Combinatorial algorithms on partially ordered sets

A C C E P T E D

by

M .M ath ., U niversity o f W aterloo, 1987

■ i .... — —

7 V i / ! Di-AN

A D isserta tio n S ub m itted in P a rtia l Fulfillm ent of tlie

oft re M - l s I . R equirem ents for th e Degree of

D O C T O R O F P H IL O S O P H Y in th e D ep artm en t of C o m p u te r Science

We accept th is thesis as conform ing to th e required sta n d a rd

Dr. M . It . Fellows, S upervisor (D e p artm en t o f C o m p u ter Science)

^— | y f | i - y r ~m—fc

l)r. J. A. EllisfTDfpagtetier^al M em ber (D e p artm en t of C o m p u ter Science)

Dr. W . J . M y ?v 5 ltf, D e p artm en tal M em ber (D e p a rtm e n t of C o m p u te r S cence)

Dr. It. Odeli, O utside M em ber (D e p artm en t of M ath em atics)

Dr. It. C. R ead, E x tern al E xam iner (D e p a rtm e n t of C om binatorics an d O p tim izatio n , U niversity of W aterloo)

© Y A SU N O R I K O D A , 1991 U niversity of V icto ria

All rig h ts reserved. T hesis m ay no t be rep ro d u ced in w hole or in p a r t, by m im eograph o r o th e r m eans, w ithout th e perm ission of th e a u th o r.

11

Supervisor: Dr. M ichael Fellows

A b str a c t

T h e m ain results of this d issertatio n are various algorithm s re la ted to p artia lly ordered sets. T h e d isse rta t'o n basically consists of two p a rts. T h e first p a rt tre a ts algorithm s th a t g enerate ideals of p artially ordered sets. T h e second p a rt concerns th e generation of partially ordered sets them selves.

F irst, we p re sen t two algorithm s for listing ideals of a forest poset. T hese algorithm s g enerate ideals in a G ray C ode m anner; th a t is, consecutive ideals differ b y exactly one elem ent. B oth algorithm s use storage O (n ), where n is th e n u m b er of elem ents in th e poset. T h e first alg o iith m traverses, a t each phase, th e c u rre n t ideal being listed and runs in tim e 0 ( n N ) , where 11 is th e n u m b er of ideals of th e po set. T h e second algorithm mimics th e first b u t elim inates th e trav e rsal and runs in tim e 0 ( N ) . T his algorithm has the p ro p e rty th a t th e am ount of co m p u tatio n betw een successive ideals is 0 ( 1).

Secondly, we give orderly algorithm s for constru ctin g acyclic digraphs, acyclic tr a n ­ sitive digraphs, finite topologies and finite lattices. For th e first tim e we show th a t th e n u m b er of finite lattices on 11, 12, and 13 elem ents are 37622, 262775, an d 2018442, re­ spectively, and the num ber of finite topologies on 8 an d 9 elem ents are 35979 an d 363083, respectively.

We also describe orderly algorithm s for generating fc-colored graphs. W e p resent, in p artic u la r, an algorithm for generating connected bicolorable graphs. W e also prove some properties of a canonic m atrix which m ight be generally useful for im proving th e efficiency of orderly algorithm s.

Dr. M . R . Fellows, S upervisor (D e p a rtm e n t of C o m puter Science)

Dr. J . A . E llis y U e ila i'trte n f^ M e m b e r (D e p a rtm e n t o f C om p u ter Science)

Dr. W . J . M yrvold , D e p a rtm e n ta l M em ber (D e p a rtm e n t of C o m p u ter Science)

Dr. R. O deh, O u tsid e M em ber (D e p a rtm e n t of M a tlie m n tic ;)

Dr. R. C . R ead, E x te rn a l E x am in er (D e p a rtm e n t of C om binatorics and O ptim ization, U niversity of W aterlo o )

A ck n o w le d g e m e n ts

George P olya said th a t new knowledge is obtained th ro u g h in d u ctio n (in th e ordinary English sense). T h is thesis contains new knowledge acquired th ro u g h observation and induction.

I w ould like to express m y th an k s to Prof. M ike Fellows for his careful supervision on this dissertation.

I would like to express m y special th an k s to Prof. R on R ead for in tro d u cin g me to the problem s, and continuous su p p o rt and encouragem ent for solving th e problem s. M y research in this thesis was actually sta rte d when I was a t th e U niversity of W aterloo.

1 would like to express my th an k s to P rof. Teofilo G onzalez for n o t providing me any hints except saying th a t if we try th e sm all cases, we can generalize an d figure o u t the solution. His teach in g gave m e a sense of th e research a ttitu d e .

I would like to express m y th an k s to P rof. F ran k H a rary for providing me w ith the o p p o rtu n ity to solve grap h problem s in his enjoyable courses and to D r. H e ath er Silverm an for providing me th e tim e to discuss m y fu tu re p a tie n tly an d logically.

I owe final th an k s to m y p aren ts and m y bro th ers for encouragem ent to continue m y research.

C o n te n ts

1.4 Previous W ork and A pplications

1.5 O v e r v i e w ... 2 M a t h e m a t ic a l P r e lim in a r ie s ii iv v v ii x 1 1 5 (j 11 13 1 4

C O N T E N T S vi

3 F o r e s t P o s e t id e a ls 21

2.1 I n tro d u c tio n ... 21

3.2 T h e Elirlich and U E It A lg o rith m s...22

3.3 T h e Existence of a G ray C ode of Lower Ideals of a Forest P o s e t ...

2-3.1 A lgorithm P ... 26

3.5 T h e Ideals G enerated by A lgorithm P ...29

4 L o c p le s s G e n e r a t i o n o f F o r e s t P o s e t I d e a l s 33 •1.1 M otivation ...33

4.2 T h e B inary Reflected G ray C o d e ... 34

4.3 T h e A l g o r i t h m ... 36 4.4 T h e V alidity of A lgorithm L ... 41 4.5 T h e Looplessness of A lgorithm L ... 44 4.0 A p p l i c a t i o n ...47 5 A c y c lic D i g r a p h G e n e r a t i o n 48 5.1 In tr o d u c tio n ...48

