AD-A252 552

THE ONR WORKSHOP ONDISCRETE STRUCTURES IN CLASSIFICATION

05 MAY- 06 MAY 1992

RAMADA RENAISSANCE HOTELWASHINGTON-DULLES

PROCEEDINGS

DTICELECTE

_______JUL 0 119921This document has been -approved 8 ADfor public rclease and sale; itsdistribution is unlimited.

Organized by: F.R. McMorris, University of LouisvilleMarc J. Lipman, Office of Naval Research

GRANT: N00014-92-J-1574R&T: 4111066---01

92-1714092 6 " IIINhImIIIII

DISCLAIMER NOTICE

THIS DOCUMENT IS BEST

QUALITY AVAILABLE. THE COPY

FURNISHED TO DTIC CONTAINED

A SIGNIFICANT NUMBER OF

PAGES WHICH DO NOT

REPRODUCE LEGIBLY.

PREAMBLE

The objective of this workshop was to determine the state-of-the-art in Discrete Structured Classification, and chart new directions ofresearch relevant to Navy needs. Thus it had two goals. The first wasfor the Principal Investigators currently funded by ONR in this core areaof Discrete Mathematics to inform each other and their Scientific Officerof their current research activities. The second was for the PIs and Navyscientists to meet each other and exchange problems and ideas with thehope of developing significant research partnerships.

The first goal was grandly achieved by direct PI presentations.Progress on the second was made through the problem session androundtable discussions.

Accesion For

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Statement A per telecon Marc LipmanONR/Code 1111Arlington, VA 22217-5000

NWW 6/30/92

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ATTENDEES

Dr. W.O. AlltopCode 3807Applied Mathematics GroupNaval Air Weapons CenterChina Lake, CA 93555-6001

Dr. Gary ChartrandDepartment of Mathematics and StatisticsWestern Michigan UniversityKalamazoo, MI 49008

* Dr. Donald W. DearholtDepartment of Computer ScienceMississippi State UniversityMississippi State, MS 39762

Dr. Pierre HansenGERAD and lcole des Hautes 6tudes CommercialesDepartement des M~thodes Quantitatives et Systames d'Information5255 avenue DecellesMontr6al, Quebec H3T IV6Canada

Dr. M.F. JanowitzDepartment of Mathematics and StatisticsUniversity of MassachusettsAmherst, MA 01003

Dr. Brigitte JaumardGERAD and lcole Polytechnique de MontrdalDdpartement de Math~matiques Appliqu6esSuccursale A, Case Postale 6079Montr6al, Quebec H3C 3A7Canada

Dr. Marc J. LipmanCode 1111Office of Naval Research800 N. Quincy St.Arlington, VA 22217

Dr. F.R. McMorrisDepartment of MathematicsUniversity of LouisvilleLouisville, KY 40292

Dr. K.B. ReidDepartment of MathematicsCalifornia State UniversitySan Marcos, CA 92069

Dr. Fred S. RobertsDepartment of MathematicsRutgers UniversityNew Brunswick, NJ 08903

. Dr. Uriel G. RothblumRUTCORRutgers UniversityNew Brunswick, NJ 08903

Dr. Ralph WachterCode 1133Office of Naval Research800 N. Quincy St.Arlington, VA 22217

ONR WORKSHOP ON

DISCRETE STRUCTURED CLASSIFICATION

Tuesday 5 May

0900 MARC LIPMAN

0930 BUCK McMORRIS

1030 PIERRE HANSENBRIGITTE JAUMARD

1200 Lunch

1300 BROOKS REID

1400 DON DEARHOLT

1500 Break

1515 Problem session - MARC LIPMAN

1600 Roundtable

1800 Conference Dinner

Wednesday 6 May

0900 MARC LIPMAN

0930 GARY CHARTRAND

1030 break

1100 URIEL ROTHBLUM

1200 Lunch

1300 MEL JANOWITZ

1400 Roundtable

1500 Wrap

Sm

THE FOLLOWING PAGES CONSIST OF ABSTRACTS ANDCOPIES OF THE TRANSPARENCIES USED FOR THEPRESENTATIONS.

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An axiomatic approach to the aggregationof trees and other discrete structures

F.R. McMorrisDepartment of Mathematics

University of Louisville

Suppose several classifications (usually structured as trees) have beenconstructed for the same set of objects. How can we (why should we) forma "consensus" classification that captures the common agreement of theoriginal classifications? During the past twenty years many methods havebeen developed relevant to the comparison and consensus of classifications.One approach based on the central ideas of Social Choice Theory pioneeredby K. Arrow, has been extended to produce a general axiomatic model toaddress the above question. This talk will present an overview of thisapproach and indicate some recent results.

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ICLUSTER ANALYSIS ALGORITHMS

Pierre HansenGERAD and Ecole des Hautes Etudes Commerciales

D6partement des Mthodes Quantitatives et Syst~mes d'Information5255 avenue Decelles

Montreal, Canada H3T 1V6

Brigitte JaumardGERAD and Ecole Polytechnique de Montreal

D(partement de Math~matiques Appliqu(esSuccursale A, Case Postale 6079

Montr6al, Canada H3C 3A7

Cluster analysis addresses the following very general problem: given a set of entities.find subsets of it, called clusters, which are homogeneous. i.e., such that entities withinthe same cluster resemble one another, and/or well separated. i.e.. such that entities indifferent clusters differ one from the other.

From the late 60's, Mathematical Programming has been applied to cluster analysis.This allowed to:

(i) formulate precisely many problems of cluster analysis as mathematical programs:i.e., optimization problems with an explicit objective function and constraints:

(ii) study the computational complexity of these problems;

(iii) obtain new polynomial algorithms for easy problems, with a low. and sometimeslowest possible, complexity by careful study of each of their steps and of the datastructures necessary to their implementation;

(iv) obtain new and practically efficient algorithms for NP-hard problems;

(v) derive theoretical properties of existing or new algorithms;

(vi) make new clustering methods available to researchers in various fields in the formof computer packages.

)I

Our work in the last few years and planned for follows and extends such lines. We exploretwo avenues of research. On one hand, we define several new problems of cluster analysisin the areas of divisive hierarchical clustering, partitioning, constrained partitioning andclustering with asymmetric dissimilarities. Each time, complexity issues are or will beexplored and new exact algorithms designed (as well as heuristics in the cases where exactsolution is too time-consuming). The main tools used will be graph theory, combinatorialoptimization (mainly nonlinear 0-1 programming) as well as, for the first time in clusteranalysis, exact methods of global optimization.

On the other hand, we complement the study of algorithms by analysis of the stepspreceding and following their use: ways to construct dissimilarity indices, automatedmethods to select "best" partitions among efficient ones, derivation of robustness andsensitivity concepts for clusters and partitions, new and more informative ways to rep-resent partitions and hierarchies. This work will be done in parallel with the analysis ofseveral data sets for real world problems.

The first part of the presentation will review steps of a cluster analysis study and il-lustrate the mathematical programming approach. Bicriterion cluster analysis with thesplit and diameter criteria will be used for this purpose. The second part will reviewrecent work on average linkage divisive hierarchical clustering, maximum split clusteringwith connectivity constraints, the Weber problem on the sphere and using espaliers torepresent results.

