Rutger s University, Ne w Brunswick, N J WANG, Hui-Y u Department o f Computer Science , Rutgers University, New Brunswick , NJ WILLIAMS, Albert RUTCOR, Rutger s University, Ne w Brunswick, NJ WINKLER, Pete r Bellcore, Morristown, N J ZHENG, Maoli n RUTCOR, Rutger s University, Ne w Brunswick, N J ZHOU, Ming-Kun g City University o f New York, New York, NY ZIMMERMAN, Gu y Michigan Stat e University, Eas t Lansing, MI This page intentionally left blank