5.2 Acyclic D igraph R e p re s e n ta tio n ... 49

5.3 A ugm entation of an Acyclic D i g r a p h ...51

5.4 G enerating Acyclic D igraphs... 53

5.5 Acyclic T ransitive D i g r a p h s ... 58

5.6 F in ite L attices and S e m ila ttic e s ... 58

5.7 F in ite T o p o lo g ie s ...61

6.1 In tr o d u c tio n ... yj

6.2 P ro p erties of a C anonic V e c t o r ... 71

6.3 A'-Colored G raph G e n e r a t i o n ... 73

6.4 G enerating Bicolored and A'-Colored G r a p h s ... 70

6.5 T h e C anonicity T est and th e C anonic M a t r i x ... S2

6 .6 T h e O rderly A lgorithm R e v is ite d ...,S 1

7 C o n c lu s io n s a n d F u tu r e R e s e a r c h g j 7.1 C o n c lu s io n s ... ^7 7.1.1 Ideal G e n e r a t i o n ... H7 7.1.2 G ra p h G e n e r a t i o n ... 7.2 F u tu re R esearch ... X8 7.2.1 Ideals G e n e ra tio n ... g# 7.2.2 G ra p h G e n e r a t i o n ... X9 A A n E x a m p le o f I d e a l G e n e r a tio n fo r a T r e e P o s e t 96

B T h e B E R A lg o r ith m for G e n e r a t in g a B in a r y R e fle c t e d G r a y C o d e 98

C T a b le s o f A c y c lic D ig r a p h s , P o s e t s , F in it e L a ttic e s a n d F in it e T o p o lo g ie s 100

L ist o f F ig u r e s

1.1 G raphs with four v e r t i c e s ... ... 2

1.2 A labeled graph G an d its spanning trees ... 2

1.3 II asse diagram s of finite lattices w ith up to 5 e l e m e n t s ... 3

1.4 B inary reflected codes o f lengths 2 an d 3 ... 3

1.5 A tree poset and its low er ideals ... 4

1.6 G ray code g e n e ra tio n ... 6

1.7 G raph generation by a classical m e t h o d ... 7

1.8 T h e code of a g r a p h ... 7

1.9 G rap h generation by an orderly a l g o r i t h m ... 8

1.10 N ecessary and sufficient conditions for th e existence of orderly algorithm s . 9 2.1 An exam ple of a tran sitiv e d i g r a p h ... 15

2.2 A n exam ple of acyclic d igraph whose vertices are ordered by d istan ce from s o u rc e s ...16

2.3 L.p Four elem ents n o t satisfying the unique bound c o n d itio n ...1.9 3.1 A lgorithm E ... 23

3.3 A lgorithm P

3.4 A rightm ost p a th in a forest p o s e t ... ... 30

3.5 A p ro p e rty of th e final ideal F ...

4.1 T h e d a ta s tru c tu re for the binary co u n ter a l g o r i t h m ...

4.2 A lgorithm B ...

4.3 T h e useful node list . ... 4.4 A lgorithm L ... 38

4.5 C ase 1. 1. D eletion in th e case not covered by C ase 1 . 2 ... 39 4.6 C ase 1.2. D eletion of th e first free node th a t is th e first sibling while the

rest of th e siblings are null an d s t u c k ... 39

4.7 C ase 2.1. A ugm entation on a node having no c h i l d r e n ... 40

4.8 C ase 2.2 . A ugm entation on a node having c h i l d r e n ... 40

4.9 D a ta stru c tu re s for a node in a forest poset and a node in th e useful node list 45

5.1 A layered acyclic digraph and its m atrix form ...

5.2 T h e canonical acyclic d ig rap h and its c o d e ...

5.o T h e conceptual schem e of a set o flists L{tj ordered by the n u m b er of vertices a n d h e ig h t...

5.4 T w o non-isom orphic acyclic digraphs th a t augm ent to isom orphic digraphs 54

5.5 A n orderly algorithm for generating acyclic d ig rap h s...

5.6 A n exam ple of checking th e unique bound condition for a p o set... 60

5.7 A n exam ple of a tran sd ig ra p h and an equivalent re p resen tatio n by an acyclic tra n sitiv e dig rap h w ith l a b e l i n g ... 61

5.8 A n assignm ent of tw o decom positions o f the n u m b er 10 to th e vertices such th a t th e resu ltin g digraphs are is o m o r p h ic ... 62

L IST OF F I C V R E S x

5.0 T i m r a n o i i i r i t v lest. ... 7 0

5.10 A p erm u ta tio n vector used for th e canonicity t e s t ... 70

0.1 T h e algorithm for generating bicolored g r a p h s ...78

0.2 T h e algorithm for generating connected bicolorable g r a p h s ...79

0.3 T h e algorithm for generatin g fc-colored g r a p h s ... 81

0.-1 A lgorithm M ...85

A.l T h e enum eration of ideals by A lgorithm P 97 15.1 T h e B EIt alg o rith m for generating a G ray c o d e ...99

C .l T h e n u m b er of tran sitiv e d ig rap h s... 101

C.2 T h e n u m b er of unlabeled acyclic digraphs w ith i vertices and height j . . . 102

C.3 T h e n u m b er o; unlabeled acyclic tran sitiv e d ig ra p h s ( p o s e ts )... 102

C.4 T h e n u m b er of finite lattices w ith i elem ents an d height j ...103

D .i T h e n u m b er of unlabeled bicolored graphs w ith non-interchangeable color classes of orders i and j ...

D.2 T h e n u m b er of up labeled connected bicolored graphs w ith non-interchangeable color classes of orders i and j ... 105

C h a p te r 1

I n tr o d u c tio n

1.1

T h e P r o b le m s

This dissertation describes various ipw algorithm s for com binatorial generation. One of the fu ndam ental com binatorial problem s, th e so-called enum eration problem , asks how m any com binatorial configurations th ere are for a given set of p aram eters. E num era­ tion sim ply means to count th e com binatorial configurations for a given set of p aram e­ ters, w hereas generation implies co n stru ctin g th e a c tu a l objects as well as co unting them . Hence, for each en um eration problem , th ere is a corresponding generation problem . In this d issertatio n , we assum e th a t th e re ad er is fam iliar w ith basic term s and concepts in graph theory (see H arary [20]) an d poset (p a rtia lly ordered sat) th eo ry (see S tanley [52]). There a re two kinds of com binatorial enum eration problem s. T h e first ty p e of enum era­ tion problem asks, for exam ple, how m any graphs th ere are for a given n u m b er of vertices (see F igure 1.1), or how m any different ways th ere are to arran g e n d istin ct objects.

T h e second ty p e of enum eration problem asks, for instance, how m any sp anning trees there a re for a given graph (see F igure 1.2), how m any m axim um m atchings th ere are for a given graphs, o r how m any d istin ct in dependent colum n vectors th ere a re for a given m atrix.