2

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Table 2. 24 psychlogi~cal tests ( A fA I

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4. Flags 2

5. General Information 2

6. Paragr Compreihension 3

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9. Word Maning 2

10. Adition e e

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19. Figure-Word

20. Deduction 5

21. Numerical Puzzles 5

22. Problem Reasoning 5

23. Series Completion 5

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258

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generated problems similar to those of the previous series of experiments are presented in Tables 9and 10. It appears that: (i) HTABU gives slightly better values than HDALC; (ii) computing timesof HTABU are smaller than those of HDALC. Therefore, a direct application of Tabu Search tothe hyperbolic Problem (P) appears to be preferable to the Tabu Search heuristic solution of thesequence of quadratic 0-1 problems given by Dinkelbach's lemma (1967).

BJ HDAL.C DALC

Pulkwa D. 77. M". D,, . M" D. MDri.F77, 7N -9 M 7w ?a 1 ,'

Pg 7 764 73 r13 r73 7w 43P 7W asr To Tn = ?l30 77 623

PS 4 .2 7sa S2 732 a W 2 542P6 S32 732 S L12 ,' SMQ W2 7=3 S32PF46S 7*7 46 465 717 4" 4"s 1- "1s 4' Ot"re. 43 -ok 439 436 704 436 4319 IM 4

pic =3 'm2 330 333 OMg 333 = O "033

tclu (- :6_) 0.01 0.02 0.01 0.10 0.11 0.11 0.31 0.32 032

Table 5: Comparison of heuristics and exact algorithm (13 psychological tests of Harman)

BJ HDALCPaition Dun V,, MD,, Don 'Dm MD.

P, 40 2 4062 40 2 4 2 4 J

P2 IM 3516 I3 2664 434 2664Pg 175 303 1755 256 37n2 No IAA6 66

P 130 3M 130 36 713Ps 131313 334 3M 34

P. 1W 304 ISIS 20 3156 264t ION 3116 IM 2216 3196 264

A3 106 3274 I=6 2112 3025 264Pg 1069 3144 1099 2M 25 24 2uPIo M 3001 M8 193 27W 28

Pic 706 2W66 706 1506 2272 223Pic 447 2653 447 719 127 141

0c.2 (0d) o. 1 0.451 0.43 4.96 151.04 51.02

Table 6: Heuristic solutions for Fisher's iris data (N=150)

14

AVERAGE DISSRIALARTIES

BJ HDALC QDAL; HQDLC

- I- 1706W 1 172412 173145 169610 172373 *T2JY 13 921 172711 1T2991 1707993 17028 170654 170257 171972 171967 164696 171990 172063 164472 1 rM66 171329 161924

100 4 19666 170743 16M66 171591 172201 13344 171644 172401 !63719 173036 171449 15756S 169532 N 1637 1635O32 171034 172421 161765 171183 172821 13473 17232 37115 1539

10 167470 ITO407 167470 16640 172107 16027 164 172324 159029 170146 170919 14022115 164597 170316 164557 166965 171470 154504 16546 171527 153028 Ii310 IT79 1249962 1410 103 171489 173W IS:W 7 171659 1YT5 171480 17240, T1OiW 1491763 171456 174515 16615 171462 173350 i474 171458 174515 166151 I7303S 17.1507 14364

200 4 171290 172602 154544 171221 17500 1512 171299 172602 1S4554 172231 17-2115 1358S 171067 I 3663 171552 1=3340 15733 171067 172697 I363 171274 172390 1563910 170031 171616 143491 166725 172011 147302 17001 171686 143491 16977. 171711 14703515 169642 171329 140615 166041 171306 145602 i6642 171329 140815 149154 1#006 1319967 14 1U, 4 11"17 11s" 147231 r117 1T23W8 1-T7W 14347 17126 1676823 1635$7 169594 109357 171417 17237 167100 171027 173107 168273 171600 172122 166424

300 4 169259 1697 1699 170747 171754 163740 170730 17246 16525,4 171410 172369 1500055 16M 169P.41 16066 1703M 172102 183619 170616 172466 155251 171976 171855 IS407210 1662= 16943 168270 19612 171573 153331 Io7 171906 143663 171245 171661 13"79515 16755 169366 167556 16"444 171069 150221 163672 171054 136313 169601 170923 126190

- 1 W8 1W0372 1U!H T0 IB4 ! 144 1"1'W 1723 171i14 122 ITY'1!4 170O489

3 163227 169346 163227 171302 169744 150428 170645 172536 166749 171417 I7076 161514400 4 1660 160324 19110 171275 170496 16SW.2 170452 171111 155441 1-0413 1 70433 161028

S386924 160320 1l24 172823 170460 153301 1702 17099 15726 1700 170296 15306510 148424 169264 166424 16S644 17052 142275 169731 170970 140173 171373 170210 1335515 167737 1 63221 167737 16P28" 170371 1396"6 162 1n693 13729 163422 170171 121299I lTWY -NOW 139113 7T33 715W -Tom 1171M 171917I T 1s 1 U 11"1(138 i0m3 160131 163235 10131 170567 171630 10770 170373 170162 IS96M I7IO I,0044 161369

S0 4 1 = 16 32 160M36 170373 171235 156738 170233 170816 1566 16996# 169353 1525315 1662 140196 1827 170731 171272 149645 170115 171044 I11156 171144 166624 14582110 166485 169170 166485 19M 170748 138304 149532 170244 12037S 166923 1630 135575

15 1 167941 19140 167941 169031 I170404 13756 166GM 170350 125766 169661 169221 126275

AVERAGE TIME

BJ HDALoC QDALC HQDAL

N D. . MD D.. V . MD. D.. 7. MD... D,. Ua. MD..-1UF -- M -- FI -- OrT -ii"Or -io 10.91 31.23 W15 U F

1.43 1.53 1.49 47.75 54.9 54-97 144.36 149.99 149.9S 139.73 144.14 143.592.14 3.37 3.22 67.21 5.94 6.84 233.46 256.61 256.50 211.8 229.12 229-0

,0 3a 6.37 ,S 130.12 163.23 183.17 342.13 384.36 384.24 326.01 364.84 364.78M 1 6.17 9.3 6.42 147.03 242.93 241.M1 602.21 679.18 6-3.06 5.00 661.33 6 0.19

Table 9: Results for large problems

17

S

Image Representation, Generalized Clustering, and Search in

Proximity Graphs and Pathfinder Networks

Donald W. DearholtDepartment of Computer ScienceMississippi State University.MS 39762

e-mail address: dearholt@cs.msstate.eduPhone: 601-325-2756

Abstract

Pathfinder associative networks (PFNETs) were originated to model humansemantic memory, and have proven particularly effective with associativeaspects of the organization of knowledge. Theoretical connections havebeen established with graph theory, path algebras, proximity graphs,computational geometry, clustering, and search procedures. The PFNET paradigmis being used in human-computer interface systems, and in the organization ofdata in a database for an experimental robotic vision system. Characteristicsof PFNETs include (1) the preservation of minimum-distance paths betweenentities, (2) the clustering of similar entities through the edge structure.(3) the consequent support of higher levels of abstraction, and (4) thecapability of generating proximity graphs, such as the relative neighborhoodgraph and the (open lune) gabriel graph. Monotonic search networks (MSNETs).closely related to PFNETs, provide for search in which no backtracking is evernecessary in domains having (objective) distance measures, and also support theclustering features of PFNETs. These studies have motivated investigations ofcluster learning and conceptual clustering from the perspective of primitivetransformations, with representation of the clusterings by means of PFNETs.Models of cluster learning in which the clusters represent either ordered orunordered sets, and which may or may not overlap, are being considered. Thesequential application of transformations on clusters to a graph-generationalgorithm is being considered as a learning paradigm, and this perspectiveappears to support a constructive view of clustering. Co-occurrence ofentities is an essential component of the process.

PROPERTIES AND APPLICATIONS OF

PATHFINDER-BASED ASSOCIATIVE NETWORKS

Don Dearholt

DEPARTMENT OF COMPUTER SCIENCE

MISSISSIPPI STATE UNIVERSITY

Developed under the sponsorship of:

AIR FORCE HUMAN RESOURCES LABORATORY

INSTRUMENTATION DIRECTORATE,

WHITE SANDS MISSILE RANGE

COMPUTING RESEARCH LABORATORY, NMSU

INTERNATIONAL BUSINESS MACHINES, INC.

NATIONAL SCIENCE FOUNDATION

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

OFFICE OF NAVAL RESEARCH

S

I

PHILOSOPHICAL STANCE: BETTER MODELING OF HUMANINTELLIGENCE WILL LEAD TO BETTER Al

THE NETWORKS WE ARE STUDYING:

DESCRIBE, SUMMARIZE, AND DISPLAY DATA

SUGGEST A PSYCHOLOGICAL MODEL ABOUT

MENTAL REPRESENTATIONS

COMPLEMENT MDS AND CLUSTER ANALYSIS

PROVIDE A PARADIGM FOR:

KNOWLEDGE REPRESENTATION

MODELS OF CLASSIFICATION

ORGANIZATION OF DATABASE SYSTEMS

SPREADING ACTIVATION (SEARCH)

D

p

OUTLINE

I. MOTIVATION, PERSPECTIVE, AND OBJECTIVES

II. PATHFINDER NETWORKS

A. DEFINITIONS AND PROPERTIES

B. APPLICATIONS

* III. CLUSTER LEARNING AND DYNAMIC SYSTEMS

A. MOTIVATIONS AND APPROACH

B. DEFLNITIONS AND PROPERTIES

C. APPLICATIONS

I

* RESEARCH OBJECTIVES

I. THEORETICAL

DEVELOP AND TEST METRICS

RELATIONSHIPS:

GRAPH THEORY

PATH ALGEBRAS

PROXIMITY GRAPHS (RNG, GG, DTG)

LEVELS OF ABSTRACTION

I. EMPIRICAL

SEMANTIC MEMORY

CLASSIFICATION MODELS

PROPOSITIONAL ANALYSIS

KNOWLEDGE EXTRACTION FROM EXPERTS

III. APPLICATION DOMAINS

ORGANIZATION OF CONCEPTS

INTERFACES--INFORMATION RETRIEVAL, HELP SYSTEMS

DATABASE ORGANIZATION

PERCEPTION--OUTLINES OF OBJECTS

p

THE BIGGEST CHALLENGE

FOR Al AND COGNITIVE MODELING:

TO DESIGN A SYSTEM WHICH DOES MANY THINGS WELL,

ALTHOUGH EACH ALGORITHM MIGHT NOT BE OPTIMAL

ASSOCIATIONAL ORGANIZATION

CLUSTERING

"p SEVERAL LEVELS OF ABSTRACTION

CLASSIFICATION

SEARCH

DESCRIPTION OF DECISIONS

p

I

DEFINITION

A PATHFINDER NETWORK (PFNET) IS A GRAPH BASED ON(

PAIRWISE ESTIMATES OR MEASURES OF DISTANCES

BETWEEN ENTITIES.

EACH ENTITY CORRESPONDS TO A NODE.

EACH PAIR OF NODES IN A PFNET IS CONNECTED DIRECTLY

BY AN EDGE WHOSE WEIGHT IS THE DISTANCE BETWEEN

THE TWO ENTITIES,

UNLESS THERE IS A SHORTER ALTERNATIVE PATH.

I

EXAMPLE OF A LABELED PFNTT

SECMCST 3 37SECo. MSTBPENGUIN

4 PR!,o MCSTA

HAWK GOOSESECo MCSTA

DIRECTED PFNET FOR NINE COUNTRIES

p

14

RUSI

FRANCE

NGLAND

USA CUBA

27 MEXICO 1

17JAMAICA

R-METRIC IS INFINITY

G-PARAMETER IS EIGHT

LDOCE: AMBIGUITY RESOLUTION

totallarge (orw

size amount sum

exchange cheque band o ioec

s a e aoss

TOUCHSCREEN DISPLAY FOR EMPIRICAL DATA

HOW SIVLAR IS A TO B?

A

1t

B

OK +

DISTANCE + SIMILARITY --- K

THE PARAMETERS OF A PFNET

R-METRIC:

RULE FOR FINDING THE LENGTH OF A PATH WITH KEDGES

K

L(P) WR]IIR

1=1

R PATH LENGTH DATA SCALE

I SUM OF WEIGHTS RATIO

2 EUCLIDEAN RATIO

00 MAXIMUM WEIGHT RATIO, ORDINAL

0

THE PARAMETERS OF A PFNET

Q-PARAMETER:

"DIMENSION" OF GENERALIZED TRIANGLE INEQUALITIES SATISFIED

B

A1 C* I

D

A <[BR-CR+DR]I lR

0

THE TRIANGLE INEQUALITY

B

Ai E C

D

E B +C

A <E +D B +C +D

THE GENERALIZED TRIANGLE INEQUALITY

-- B

A c

D

A _ [BR+CR+DR]IlR

PURPOSE: TO PRESERVE MINIMAL-DISTANCE PATHS

p

THEORETICAL RESULTS

FOR A GIVEN DISTANCE MATRIX,

PFNET(R ,Q):

IS UNIQUE,

PRESERVES GEODETIC DISTANCES,

LINKS NEAREST NEIGHBORS, AND

CONTAINS THE SAME INFOPMATION AS THE

MINIMUM METHOD OF HIERARCHICAL CLUSTERING

PFNET(R =-,,Q =N-l) IS THE UNION OF ALL MINTREES

PFNET(R 2,Q) IS A SPANNING SUBGRAPH OF PFNET(R I,Q)

IFF R 1_R 2

PFNET(R ,Q 2) IS A SPANNING SUBGRAPH OF PFNET(R ,Q 1)

1FF Q 1<Q 2

MONOTONIC TRANSFORMATIONS PRESERVE

STRUCTURE FOR ALL PFNET(R =oo,Q)

MULTIPLICATIVE TRANSFORMATIONS PRESERVE

STRUCTURE FOR ALL PFNET(R, Q)

S

S

~1

~i1

S

0 CO 0

A M

-n -n un 'n-

.-- ~ 0

E- ..- v~ LIC)

w- 4)O A2

C)3

on -LLu~ .. ~-.n c

VZ 0T c) r-

o C3C:)W~* II 0

so C.