In th is thesis, we tre a t the above tw o types o f enum eration problem s. We will fur­ ther ask how to elliciently g enerate those com binatorial objects. In 1960 F rank H arary

i

i

u

n

k :

Figure 1.1: G raphs w ith four vertices

published a list of 27 unsolved enum eration problem s [18], which has been subsequently u p d ated and revised [19] [21]. A m ong the “surviving” unsolved problem s are th e enum er­ atio n of acyclic tran sitiv e digraphs (o r p artially ordered se ts), finite lattic es, and tran sitiv e digraphs (also known as finite topologies). See Figure 1.3 for a graphical representation of all finite lattic es w ith up to 5 elem ents.

T h e re exists tran sitiv ity am ong th e vertices in acyclic tran sitiv e digraphs, and the graphical forms of finite lattices an d tran sitiv e digraphs. As of y et, there are no known closed an a ly tic form ulae for th e enum eration problem of these dig rap h families involving tran sitiv ity . T h u s, in ord er to c o u n t those graphs we have to reso rt to a constructive m ethod. We use orderly algorithm s to c o n stru c t or to g enerate these graphs. O rderly algorithm s g en e rate a list of th e com binatorial configurations of a given p aram eter value (such as th e n u m b er of vertices or edges) from lists of th e configurations having a sm aller p a ram eter value.

W e will find la te r t h a t the rep resen ta tio n of A:-colored graphs is th e sam e as the

F igure 1.3: Ilasse diagram s of finite lattices w ith up to 5 elem ents

representation of acyclic tran sitiv e graphs; hence, we can also develop orderly algorithm s to co n stru c t bicolored graphs, and m ore generally fc-colored graphs.

If we represent n d istin ct objects by n binary bits, th a t is, 0 stan d s for th e absence of an o b ject and 1 stan d s for th e presence of an o b ject, all th e subsets of n d istin c t objects arc easily represented by all distinct 2n n —b it strings. O ne of th e m ost efficient ways to generate these n —b it strings is to generate th em a,s a G ra y code. See th e exam ple of a G ray code in F igure 1.4.

0 0

0 0 0

0 1

0 0 1

1 1

O

i l

1 0

0 1 0

1 1

0

1 1 1

1

0

1

1

0 0

Figure 1.4: B inary reflected codes of lengths 2 an d 3

T his G ray code has th e p ro p erty th a t ad jac en t strings in a list have exactly one b it different. A G ray code can be viewed as a linearly ordered set of all com binations chosen from a se t of n un related elem ents ( antichain) such th a t tw o successive com binations differ

t

t f

V . u v v /

F igure 1.5: A tree poset an d its lower ideals

by one elem ent.

Now, we consider th e m ore general case in w hich the set of elem ents from which a com bination (a code) to be chosen is stru ctu re d . In o u r case, th e elem ents are chosen from a poset V whose H asse diagram is a forest. T he com binations th a t we g enerate are lower ideals o f V .

A low er ideal 7 of a poset V is a subset of V satisfying th e conditions th a t if x 6 I and

y < x , th e n y € I . T h e m inim al elem ent a: of a p o set V is an elem ent of V such th a t there is no elem ent y, y < x. If t le H asse diag ram of "P is a forest J-, th e m inim al elem ents are called th e roots a n d z set of lower ideals p has a one-to-one correspondence w ith the set of subtrees w hich contain th e roots o f V . In th is d issertation, an ideal m eans a lower ideal. A Gray code o f ideals is a linearly ordered se t of ideals such th a t successive ideals differ by a single poset elem ent. In F ig u re 1.5 we show a poset and all its lower ideals. T h e e m p ty set is always an ideal by definition.

W e will describe new efficient algorithm s for generatin g the lower ideals of a forest poset. In o ther w ords, a generalized G ra y code is a list of a n y com binatorial objects with th e sam e p a ram eter in which tw o consecutive ob jects differ by sm all am o u n t of changes of th e elem ents w hich co n stitu te an object. T h u s, we m ay consider a list of lower ideals as a generalized G ray code. Such a generalized G ray code has recently been an actively studied topic in th e area o f com binatorial algorithm s. An excellent survey of recent developm ents concerning various generalized G ray codes is given in W ilf [56].

5

1.2

A G ray C o d e

An ra-bit Gray code is a n on-repeating sequence of the 2n d istin c t n —b it strings (codew ords) over {(), 1} such th a t successive codewords in th e sequence differ b y th e com plem entation of a single bit. T h e French engineer B audot invented this code as a solution to th e problem of m inim izing e rro r in reading a b in ary telegraph transm ission message. A tra n s m itte r keyboaid em ploying a five digit code was exhibited a t th e U niversal E xposition in P aris in 1878. F or an account of th e work of B audot, see H eath [26]. F ran k G ray [16], ap p a re n tly w ithout knowing th e work of B audot, first published in 1953 a description of th e code for use in p reventing errors in certain signal transm issions [16].

G ra y codes were, in fa ct, known in th e n in etee n th cen tu ry in th e form of a solution to a n u m b er of puzzles such as “T h e Chinese R ing” an d “T h e Tower of H anoi” (See th e paper b y M artin G ardner [13]).

If you consider the n —cube whose vertic s rep resen t all th e b in a ry strings of len g th n and w hose edges connect a pair of vertices if th e ir b in ary strings differ by one b it, th en the G ra y code gives a H am iltonian path or cycle in th e n —cube. In fact, an y H am iltonian path in th e n —cube graph gives a G ra y code. C ounting th e H am iltonian cycles an d path s in the n —cube is still an open problem . T h e first w ork on this problem was b y G ilbert [1-1]. T h e most recent results on th e problem a re given in Douglas [10].

G ray codes have num erous im p o rta n t applications in engineering and science. F or a survey see Godclyn et al. [15]. T h ere are m any in terp re tatio n s of a G ray code. Here, we describe a G ray code as a list generated recursively. From F igure 1.4, we can easily deduce th a t a G ra y code of size n can be constructed from o re of size n — 1. L et L n den o te th e list of all n —bit strings, ordered as a binary reflected G ra y code. T h e recursive procedure to c o n stru c t such a B inary Reflected Gray code is sum m arized in F igure 1.6.