C _D

co1

1 0am

IW 4w -

00-0

Op" M

*0v

COMPUTER VISION

GOAL: SCAN THE ENVIRONMENT AND MAKE DECISIONS

WITHOUT HUMAN INTERACTION

REQUIRES: KNOWLEDGE REPRESENTATION

CLASSIFICATION

, ABILITY TO DESCRIBE SCENE

RECONSTRUCT SCENE

ENHANCE SCENE

MODIFY SCENE

IA

a L

m 4

0-

0 94-LL 0

0

4-

zz

u-a

00

02 CD

QE

Ms0Io

A HIERARCHY OF

EMPTY NEIGHBORHOOD GRAPHS

PFNET (L 2, oo, N-1)

RNG (L 2 ) = PFNET (L 2' oo 2)

PFNET (L 2, r, 2)

GG (L 2 ) MSNET(L 2 )

MGG (L 2 ) = PENET (L 2' 2, 2)/

DTG (L 2)

PFNET(L 2, 1, 2)

EACH GRAPH IS A SPANNING SUBGRAPH

OF THE GRAPH BELOW IT

IC

oo<>

II

I

U

U

0

0

I -

U

/

I

I

p 7/

bQQ

~o-o/I

/ C>\OQ//

/

I /

~-JLJ

0

~42~ ________~-fKu~A/

/ z

//'r~fl

it ~j

U

,,/I _____ ii

MOTIVATIONS FOR STUDY OF CLUSTERINGS

LEARNING VIA CLUSTERINGS

MODEL DYNAMIC SYSTEMS

UTILIZE PATHFINDER NETWORKS

ASSOCIATIVITY

CLUSTERING

LEVELS OF ABSTRACTION

p

CONCEPT/CLUSTER LEARNING MODEL

SHOULD SUPPORT:

SETS OF CLUSTERS AS THE DOMAIN

PROGRESSION THRU CLUSTERINGS AS LEARNING OCCURS

p A DISTANCE MEASURE BETWEEN CLUSTERINGS

INTERPRETATION WITHIN THE PATHFINDER PARADIGM

p

I

MODEL FOR DYNAMIC SYSTEMS

SHOULD SUPPORT:

SETS OF CLUSTERS AS THE DOMAIN

PROGRESSION THRU CLUSTERINGS AS SYSTEM CHANGES

A DISTANCE MEASURE BETWEEN CLUSTERINGS

PATHFINDER OR PROXIMITY GRAPH PARADIGM

I

ADVANTAGES

PSYCHOLOGICAL FOUNDATIONS

REPETITION OR REHEARSAL CAN BE MODELED

REPRESENTATIONS AS ASSOCIATIVE GRAPHS

KNOWN GRAPH-THEORETIC PROPERTIES

VERIDICAL GRAPHICAL DISPLAY

I

ASSUMPTIONS

NOTATION DENOTES CLUSTERS VIA ()s

CO-OCCURRENCES DERIVED FROM ()s NOTATION

FEATURES OF ENTITIES WON'T BE CONSIDERED

S REPETITION INCREMENTS CO-OCCURRENCE

CO-OCC + DISSIM = CONSTANT

I

I

TWO MODELS:

1. SEQUENTIAL LEARNING PARADIGM

i 2. CONCURRENT (SNAPSHOT) PARADIGM

I

I

METRIC AXIOMS

REFLEXIVITY

SYMMETRY

TRIANGLE INEQUALITY--INVALID FOR CO-OCCURRENCES

IPFNs IMPOSE THE GENERALIZED TRIANGLE INEQUALITY

I

COMPONENTS OF THE MODEL

DOMAIN D = (A, B, C, D, ... I

SAMPLE CLUSTERING Ci = ((A, B, C), D, E)

PRIMITIVE OPERATIONS

ADD ENTITY TO D

REMOVE ENTITY FROM D

MERGE TWO CLUSTERS INTO ONE CLUSTER

SPLIT ONE CLUSTER INTO TWO CLUSTERS

WHERE AN ENTITY IS A CLUSTER

EXAMPLE: THE LEARNING PARADIGM

CO = [AB, C, DEqF)

so= -000000 -000000 -000000-0 00000 -00 0000-

Cl ( (A, B, C), D, E, F)

S1= -110001 -100011 -00000 0-000000 -0

A 1 B @ 000000-I

EXAMPLE CONTINUED:

C2 =((A, B, C)2, (D, E), F}

S2= -220002-200022-000

I0 0001-0A 0000 0-

C3 =((A, B, C)3, ((D, E), F))

S3= -330 003-300033 -000000-21

A BD E 001

11 33C F

EXAMPLE CONTINUED:

C4 = (A, B, C)4, ((D, E), F)2, (B, D)}

S4= -440004-410044-0000 10-3 2

4 2 000320-

EXAMPLE: THE CONCURRENT PARADIGM

C = [(A, B, C), ((D, E), r'), (B, D)}so= - 11 00 0

1 -1 10 0

0 10-21

A DE 0 00 10-

APPLICATIONS

METRIC FOR DIFFERENCES IN EXPERTISE

IDENTIFICATION OF ERRORS

CONSENSUS

AUTOMATIC SYSTEM ADAPTATION WITH LEARNING

HUMAN-COMPUTER INTERFACE

ROBOTICS VISION DATABASE

TRANSFORMATIONS, DISTANCE,

AND DISTANCE GRAPHS

Gary ChartrandWestern Michigan University

ABSTRACT

Several transformations are described - between graphs and between subgraphs in a graph.Each of these transformations gives rise to a distance (between graphs or between subgraphs in a

graph). The relations between collections of graphs or between the subgraphs of a specified size

within a (connected) graph can be described by graphs themselves, called distance graphs. In.addition to describing these concepts, another distance between induced subgraphs of a specified

order and the corresponding distance graphs are also discussed.

2

Introduction

The distance between two vertices in a connected graph is the length of the shortest path

connecting the vertices. Distance is one of the most fundamental concepts in graph theory.

Algorithms for determining distance in graphs are well known while applications involving

distance in graphs are varied and numerous. Indeed, so much work has been done on this subject

that Buckley and Harary wrote a book in 1990 entirely devoted to distance in graphs. Distance in

graphs has been generalized in several ways, most notably perhaps to Steiner distance.

The combination of distance and graphs occurs in many other ways. One of these concerns

distance between graphs. It would be more accurate to speak of distance between certain pairs of

graphs because in many instances distance is defined between graphs having some specified

properties. In all such distances, the distance between two graphs is 0 if and only if they are

isomorphic. Hence, the distance between two graphs is a measure of the structural difference

between the graphs.

Transformations

Some distances between graphs involve the idea of transformations. Let G and H be

two (p, q) graphs. We say that G can be transformed into H by an edge rotation if Gcontains distinct vertices u, v, and w such that uv r E(G), uw e E(G), and H - G - uv + uw.

More generally, we say that G can be r-transformed into H if there exists a sequence G = G0,

G 1 ... , Gn = H (n > 0) of graphs such that Gi can be transformed into Gi+ by an edge

rotation for i = 0, 1, ..., n - 1.

An edge slide is a restricted version of an edge rotation. A graph G can be transformed

into a graph H by an edge slide if G contains distinct vertices u, v, and w such that uvE

E(G),vw E E(G), uw 0 E(G) and H = G - uv + uw. If a graph H is isomorphic to a graph G

or H can be obtained from a graph G by a sequence of edge slides, we say that G can be s-

transformed into H. For example, the graph H of Figure 1 can be obtained from the graph G

by an edge rotation, but H cannot be obtained from G by an edge slide. On the other hand, the

graph H' can be obtained from G' by an edge slide (as well as by an edge rotation).