N ote th a t in th e algorithm show n in Figure 1.6, a list of sm aller size is read twice, th e second tim e in reverse order. In a la te r ch a p te r, we in tro d u ce th e alg o iith m by E hrlich [11] w hich generates th e code w ith o u t using a list. T h is alg o rith m has been im proved by Ilitn e r, E hrlich and Reingold [4] so th a t th e tim e to d eterm ine th e next codew ord is bounded by sem e co n stan t.

1. T h e list Lq is em pty.

2. For each n = 1, 2 , . . having o btained L n- 1, then 2.1 w rite out licit T n_ i an d prefix each strin g w ith

a b it “0” ;

2.2 w rite out list Z n_ i in reverse o~der, and prefix each strin g w ith a b it “ 1” ;

3. T h e list L n is th e result of co ncatenating th e tw o lists th a t w ere form ed in step 2 .

F ig u re 1.6: G ray code generation

E h rlich [11] views algorithm s for g en eratin g p erm u ta tio n s, com binations, p artitions of n , a n d G ray codes as applications of th e generalized Jo h n so n -T ro tto r algorithm . T he J o h n so n -T ro tte r alg o rith m generates all p erm u ta tio n s in which a p erm u ta tio n differs from its predecessor by th e in terchange of tw o ad jacen t elem ents [27] [55]. From an algorithm ic p o in t of view, we will see t h a t our algorithm s for generating th e ideals of a forest poset are of th e sam e basic kind.

1.3

O rd erly A lg o rith m s

O rderly algorithm s were in tro d uced b y R ead in [41] and [42] in efforts to im prove naive grap h generation techniques such as th o se used, for exam ple, in Heap [25]. A typical clas­ sical alg o rith m for grap h generatio n m ay be illu stra te d as follows. C onsider the problem of m aking a list of graphs w ith q + 1 edges from a list of those w ith q edges. See F igure 1.7. In th e classical algorithm , we n o te th a t in step 2, isom orphic graphs are usually generated m any tim es, and in step 3, we have to search for G in th e en tire list L q+1; conducting an isom orphism test o f G again st all graphs already in th e list L q+i.

7

1. S ta rt witli a list L q o f all graphs with q edges. 2. Pick each graph of L q in tu rn and from it generate

candidates for th e list L q+\ by th e addition of a new edge ' ail possible ways.

3. As each candidate g ra p h G for L q+\ is produced

determ ine w hether L q+1 contains some grap h isom orphic to G. If it is th e case, reject G and continue

processing th e next candidate; if n ot, ad d G to L q+\.

F igure 1.7: G raph generation by a classical m eth o d

i

2

F igure 1.8: T h e code of a grap h

consider th e “code” of a graph, th e canonical graph. T h e canonical grap h rep resen ts th e isom orphism class to which it belongs. A ty p ical code for graphs is t h a t defined from th e adjacency m atrix as in Figure 1.8. T h e up p er trian g u lar elem ents o f an adjacency m a trix are read off by colum ns to give a bin ary string. T his strin g changes according to how th e vertices of th e grap h are num bered. A m ong all th e possible ways to n u m b er th e vertices, we choose th e one which makes th e strin g m axim al (e.g., lexically). T h e m axim al strin g is called th e code of th e graph. If a grap h is encoded in th is way, th e ta sk in ste p 3 of th e classical generation m ethod reduces to checking if th e code is alread y in

F u rth e r we define an ord er relation (th e ulisto rd e r” d enoted by “ -<” ) over th e elem ents of L q. If A -< B , we say th a t “A comes before B ” or “A precedes B ” o r “B comes after A ” .

1. S ta rt w ith L q of canonical graphs arranged in list order “ -<” . T h e list L q+i is initially em pty.

2. T ake each graph of L q in order an d apply on it w ith

th e au gm enting operation to get a sequence of candidates for L q+i 3. As each can d id ate G is produced, test if the candidate graph G is canonical, and if it comes after th e last grap h a t present in L q+i If it is so, add it to L q+1; otherw ise ignore it.

F igure 1.9: G rap h generation by an orderly algorithm

A typical exam ple of such an ordering is th e lexical ordering. If each graph is represented by its code defined as in th e above exam ple, th e lexical order of th e codes of L q is easily u n derstood. N ote th a t as opposed to th e classical algorithm s, o rd 'v ly algorithm s do not require a search of th e list L q+i in order to d eterm in e w heth er or no t a graph has already been generated.

T h ere are th re e things w ith which we have to be concerned w hen we design orderly algorithm s. F irst, we m ust define th e code for each isom orphism class over th e set of (labeled) graphs w ith th e p a ra m e te r q. Secondly, we m u st define the n a tu re of the aug­ m enting operatio n s. T hirdly, we m ust specify th e th ree orderings: (1) th e order in which th e codes of th e graphs a p p e a r in th e lists L q an d L q+1; (2) th e order in which the aug­ m en tin g process creates candidates for f/g+ i; (3) th e ord er used in th e definition of the code (th e ord er o f isom orphs).

Now, suppose we have an orderly algorithm creatin g L p+i from L p. C onsider a m apping / from a canonical g ra p h G in L p+j to a canonical grap h G ' in L p such th a t G ' is the first g ra p h in L p v'hich produced G. Such a m apping m ust satisfy th e following three conditions: (1) an y canonical g ra p h in L p+j can be produced by th e au gm enting operations from a t least one canonical grap h in L p; (2) th e m apping / is weakly m on o to n ic; th a t is, if X , Y £ Tg+i an d I x F , th en f ( X ) -< / ( F ) ; (3) th e ord er of au gm enting a canonical

A weak monotonic function f: L p+I —:» L p f(X>

' /

Figure 1.10: N ecessary and sufficient conditions for th e existence of orderly algorithm s

graph in L p conforms to th e order of graphs in th e list (S trictly speaking, graphs th a t are no t added to L pJri can b e generated in a n y order). T hese conditions a re illu stra te d in F igure 1.10. R ead [41] proved th a t these are not only sufficient b u t also necessary conditions for th e existence of orderly algorithm s. T h u s, designing an orderly algorithm is equivalent to finding such a weak m onotonic function.