3

V W V wG: V2 H: ZU Z U Z

V W V W

U Z U Z

Figure 1

It was shown by Chartrand, Saba, and Zou that every (p, q) graph G can be r-

transformed into any other (p, q) graph H. It was also shown by Johnson that s-transformation

preserves connectedness. Further, a graph G can be s-transformed into a graph H if and only if

G and H have the same number of components and corresponding components of G and H

have the same order and same size.

SMetrics Based on Transformations

Associated with these transformations are two metrics defined on graphs. Let G and H

be two graphs having the same order and same size. The edge rotation distance or, more simply,

the r-distance dr(G, H) between 6 and H is the smallest nonnegative integer n for which

there exists a sequence G 0, G 1, ..... G n of graphs such that G6= G0, H =G, and G i can be

obtained from G_ by an edge rotation for i =1, 2, ..... n. For example, the edge rotation

distance between graphs G and H shown in Figure 2 is dr(G, H) = 3.

G: p OH:C 0

Figure 2

The following properties of edge rotation distance were established by Chartrand, Saba,

and Zou.

S

4

Proposition 1 If G and H are two graphs having the same order and same size, then

dr(G, H) = dr(G, H).

It was shown that every nonnegative integer is the r-distance between some pair of graphs.

Proposition 2 For every nonnegative integer n, there exist graphs G and H such that

dr(G, H) = n.

Prior to presenting an upper bound for the r-distance between two graphs, we introduce

another concept. For nonempty graphs G1 and G2, a greatest common subgraph of G1 and G2

is defined as any graph G of maximum size without isolated vertices that is (isomorphic to) a

subgraph of both G1 and G2.

Proposition 3 Let G and H be two (p, q) graphs with q > 1, and let s be the size of a

greatest common subgraph of G and H. Then dr(G, H) < 2(q - s). Moreover, this bound is

sharp.

Another distance between graphs is associated with edge slide and was discussed by

Johnson and by Benadd, Goddard, McKee, and Winter. Let G be a graph with components Gi,

* 1 i < k, and H a graph with components Hi, 1 < i < k, such that Gi and Hi have the same

order and same size. We define the edge slide distance or, simply, the s-distance ds(G, H)

between G and H as the smallest nonnegative integer n for which there exists a sequence GG0, G1 = .... Gn H of graphs such that, for i = 1, 2, ... , n, Gi can be obtained from Gi 1 by

an edge slide. If G and H are the graphs presented in Figure 3, then the edge slide distance

between G and H is ds(G, H) = 2.

G: H:

<XFigure 3

Note that dr(G, H) = 1 for the graphs G and H of Figure 3. It is straightforward to

show that dr(G, H):5 ds(G, H) for every pair G, H of connected graphs having the same order

and same size. The following result is perhaps less obvious.

5

* Proposition 4 For every pair m, n of positive integers with m < n, there exist graphs G andH such that dr(G, H) = m and ds(G, H) = n.

Distance Graphs

Let S be a set of (nonisomorphic) (p, q) graphs. Then we define the edge rotation

distance graph Dr(S) of S as the graph with the vertex set S such that two vertices G and H

of Dr(S) are adjacent if and only if dr(G, H) = 1. A graph F is an edge rotation distance graphif F = Dr(S) for some set S of graphs.

Let S' be a set of (nonisomorphic) graphs having the same number of components,

labeled in such a way that the ith components of all graphs have the same order and same size.Then we define the edge slide distance graph Ds(S') of S' analogously.

It was shown by Chartrand, Goddard, Henning, Lesniak, Swart, and Wall that every

graph is an edge slide distance graph and it was conjectured that all graphs are edge rotationdistance graphs. A number of classes of graphs are known to be edge rotation distance graphs.

The next two results are due to Chartrand, Goddard, Henning, Lesniak, Swart, and Wall.

* Proposition 5 Complete graphs, cycles and trees are edge rotation distance graphs.

Proposition 6 Every line graph is an edge rotation distance graph.

Proposition 7 (Faudree, Schelp, Lesniak, Gyrfds, and Lehel) The complete bipartite graphsK3,3 and K2,p (p _ 1) are edge rotation distance graphs.

Proposition 8 (Jarrett) For every pair m, n of positive integers, the graph Kmn is an edge

rotation distance graph.

F-Transformations

Let G and H be two (p, q) graphs, both containing a subgraph isomorphic to a givengraph F of order at least 2. We say that G can be transformed into H by an F-rotation (orsimply, G can be F-rotated into H) if there exist distinct vertices u, v, and w of G and asubgraph F' of G isomorphic to F such that u e V(F'), (v, w) 9 V(F'), uv e E(G), uw eE(G), and H = G - uv + uw. For example, if F - K1,3, then the graph G of Figure 4 can beKi, 3-rotated into H and H'.

G: H:v W

x x x

U ZU Z U Z

Figure 4

More generally, we say that a graph G can be F-transformed into H if either (1) G =_H

or (2) there exists a sequence G = Go, GI, ..., Gn = H of graphs such that, for i = 0, 1, ... ,

n - 1, the graph Gi can be F-rotated into Gi+1 . For instance, the graph G of Figure 5 cannot

be K1,4-rotated into H, but G can be K1,4 -transformed into H.

G( = Go): G1• H(= G2):

Figure 5

Observe that K 2-rotation and K2 -rotation are edge rotation and edge slide, respectively.

Clearly, if a graph G can be F-transformed into a graph H, then G and H have the same

order, same size, and both contain a subgraph isomorphic to F. Unfortunately, the converse is

not true, in general. For instance, the graphs G and H of Figure 6 have the same order andsame size, and both G and H contain a subgraph isomorphic to C4 , but G cannot be C4 -

transformed into H. In fact, G can be C4 -transformed only into itself.

p

7

G: ~H

:C

Figure 6

One may ask the question: What are necessary and sufficient conditions for one of two

graphs G and H to be F-transformed into the other? We have already seen the answer to this

question if F = K2 or F = K2. The following results are due to Jarrett.

Proposition 9 Let F be any nontrivial connected graph. If a connected graph G can be F-

transformed into a graph H, then H is connected.

Corollary 9a Let F be a nontrivial connected graph. A graph G can be F-transformed into agraph H if and only if the graph G has components G1, G2, ..., Gk, the graph H has

components H1, H2, ..., Hk, and Gi can be F-transformed into Hi for every i (1 <i <k).

Proposition 10 Let F be a connected graph of order p' with 8(F) = I, and let G and H

be two (nonisomorphic) connected (p, q) graphs, each containing an induced subgraph

isomorphic to F. Then G can be F-transformed into H.

Next we show that if 8(F) > 1 or F is not an induced subgraph of G or H, the result

does not necessarily hold. For example, for the graph F of Figure 7 we have 8(F) = 2 > 1.

Although the graphs G and H contain F as an induced subgraph, G cannot be F-transformed

into H. In fact, G cannot be F-transformed into any graph different from G.

F: G: H:

Figure 7

For the graph F of Figure 8 we have 8(F) = 1, but F is not an induced subgraph of G,

and H cannot be obtained from G by an F-transformation. Indeed, G can be F-transformedP only into itself and graphs G' and G" of Figure 8.