Let us clarify th e com plexity of orderly algorithm s. L et L v den o te th e list of all non- isom orphic canonical graphs w ith a p a ra m e te r p. L et I ( p ) be th e tim e required to do an isom orphism test for two graphs w ith a p a ra m e te r p and A {p) th e tim e required to do a canonicity te st for a grap h w ith a p a ra m e te r p. A canonicity te st basically involves canonically labeling th e vertices of a grap h an d determ in in g w h e th e r o r n o t th e original labeling of the given graph is sam e as this canonical labeling. F in d in g a canonical labeling usually involves com paring th e labelings for a ll th e autom orphism s of a given graph. Let L'p be th e list of graphs augm ented from each canonical g ra p h of L p. (Hence, a n augm ented graph in is e ith e r canonical or non-canonical; therefore, precisely speaking L p is not

a list b u t a set. However, o u r m ain concern is th e cardinality of a set. T h u s, we use the te rm lis t instead of se t for convenience.) We denote th e cardinality of a set A' by |.\'|. In a classical generation scheme, each augm ented elem ent of L p is tested again st all the graphs in the c u rren t L p+1 being created. Therefore, th e to ta l tim e to com plete the list L p+1 is a t m ost \L p\\Lp+i \ I ( p + 1). O n th e o th er hand, th e to ta l tim e to create the list by an orderly algorithm is a t m ost \L “\A (p + 1). N oting th a t [Z,r,( is typically exponential or super-exponential, we realize th a t th e orderly algorithm shows a significant im provem ent over th e classical one. N ote th a t a t th e m om ent of w riting th is dissertation, it is nov known w hether or no t isom orphism te st can be done in polynom ial tim e in th e size of a graph. However, we can replace an isom orphism te s t by a procedure to find a canonic foim of a graph. I t is no t necessary t^ find a canonic form of a grap h in order to d eterm in e w hether a re p resen tatio n of a grap h is canonic, however, a procedure to find a canonic form of a grap h provides one m eans b y w hich a canonicity te s t can be perform ed.

O u r oraerly algorithm s described initially in this d issertatio n are slightly different from R e a d ’s original orderly algorithm w ith respect to th e order w ithin a list. In R oad’s algorithm , all th e graphs were ordered in some order w ithin a list. Suppose th a t wo do not concern ourselves w ith th e order of graphs in a list b u t all th e graphs in th e list have th e sam e p a ram eter, say, p. So, precisely speaking, th e te rm “list” is no t correct, b u t for convenience we use “list” instead of “set” . C an we g enerate a list L p so th a t the order of th e graphs in th e list does no t m atter? T h is is feasible since a canonicity test depends solely on th e graphs, b u t n o t on th e order of graphs in th e list L p. (If one needs an ordered list, we m ay so rt th e list la te r.) Such an orderly algorithm w ith o u t th e condition of linear order w ith in a list is called a weak-orderly algorithm . In th e final ch a p te r, we will describe a technique th a t can be used to transform w eak-orderly algorithm s to orderly algorithm s w ith th e p ro p e rty th a t all graphs in each generated list are ordered.

A n o th er difference betw een o u r orderly algorithm s an d R ead’s original orderly algo­ rith m is th e m eth o d of au g m en tatio n . A grap h can be classified by tw o n atu ra l p aram eters, nam ely, th e n u m b er of vertices p an d th e num ber of edges q. T h e original orderly algo­ rith m b y R ead [41] uses q as th e p a ra m e te r of th e list while p is fixed. In our orderly

11

algorithm s in this d issertation, we use th e n u m b er of vertices p as th e p a ra m e te r of the list and q varies as well. T h e num ber of edges varies since adding one v erte x can induce a different num ber of new edges (0 to p) incident w ith th e augm ented vertex.

Finally, we briefly describe th a t th e relationship am ong orderly algorithm s, the grap h coding problem , and the isom orphism problem . G raph generation ac tu a lly constructs graphs as well as counts them . T h u s, g ra p h generation requires effi :ient d a ta re p resen ta­ tion to avoid a m em bership te st, i.e., a te st to d eterm ine w hether or no t a given grap h has th e desired properties such as acyclicity, p la n a rity an d colorability (see C olburn and R ead[6]). F inding a good code for a specific graph family involves finding an efficient d a ta representation o f th e graph. T h u s, orderly algorithm s are also closely related to th e graph coding problem (see R ead and C olburn [45]). T h e grap h coding problem is, practically speaking, the problem of canonically labeling th e vertices of a given grap h by integers so th a t tw o graphs are isom orphic if and only if th ey have th e sam e labeling (th e code). Hence, if the grap h coding problem is solved efficiently, we can also solve th e isom orphism problem efficiently. A pplications of grap h coding can b e found in th e field of chem istry (see R ead [43], [44]).

1.4

P r e v io u s W ork an d A p p lic a tio n s

G en eratin g the ideals of a poset has several applications in O perations R esearch and O ptim ization. S teiner [53] tre a ts th e problem o f generatin g ideals as an in teg ral p a rt o f dynam ic program m ing algorithm s for precedence-constrained scheduling problem s, as­ sembly line balancing, p roject scheduling, and reachability in reliability netw orks. T h e problem of generating th e ideals o f a tre e poset lexically was first considered by R uskey [49]. T h e algorithm for ideal generation of [49] runs in lin ear average tim e. T h e algorithm o f S teiner [53] for generating all ideals of a n a rb itra ry poset runs in linear average tim e.

A lgorithm s for generating graphs axe useful in exploring certain conjectures in g ra p h theory a n d in o btaining census inform ation for graphical en u m eratio n problem s. L isting some families of chem ical com pounds can be done by generating graphs representing th e

chemical stru ctu re s. For exam ple, an alternant molecule is essentially represented by a bicolorcd grap h (see B alaban [1]). Chem ists are often interested in having som e com put­ erized sy stem for cataloging chem icals so th a t a given chem ical can be easily identified, m aking chem ical p a te n t searching, for exam ple, less tedious. T h e relationship between C hem istry and G ra p h T heory is exposited a t length in B alaban [1]. Bicolored graphs w ithout isolated vertices are also used to represent sim plical complexes which in tu rn represent some arch itectu re design problem s. O ne arch itectu ral application is concerned w ith th e m inim al re p resen ta tio n of p a tte rn s based on a re ctan g u lar grhl. Since a grid is a bicoiored graph, th ese p a tte rn s can be also represented b y bicolored graphs. See H arary et al. [22] for an application of enum eration techniques to arch itectu ra l design. D eter­ m ining th e value o f p erm anent of a m a trix can be done by an algorithm for counting all m axim um m atchings in a b ip a rtite graph corresponding to th e m a trix (see Nijenhuis and W ilf [38]). A technique for generatin g th e m axim um m atchings in a general grap h is used for analyzing some biom edical im ages (see T anim oto [54]). For developm ents in the field of graph generation, see R ead [42]. T h e problem of graph generation culm inated in 1981 w ith th e com pletion of th e ca ta lo g of all graphs w ith 10 vertices. F or an account of this cataloging, see C am eron e t al. [5].

T h e en u m eratio n by m eans of closed an aly tic form ulas of acyclic tran sitiv e digraphs (p a rtia lly ordered sets), finite topologies (tra n sitiv e d igraphs), and finite lattices are u n ­ solved problem s (see H arary a n d P alm er [23] an d also BirkhofT [3]). T h u s, in order to count th ese graphs w ith a given n um ber of vertices p, we m ust cu rren tly rely on gener­ ative m eth o d s. For acyclic tra n sitiv e digraphs and finite topologies, Evans, H arary, and Lynn [12], using a com puter, counted th e num bers of labeled cases on up to 7 vertices. For th e unlabeled case, graphs on up to 7 vertices have been generated and recorded in a book by Sloane [51]. C ulberson an d R aw lins [7] recently generated th e acyclic transitive digraphs u p to 11 vertices. L astly, K yuno [33] an d K yuno an d Ito [34] generated the finite lattices u p to 10 vertices.

In th e above described research, Evans e t al. and K yuno and Ito used an expensive isom orphism test to elim inate duplicates. B y using an orderly algorithm , we can avoid

13

the repeated use of an isom orphism test (see Read [41]).

A k - "d o red graph is represented by exactly th e sam e canonical form as can be used for acyclic digraphs. T h u s, we can obtain an orderly algorithm for generatin g A:—colored graphs from an orderly algorithm for generating acyclic digraphs. For labeled ^-colored graph enum eration, see Read[40] and for bicolored an d bicolorable graph enum eration, see Ilarary[17], [20].

1.5

O v erv iew

T h e plan of this dissertation is as follows. In C h ap ter 2, various definitions and notatio n which are used in la te r chapters are defined. In C h a p te r 3, th e algorithm for generating a binary reflected code by Ilitn e r E hrlich and R eingold is intro d u ced . T hen, th e exis­ tence? of a C ray code of lower ideals of a forest poset is given. Finally, th e alg o rith m for generating such lower ideals in a G ray code m anner is given and som e m ethods of lower ideals generated by the algorithm are also given. In C h a p te r 4, th e previous algorithm is im proved to an algorithm w ith theoretically o p tim al tim e com plexity. In C h a p te r 5, we describe orderly algorithm s for g enerating acyclic digraphs, acyclic tran sitiv e digraphs (posets). finite lattices, an d finite topologies. In C h a p te r 6, we give o rderly algorithm s for generating A:—colored graphs. We also p re sen t som e properties by w hich we m ay im prove the efficiency of orderly algorithm s in general. In th e last ch a p te r, we sum m arize our results and ■'escribe open problem s related to th e topics tre a te d in th is d issertatio n .

M a th e m a tic a l P r e lim in a r ie s

G rap h -th eo retical term s used in this dissertation m ay be found in Ila ra ry [20], except th a t in this d issertatio n , p o in ts are called vertices and lines are called edges and similarly, p oset-theoretical term s m ay be found in S tanley [52]. F or th e convenience of the reader, however, some o f th e ce n tral term s are recalled in th is ch a p te r. We consider a set of elem ents P an d a binary relation < defined on P . In this d issertatio n , the nu m b er of elem ents in P is always finite. By an abuse of n o tatio n , we denote this as P = (P , < ). We say t h a t x and y a re comparable if eith er x < y or y < x \ otherw ise, they a re incomparable. We w rite a: < y if x < y b u t x ^ y. T his P can be represented as a grap h as follows. Let each elem ent x in P be represented by a vertex. Let tw o vertices x and y be connected by a n arc from y to x if th e corresponding tw o elem ents x an d y are such th a t x < y. T hen, th e g ra p h constructed in such a way is called a digraph, denoted also by P , for convenience.

A d igraph P is called a transitive digraph if P satisfies th e condition: for x an d y in -P) x < U and y < z im ply x < z (tra n s it vity). Refer to F igure 2.1 for an exam ple of a tran sitiv e digraph.

A sequence o f d istinct elem ents in P , C = x x < x2 < . . . < x n is called a chain or directed path in th e corresponding graph if any consecutive two elem ents of C are com parable w ith respect to th e binary relation in P . T h e length o f the chain C is denoted by 1(C) = | C | - 1 . If x i and x n are equal in th e chain, th e chain is called a directed cycle.

15

A digraph P which has 110 directed cycles is called an acyclic digraph.

A source is a vertex w ith no incom ing arcs. If a: is a vertex of a digraph P having no directed cycles, we define th e height h( x ) as follows. h( x ) = s u p { l{ C ) : C ch a in £ C(x) }, where C(x) = { y £ P : 3 ch a in f r o m y to a:}. In p artic u la r, if a: is a source, h( x ) = 0 . Similarly, h ( P) — su p { h (x ) : x £ P ] is th e height of P . If th e underlying digraph is clear, wo sim ply w rite h (w ith o u t a param eter) instead of h ( P) . A layer is a set of elem ents in P which have th e sam e height. See F igure 2.2.

A digraph P is called a partially ordered set (poset) if th e b in ary relatio n satisfies the following tw o ad d itio n al conditions as well as tran sitiv ity : ( F a; < x for all x £ P (refiexivity), and (2) x < y an d y < x im ply x — y (an tisy m m etry ). A graph representing a p artially ordered set is called an acyclic transitive digraph. A n elem ent a: of a poset P is m c n m a l (m inim al) if x < y (x > y ) implies x — y for all y £ P \ x is greatest (least) if y < (y > x) for all y £ P .

We generally denote a poset by V , an d its underlying graph by P . We den o te th e set of elem ents in V by S ( V ) an d relation 1 h a t defines V by R ( V ) C S ( V ) X S ( V ) . If x, y G V , then we say y c o v rs x if x < y and no elem ent z 6 P satisfies x < z < y. We also say th a t .r is a child of •/ and y is a p a r e n t of a: if x < y. A n ideal I of a poset V is a subset of the elem ents of V such th a t x £ I an d y < x implies th a t y £ I .

An antichain, A, is a subset of th e elem ents of V in which every tw o elem ents are

2 4 6

5 3

Figure 2.2: A n exam ple of acyclic digraph w hose vertices are ordered by distance from sources

incom parable; t h a t is, if a:, y € A w ith x ^ y, th e n neither x < y nor y > x holds. Observe th a t th e set of m axim al elem ents M in an ideal I forms a n antichain and M fixes I in the sense t h a t I — {a:|3m £ M , x < m ). T h u s, th e re is a one-to-one correspondence between th e ideals and th e antichains.