. °<88ID ~~~F: o>

G ( H:C>

Figure 8

With each F-transformation another metric can be defined. Let F be a graph of orderp' : 2 and let S be a set of (p, q) graphs such that for every pair G, H of graphs in S, the

graph G can be F-transformed into H. The F-distance F-d(G, H) between G and H is

defined as the minimum number of F-rotations necessary to transform G into H. For the graphs

F, G, and H of Figure 9, we have F-d(G, H) = 2.

l* F: G: H:

Figure 9

If a graph G cannot be F-transformed into a graph H we set F-d(G, H) = 00. It is

obvious that K2-d(G, H) : F-d(G, H).

Proposition 11 Let F be a graph of order p ( 2) and let n be a nonnegative integer. Then

there exists a pair G1, G2 of graphs such that F-d(G 1, G2) = n.

Let F be a graph of order p' (> 2) and let S be a set of (p, q) graphs, each containinga subgraph isomorphic to F. Then the F-distance graph D(S) of S is that graph whose vertex

set is S and in which two vertices G and H are adjacent if and only if F-d(G, H) = 1. Forexample, if F - K3 and S is the set of graphs G1, G2, G3, and G4 shown in Figure 10, then

S(S) =- K4 - e.S

G2G:

G3

Figure 10

Proposition 12 Let F be a nontrivial connected graph. Then every graph is an F-distancegraph.

Subgraph Distance

Let G, and G2 be edge-induced subgraphs of the same size in a graph G. The subgraphG2 can be obtained from G1 by an edge rotation if there exist distinct vertices u, v, and w suchthat uv = E(G1), uw O E(G1), and G2 = G1 - uv + uw. More generally, G1 can be r-

transformed into G2 if GI = G2 or G2 can be obtained from G1 by a sequence of edge

rotations. It was shown by Chartrand, Johns, Novotny, and Oellermann that every edge-inducedsubgraph of a connected graph G can be r-transformed into any edge-induced subgraph of Ghaving the same size. The edge rotation distance dr(GI, G2) between G1 and G2 is theminimum number of edge rotations required to r-transform G1 into G2. For the graph G ofFigure 11, the subgraph G3 can be obtained from G1 by an edge rotation so that dr(Gl,G3) "-1.On the other hand, G3 cannot be obtained from G2 by an edge rotation, but G3 can be obtained

from G2 by an r-transformation and dr(G2, G3) = 3.

10

z z v

x Y u t yU

G2 : z v G3:

x y y u

Figure 11

One can also define a subgraph transformation based on edge slide. Let G1 and G2 be

two edge-induced subgraphs of the same size in G. We say that G2 can be obtained from G,by an edge slide if there exist distinct vertices u, v, and w of G such that uv E E(G1 ), uw 4

E(G1), vw E E(G), and G2 = GI - uv + uw. For example, for the graph G of Figure 12, the

subgraph G2 can be obtained from G, by an edge slide. More generally, we say that G, can

be s-transformed into G2 if either (1) G, = G2 or (2) G2 can be obtained from G 1 by a

sequence of edge slides.

w wG: GI: G2:

v U v u v U

z z z

Figure 12

As we mentioned earlier, for every pair H, H' of edge-induced subgraphs of the same size

in a connected graph G, the subgraph H can be r-transformed into H'. Unfortunately, this is

not the case for s-transformations. For example, if G is the graph of Figure 13, then H cannot

be s-transformed into H'. In fact, H can be s-transformed only into itself.

I

11

G: H. H': V\ X

u w y u w w y

Figure 13

Let e and f be edges of a graph G. A triangular e-f walk of G is a finite, alternating

sequence e = e0 , T 1 , e l , T 2, ..., en_ 1, Tn , en = f of edges and triangles such that ei_1 and e i

belong to Ti (1 < i < n). A triangular e-f path is a triangular e-f walk in which no edges or

triangles are repeated. The number n of triangles in the triangular path is called its length. In the

graph G of Figure 14 there exists a triangular e-f path (with Ti = ({ei_ 1, el}), i = 1, 2, 3), but

there is no triangular e-g path.

e = e el e2 e3 =f g

Figure 14

It is straightforward to show that every triangular e-f walk in a graph contains a triangular

e-f path.

Observe that for every two edges e and e' of a triangle T, the subgraph ({e')) can be

obtained from (fe)) by an edge slide. Therefore, if in a graph G there exists a triangular e-f

path, then G1 = ({e)) can be s-transformed into G2 = ({f).

Whenever edges e and f belong to a 3-cycle in G, we denote this triangle by T(e, f)

and call it a slide induced triangle. With every edge slide there is associated a unique triangle T,

namely, if G2 = G, - e + f, then T = T(e, f). These observations are useful in proving the

following result by Jarrett.

Proposition 13 Let G be a connected graph of size q a 1, and let q' be an integer with

1 < q' < q. For every pair G1, G2 of subgraphs of G having size q', the subgraph G, can be

s-transformed into G2 if and only if every two edges of G are connected by a triangular path.

I

12

Triangular Line Graphs

For a given graph G, we define its triangular line graph 7(G) as that graph with vertex set

E(G) such that two vertices e and f of '2(G) are adjacent if and only if T(e, f) is a triangle of

G. For G = K4 - e, the graph 7(G) is shown in Figure 15.

G: e3 "(G): e2 e3

eee

el el e4

Figure 15

It follows from the definition that '7(G) is a spanning subgraph of the line graph L(G).

The next result is perhaps less obvious. These are due to Jarrett.

Proposition 14 Let G be a connected graph of order p > 2. Then '1(G) = L(G) if and only

if G = Kp.

Proposition 15 Let G be a connected nontrivial graph. For every pair G1, G2 of edge-

induced subgraphs of G having size 1, the subgraph G, can be s-transformed into G2 if and

only if '(G) is connected.

Corollary 15a Let G be a nontrivial connected graph. Then for every pair G 1, G2 of edge-

induced subgraphs of G having the same size, G, can be s-transformed into G2 if and only if

'7(G) is connected.

For integers n > 2, the nth iterated triangular line graph T'n(G) of a graph G is defined

to be '7(Tn-l(G)), where T'(G) denotes 7(G) and Tqn-(G) is assumed to be nonempty.

Clearly, Tn(G) is a subgraph of the nth iterated line graph Ln(G) of G. In fact, for n =

1, 'T1(G) = "7(G) is a spanning subgraph of L1 (G) = L(G).

Note that every triangle T in G gives rise to a triangle T' in '2(G) with a one-to-one

correspondence between the edges of T and the vertices of T'. Moreover, if T1 and T2 are two

triangles of G, then the corresponding triangles T' and Tj of '1(G) are edge-disjoint. For

suppose, to the contrary, that T' and Tj have an edge in common or, equivalently, T' and T'

13

have two common vertices, say e and f. Necessarily, e and f are common edges of T1 and

T2, which implies that T1 = T2 . Thus '7(G) has at least as many triangles as G has. We show

that only for K4 -free graphs G are the number of triangles in G and '7(G) equal.

Proposition 16 Let t(G) denote the number of triangles in a graph G. Then t(G) = t(7(G))

if and only if G is K4-free.

Proposition 17 Let G be a K4-free graph. Then Tn(G) = T 2 (G). for n > 2.

The previous result does not hold for a graph G = K4 . However, T 3(K4 ) = 8K3 and,

therefore, for n > 3 we have Tn(K4 ) = T 3(K 3). The graphs G = K4 and Ti(K4 ), 1 < i < 3,

are shown in Figure 16.

Conjecture For every graph G containing at least one triangle, there exists an integer k > 0,

such that for n > k, Tn(G) = Tk(G).

G'7(G):

T 2 (G): T 3(G):

Figure 16

n-Subgraph Distance Graphs

The n-subgraph distance graphs were introduced by Chartrand, Hevia, Jarrett, Saba, and

VanderJagt. Let G be a graph of size q (> 1) and let n be an integer with 1 < n < q. The n-

subgraph distance graph Ln(G) of G is that graph whose vertices correspond to the edge-

14

induced subgraphs of size n in G and where two vertices of Ln(G) are adjacent if and only if. the edge rotation distance between corresponding subgraphs is 1. It is convenient to label thevertices of Ln(G) by the edge sets of the corresponding subgraphs or simply by listing the edges.

Each edge in a vertex label is called a coordinate. Since the coordinates are elements of a set, the

order in which the coordinates of a vertex are listed is irrelevant. For example, if a vertex ofLn(G) corresponds to the subgraph of G induced by the edge set (e l , e2, ..., er}, then we may

label this vertex as e l , e2, ..., en or ei u X, where X = (ejI 11 _j < n,j * i, or simply as eiX.

For the graph G K1 + (K1 u K2) of Figure 17, the graphs Li(G), i = 1, 2, 3, 4, are shown.

aa

G: 0L()

bb c

dd

L2b L3 (G): bcd

a, acd Cabd

abc

bd cd

L4(G): abcd0

Figure 17

Observe that L4 (G) = K1 for the graph G of Figure 17. In general, Lq(G) = K1 for a

graph G of size q. Furthermore, LI(G)= L3 (G) for the graph G of Figure 17. This fact

illustrates the following result.

Proposition 18 Let G be a graph of size q, and let n be an integer with 1 < n < q. ThenLn(G ) = Lq-n(G ).

15

The graphs Ln(G), 1 < n < q = E(G), are also called generalized line graphs since the 1-

subgraph distance graph LI(G) is the line graph of G. We shall also refer to these graphs as n-

subgraph rotation distance graphs to distinguish them from the n-subgraph slide distance graphs,

which we are about to describe. We begin with the definition of n-subgraph slide distance.

Let G be a graph of size q (> 1), and let G1 and G2 be two edge-induced subgraphs of

G having the same size n (1 < n < q). We define the n-subgraph slide distance ds(G 1, G2)

between G, and G2 as the smallest nonnegative integer k for which there exists a sequence

H0 , H I, .... Hk of subgraphs of G such that G, = H0,G 2 = Hk and, for i= 1,2 ... k,H i

can be obtained from Hi_1 by an edge slide. If no such k exists, we define ds(G1, G2) = 00. If

G = K4 - e, and G1 and G2 are two subgraphs of G shown in Figure 18, then

ds(GI, G 2 ) = 2.

G: b GI" b G2:

a c

Z I 0____d d

Figure 18

We define the n-subgraph slide distance graph Sn(G) of G as the graph whose vertices

correspond to the edge-induced subgraphs of size n and where two vertices G1 and G2 of

Sn(G) are adjacent if and only if ds(G 1, G2 ) = 1. It is straightforward to see that SI(G) = !(G),

and, therefore, SI(G) is a spanning subgraph of LI(G). In general, for n >_ 1, Sn(G) is a

spanning subgraph of Ln(G). For the graph G = K4 - e, the graphs Si(G), 1 < i ! 5, are shown

in Figure 19.

16

G: b SIG: a b

azcZ

d d c

ceabd ade acd

bd ace

S4(G): S5(G) =-K1 :

bcde acdeabcde

abee abde

Figure 19

Observe that for the graph G =-K4 - e, we have S2(G) S3 (G) and SI(G)- S4 (G).

This fact can be generalized as follows.

Proposition 19 Let G be a graph of size q Q: 1) and let n be an integer with 1 n < q.

Then Sn(G) =-Sq-n(G).

17

Subgraph Distance for Subgraphs of the Same Order

For a connected graph G of order p and an integer n such that 1 < n < p, let F and H

be induced subgraphs of G of order n. We define a pairing 7C from the set V(F), say {v1, v2 ,

.... vn }, to the set V(H) as a one-to-one correspondence that associates a vertex of V(F) with

one of V(H). The distance induced by 7c between F and H is defined as

n

dg(F, H) = d(vi, it(vi))i= I

and the subgraph distance between F and H is

d(F, H) min 4(F, H).

This concept was introduced by Chartrand, Johns, Novotny, and Oellermann. Observe that if F

consists of a single vertex, say u, and H consists of a single vertex, say v, then d(F, H) = d(u,

v). Thus d(F, H) is a generalized distance defined in terms of subgraphs. As an example, Figure

20 gives a connected graph G, two induced subgraphs F and H of G, a listing of all pairings.from V(F) to V(H), and d(F, H).

18

Vi Ul Vi

G: F: _H: 0u1 x y

U2 (

U3 V2 V3

U3 W V2 V3

Pairings ui Vi d(u i, vi) dnk(F, H)u1 v1 2

U u2 V2 3 8

U1 V1 2It2 U2 V3 4 8

U3 V2 2Ul V2 3

7C3 U2 V 1 3 9

_ _u3 v3 3

Ul V2 3714 u2 v3 4 10

u3 Vl 3

uI v3 45 u2 Vl 3 9

U3 v 2 2uI v3 4

6 u2 v2 3 10

I u3 V1 3

d(F, H) = 8

Figure 20

19

n-Vertex Graphs

Let G be a connected graph of order p and let F and H be subgraphs of order n with

1 _< n _ p - 1. Then d(F, H) = 1 if and only if there exist adjacent vertices u r V(F) and v E

V(H) such that V(F - u) = V(H - v). We define the n-vertex graph of G as that graph Gn

whose vertices are the induced subgraphs of order n in G and two vertices F and H of Gn are

adjacent if and only if d(F, H) = 1. A graph and its 2-vertex graph are shown in Figure 21.

G j: G2

Figure 21

01,

The optimality of clustering and monotone optimal assembliesbyUriel G. Rothblum

The purpose of the study is to present recent resultswhich provide sufficient conditions for the optimality ofclustered partitions. In particular, we Introduce a formaldefinition of clustering by calling a partition clustered ifconvex hulls of the sets of the partition are disjoint* It isshown that if given vectors Al , so. , An In Rk are to bpartitioned into m groups, of predetermined sizes, so as tomaximize an objective which is a quasi-convex function ofthe sums of the vectors in each set, then a clustered optimalpartition exists. Further, if quasi-convexity is replaced bystrict quasi-convexity, every optimal partition is clustered.Computational implications of the results are discussed.

The techniques we developed enabled us todetermine sufficient conditions for the optimality ofmonotone assemblies. Here we consider the probles ofS identifying multipartitions which occur when items ofdifferent types are to be partitioned into sets, for example,in system assembly, components of different type areassigned to modules which compose the system. We showthat if the system is coherent and if the components in eachmodule are In series, there Is a reliability maximizingassembly which is monotone, i.e., it has a single modulewhich gets the best parts of each type (according to Itsspecification), there is another module which gets thesecond best parts of each type (according to itsspecification), and so on, till finally, there is one modulewhich gets the worst parts of each type (according to itsspecification).