T h e poset th a t is an an tich ain o f n elem ents is denoted A n . For tw o sets A and B, the H am m ing d istan ce betw een th e tw o sets A an d B is th e n u m b er of elem ents in which they differ, th a t is, J { A , B) — |(A - B) U (B — A )|.

A p o set can b e represented b y a Ilasse diagram in w hich d istin c t elem ents are repre­ sented b y d istin ct points, in which x is placed above y w henever x > y, and x and y a r e joined b y an arc from x to y w henever x covers y. A ny finite poset is determ ined up to isom orphism by its diagram .

If th e H asse diagram of a poset V is a tre e rooted a t a unique m inim al elem ent, we sim ply say th a t th e poset is called a tree poset and th e Ilasse diagram is term ed a tree. If th e H asse diag ram of a po set is a disjoint union of trees, we term th e poset a forest poset. W e m ay tr e a t this tree poset as an ord ered tree w ith th e m inim al elem ent being the ro o t. In general, there are m any ordered trees corresponding to a given poset. We will pick up one of th em as a rep resen tativ e o f th e poset. If a poset is a tree poset, a tree is an o rie n te d ordered tree, w here th e direction of arcs a re oriented tow ards th e root, the m inim al elem ent. By ordered tree we m ean its subtrees a re ordered. An ordered forest is sim ilarly defined. Nodes in an ordered forest a re num bered in preorder.

17

C hildren of a com m on p aren t are called siblings. Rooted-subtrees are subtrees whose m inim al elem ent is th e ro o t of a tree poset. T hen, we can easily see th a t th ere exists a one-to-one correspondence betw een th e set of lower ideals of a tre e poset T and th e set 01 rooted-subtrees. By th e definition of an ideal, th e em p ty su b tree is also an ideal of a tree. Each ideal of a tree is represented by a codeword of a b it-strin g w here “0” m eans the corresponding elem ent is n o t in th e ideal an d “1” m eans th e corresponding elem ent is in th e ideal. In this paper, su b tree m eans rooted -su b tree unless otherw ise stated .

A fo rest poset is one whose Hasse diagram is an ordered forest. T h e ro o ts of th e trees of th e forest are th e m inim al elem ents of th e poset. B y x -< y or y y x we m ean th a t the elem ent x is th e predecessor of y or y is th e successor o f x in th e p reo rd e rin g of th e elem ents in a forest poset T .

In R uskey [49] th e ideals of a t ’-ee poset are called r-subtrees. T h e se t of ideals of a poset V , ordered by set inclusion, form a d istrib u tiv e la ttic e J { V ) . It is well know n t h a t for every finite d is t”ib u tiv e la ttic e L, th ere is a unique poset V for w hich J { V ) is isom orphic to L. For m ore inform ation an d th e definition on a d istrib u te d lattic e, see S tan ley [52J. In a slight abuse of no tatio n , we also use J ( P ) to denote th e H asse diagram o f th e la ttic e of ideals of 'P. In this con tex t we refer to J( f P) as the ideal graph of P . W e a re in terested in finding H am ilton cycles an d p a th s in th e underlying u n d irected graph of H asse diagram s of d istrib u tiv e lattices (such a cycle or p a th yields a G ray code of th e ideals, th e o b ject of our algorithm s). For m ore inform ation on posets an d ideals, see S tanley [52].

N ote th a t J ( P ) is connected and b ip a rtite . I t is connected since every d istrib u tiv e lattic e lias a m axim um and a m inim um elem ent. T h ere is a b ip a rtitio n o f th e vertices of J ( ’P) in to tw o sets, E ven{V ) and 0 d d ( V ), depending on w h e th e r th e ideal represented by a vertex has an even or odd n um ber of elem ents. T h e p a rity difference d ( V ) of a poset V is defined to be th e difference \E ven(V )\ — 10 dd{P)\.

T h ere are two sta n d a rd ways of com bining posets to get a new poset. T h e direct sum of tw o disjoint posets V and Q is th e poset V + Q on th e union S ( V ) U £ ( 2 ) such th a t x < y in V + Q if an d only if eith er x , y £ V an d x < y in V , or x , y € Q and x < y in Q. T h e ordinal sum of tw o disjoint posets V an d Q is th e poset V ® Q on th e union

S (V) U S ( Q ) such th a t a: < y in V 0 Q if x , y £ V a n d x < y in V , or x , y £ Q and x < y in Q, or if x £ V and y € Q. N ote th a t a tree p o set can be built up by applying the operations of direct sum of tw o forest posets and th e ordinal sum of a forest poset w ith a single elem ent.

A g ra p h G = (V (G ), E ( G )) consists of a set of vertices V { G ) and a subset of unordered pairs of vertices called edges E ( G ) C V ( G ) X V (g). T h e product of th e graphs G and H is the g ra p h G x H w ith vertex set Vr(G ) X V ( I I ) , in w hich [v, w] is ad jac en t to [?>', «>'] if and only if v — v' and [w, w '] £ E { I I ) or w = w ' and [u, v'] £ E ( G ) . We can easily see th a t

J(7>! + Vi ) “ J (V i ) x J {V2) (‘2.1)

where = denotes g ra p h isom orphism .

W e a re in terested in efficient algorithm s for generating com binatorial objects. Suppose th a t th e objects a re represented b y sequences of n elem ents an d th a t th e to ta l num ber of objects is N . An alg o rith m for g enerating those o b jects is said to ru n in constant average time if th e to tal a m o u n t of c o m p u tatio n (excluding o u tp u t) is 0 ( N ) . If the to ta l am ount of com p u tatio n is 0 ( n N ) , th e n th e algorithm is said to ru n in linear average time.

A c o n sta n t average tim e alg o rith m is said to be loopless (o r c o n sta n t w orst-case tim e) if (a) th e am o u n t o f co m p u tatio n used in going from one o bject to th e next is 0 ( 1); (b) the am o u n t of co m p u tatio n needed to produce th e first o b ject is 0 ( n ); an d (c) th e am ount of co m p u tatio n needed to decide w h eth er an o bject is th e last one is 0 ( 1). Up to a co n stan t factor, no algorithm can be fa ste r th a n a loopless algorithm . Sim ilarly, a lin ear average tim e alg o rith m is called linear worst-case tim e if (a) and (c) above can be done in 0 (n ).