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0 G. D. Crown and M. F. Janowitz

Abstract. Let G be a finite nonempty set. A consensus function

can be viewed as a mapping F:Gk - G. For n e Gk, F(n) is often

constructed from certain building blocks. For a family of

relations on a set X, these might be the pairs (xy) that

constitute the output relation. In its most general setting, one

can work with a triple (G,S,X) where S is the set of building

blocks and :G --)P(S) is a one-one function. A neutral consensus

function F is characterized by the existence of a family VF of

subsets of Vk = {1,2,...,k) having the property that for any k-

tuple n4 Gk, s e z(F(W)) if and only if {i: s e ' (gi)} DF-

Assuming that Vk e DF, and o * VF, we investigate conditions on

that are equivalent to: A e VF implies Vk\A E VF; A VF impliesVk\A e DF; VF is an order filter (principal filter, ultrafilter)

of Vk. The work is in progress, but here are some sample results:

Theorem. Assume G has elements g,h such that 7(g)vy(h) is not of

the form ,(v), while ?(g)c(h) - 7 (w) for some w e G. Than forany neutral F, it is true that A,B EDF implies AB *o.

(Included here would be any nondistributive lattice, as well as

any meet semilattice that is not a lattice). There are also some

general results that imply the following: If G is a lattice

having a pair of comparable sup-irreducibles, then DF is an order

filter of Vk. If G is an atomistic lattice having distinct atoms

a,b,c such that c s avb, then 2F is a principal filter (so any

neutral consensus method is an oligarchy). If G is an atomistic

join semilattice that is not a lattice having distinct atoms

a,b,c such that c s avb < 1, or if G is a nondistributive join

semilattice that is not a lattice having at least 4 atoms, such

that every line c-ntains exactly 2 points, then any neutral

consensus method is a dictatorship.

0

*ORDINAL CONSENSUS METHODS

G. D. Crown and M. F. Janowitz

(Preliminary Report)

What is a consensus method?

G = finite set of objects

(the objects to be summarized)

V = indexing set (the "voters")

(fixed, possibly infinite)

P = G = set of profiles.

Thus for R E P,

= (go) CEV or n:VH--G.

Consensus method: F:Pi -G.

Kon )ceT Rw.j,#- \J.,-Ue)

\Jjer- Or A A-V . -eL.'& FT'r- -c

~ Q~e-T

c,,- C oSeL- TejZc4..

6L-' oI-a r-j t 41C e

*For a profile n, will approach the

construction of F(n) by means of a'stability" family.

Pair (S,T), where S is a set and

: :Gi->P(S) is a one-one

mapping.

Neutral consensus method F:There is a family D = DF of

subsets of V such that

(i) o i D, but V E D.(ii) s E 7 (F (n)) if f s E 2),I

where s = {a: s E (n (a)}.

Goal: Relate properties of D with

properties of -.

* Examples of stability families

1. G = a set of binary relations

on a set X. (weak orders,

linear orders, partial orders,

quasiorders, etc.) S = XxX, and

(R)= {(x, y): xRy}

or

(R)= {(x, y): xRy fails}

2. G = finite join (or meet)

semilattice.

S = sup-irreducibles in that

a,b < s = avb < s.

T(g) = {s E S: s < g}, or

T(g) = {s E S: s $ g}.

(or the dual of this example)

3. Trees and Pyramids. X = set

with Ixi a: 5. A set 9 of

subsets of X is a trE-, if

(a) X E T, and 0 o i.

(b) {x} E 9T for all x E X.

(c) For A, B E T, AnB E {o,, A, B}

YTi s a pyramid i f (c) i s

replaced by

(d) For A,B E Y, AnB E T.

Take S = proper subsets of X,

and z (Y) = {A: A E Y} . There

are many similar examples.

) I~c~~ elA &zE cA %3 3

Cc-F-cr,- c44 c- 1 - 3 -T

Dho1 T,,4Ier i- V

9. A? ==:-P A~ n

~~r-Aur-c,i ( ~ A-&~s

(TT -r_

*Some Sample Results

F = neutral consensus method withF"

(S, -) is a stability family.

Theorem 1. If IVi a 2, (a) * (b)

(a) For any such F,

A E D implies A' i D.

(b) 3g,h E G such that y(g)uy(h)

is not of the form x'(v) for

any V E G.

I f I V 3, (c) * (d)

(c) For any such F,

B 4 D implies B' E D.

(d) There exist g, h E G such that

z(g) n (h) is not of the form

*Z(w) for any w E G.

~- -. (6~ SC) -4 t;I L .t(

\A-) it~d

(&bc, vc (b4Cj

(u-bL~ c-)

Theorem 2. If there exist g,h E G

such that

(g) uz (h) is not of the form (v) ,

but

(g) n (h)= y(w) for some w E G,

then AB E D implies AnB 0 o.

Corollary. Let G be any non-

distributive lattice, or any join

*semilattice that is not a lattice.

Take S = sup-irreducibles, and

T(g) = {s e S: s -< g}.

Then A, B E D implies AnB # 0.

-- A VERSION OF ARRow's THEOREM

a1 ,a2, a3 E S have the following

properties:

(1) 3glg 2 ,g3 E G I (i, j,k) any

permutation of (1,2,3), then

a. E y (g.) {a a k } n (gi) 0

(2) 3z E G 9 ui,(ai) _; z(z).

i (3) 3w E G 9 {al,a 2 ,a 3 }nz(w) = 0-.

(4) {a I ,a 2 , a 3 } is transitive in

the sense that a 1 ,a 2 E 'Y (g)

implies that a 3 E T(g).

Theorem 3. If the above conditions

are satisfied and F is a neutral

consensus method, then D is a

*- lattice filter of V.

Remark. The above notion of

transitivity of course comes from

transitivity of relations:

xRy, yRz implies xRz.

But what makes it of further

interest is the fact that in a

join semilattice, c -5 avb makes

(a,b,c) a transitive triple.

SFor relations, can talk about

a triple { (a,b) , (c,d) , (x,y)} being

transitive if

aRb,cRd implies xRy.

In general, if T _ S and S E S,

can define (T,s) to be transitive

if T _; T(g) implies s E 'Y(g)

Theorem 4. Let a, b E S; x, y, z E G.

Assume:

(1) a e (g) b e (g).

(2) a E (x).(3) a (y), b r= y(y).

(4) b (z).

Then F neutral implies ) an order

filter.

S Theorem 5. Assume x, y,g,z E G, and(1) (x)uT(y) # z (w)

(2) (x) nry(y) = 0.

(3) (x) uy (y) c x (g)

(4) -(z) = 0.

Then F neutral implies D is closed

under intersections.

Theorem 6. Assume x, y, g E G, and

(1) (x) uv (y) z (v) for any v.

(2) z(x)rnz(y) - (w) for some w.(3) y (x) ux (y) c y (g).

(4) D is an order filter.

Then D is a lattice filter.

Apply these and similar results to

lattices having more than 1

*member:

Theorem 7. G is a lattice, and

S = sup-irreducibles. These are

equivalent:

(1) G is not distributive.

(2) For any neutral F, ) is a

lattice filter.

Defn: A join-semilattice G is

distributive if every principal

filter of G is a distributive

lattice.

Theorem 8. G is a nondistributive

join semilattice not a lattice,

with S its sup-irreducibles.

For any neutral consensus method F

*on G, V is an ultrafilter.

In progress: Work with

distributive join-semilattices.

The setting: A stability

family (S, X) such that:

(1) x(a)u?-(b) = z(v).

(2) Y(a)nz(b) is either z(w) or 0.