A lattice L = (L, V, A) is a poset L such th a t for every two elem ents x , y in L, the poset { z £ L : z > x , z > y } has a least elem ent x V y € L, th e jo in of x an d y, and the poset { z £ L : z < x , z < y } has a g re atest elem ent x A y € L , th e meet of x and y. We call th is lattic e p ro p e rty the unique bound condition. If only eith er a g re atest or a least elem ent is defined for any tw o elem ents x and y, th e lattic e is called a semilattice. For sim plicity, the H asse diagram of a la ttic e is also called a lattice.

19

d c

Figure 2.3: L4: F our elem ents n o t satisfying the unique hound condition

PROPOSITION 2 . 1 I f a poset P is a lattice, then P does no t contain the poset L 4 o f figure

2.3 as a subposet.

□

If th ere is such a set of four elem ents in P , w e say th a t th e four elem ents violate the unique bound condition.

A digraph is strongly connected, or strong, if an y two vertices are m u tu a lly reachable by a directed p a th . A strong component of a d ig rap h G is a m axim al stro n g subgraph. Let S\ , S 2, ■ ■ ■, S m be the stro n g com ponents o f G. T h e condensation G* o f a d ig rap h G

is a d ig rap h th a t has the stro n g com ponents of G as its vertices, w ith an arc from S i to

Sj w henever th ere is a t least one arc in G from a vertex of S{ to a v e rte x in Sj. T h e condensation G* of any dig rap h G has no cycles, and hence is an acyclic digraph. Tw o vertices u and v of th e d igraph G are similar if th ere is som e au to m o rp h ism a of G, for which a ( u ) = v. Tw o edges x \ = (u i,U i) an d x2 = ( u2, v 2) are similar if th ere is an autom orphism a of G such th a t a ( ( u i , u i ) ) = ( u 2, v2). A d ig ra p h is sym m etric if every p air of vertices is sim ilar an d every p air of edges is sim ilar.

We s ta te tw o lem m as w ith o u t proofs. For th e proofs, see H a ra ry [12](page 200).

Le m m a 2 . 1 Every strong com ponent o f a transitive digraph is complete and symm etric.

LEMMA 2 . 2 The condensation o f a transitive digraph with m strong components is itself an acyclic transitive digraph on m vertices.

r sucli th a t u> is th e r t h m em ber of th e family. T he lexical order of two vectors v and w is defined as follows. If v = (i>i, «2, • • •»v n) an d w = (u q , w2, - . . , w rn), th e n v is lexically larg e r th a n w (denoted by eith er v X- w or w -< v) if an d only if th ere is som e k < m , n such th a t V{ = Wi for all i < k , and vk > Wk for som e k < m , n , or n > m an d for I < i < m , V{ = If v could be equal to w, we w rite v y w or w < v.

F or a given vector of size n , v = (e i, e2, . . . , c n), a prefix v ' is defined to b e th e subvector com posed o f th e leading i elem ents of n; th a t is, v ' = ( e i, e2, . . . , e;), 1 < i < n.

A perm utation m a trix S is a square m a trix which has precisely one “1” in each row and precisely one “ 1” in each colum n. T h e p erm u ta tio n m a trix S is obtained from the identity m a trix by in terch an g in g th e fcth row w ith th e r t h row such th a t k < r and c rk ^ 0. Let M b e a square m atrix . T hen, M ' = S X M , where “ x ” is the usual m atrix m ultiplication, is th e square m a trix ob tain ed b y interchanging the fcth row and th e r t h row in th e m atrix M an d M " = M X S is th e square m a trix o btained by interchanging th e fcth column an d th e r t h colum n in th e m a trix M . If M is an adjacency m a trix for a graph G, then M ' = S X M x S is th e adjacency m a trix for th e grap h G obtained b y interchanging th e labelings of th e vertices k an d r. T h e m ultiplication of tw o p erm u ta tio n m atrices S ' = S i X S2 is obviously also som e p erm u ta tio n m atrix .

A A’-colored g ra p h is a g ra p h G = ( X i , X2, . . . , X k , V , E ) w here Xi , i = 1 , . . . , k are m u tu a lly disjoint subsets of th e vertices V , w ith u£_x X,- = V a n d E is th e set of edges,

w hich is a su b set of unordered pairs of vertices (a;,-, Xj ), Xi £ X i an d Xj £ X j , 1 < i < j < k.

W e consider in this p a p e r only th e /s-colored graphs for which each of th e X i , i = l , . . . , k is n o t em pty, i.e., |X i| = m,- > 0 .

C h a p te r 3

F o rest P o s e t id ea ls

3.1

In tr o d u c tio n

A m ong all th e G ray codes th e one m ost widely know is called th e B in ary R eflected G ray C ode (B R G C ). A lgorithm E due to E hrlich [11] for generating th e B R G C is described in Figure 3.1. T his algorithm runs in 0 ( n ) tim e from one codew ord to th e n e x t. T h e algo­ rith m can be viewed as analogous to th e well know n Jo h n so n -T ro tte r algorithm [27][55] for generatin g p erm u ta tio n s. In th e Jo h n so n -T ro tter algorithm , each elem ent has an asso­ ciated priority, an d a sta tu s. If th e s ta tu s of an elem ent is movable, it is m oved(sw apped w ith th e ad jacen t elem ent), otherw ise, th e elem ent of th e n ex t highest p rio rity is m oved. Once an elem ent of some p rio rity is moved, all th e elem ents w ith higher p riorities becom e movable. T h e process continues u n til all th e elem ents are n o t m ovable.

In th e case of o u r algorithm s, th e p riority of th e elem ents of a forest poset is defined according to th e preorder of th e elem ents in th e forest p o set (th e ro o t elem ent has th e highest prio rity ). All th e elem ents are considered according to th is p rio rity as in th e Jo h n so n -T ro tter algorithm . In add itio n , all th e elem ents have predecessor or successor relationships in th e poset. Unlike th e Jo h n so n -T ro tter algorithm , only th e elem ents in the c u rre n t ideal or having a predecessor in th e c u rre n t ideal are considered. M oreover, no elem ents having a successor in th e ideai can be an ca n d id a te to b e rem oved from th e c u rren t ideal.

Algo- ith m E lias been im proved by B itner, Elirlicli and Reingold to th e one which runs in 0 ( 1 ) tim e from one codew ord to th e next. We refer to th eir algorithm as Algorithm B ER.

We ad o p t a different appro ach to im prove A lgorithm E. A lgorithm B, our version of A lgorithm B E R (to be explained in C h ap ter 4), is used as a subprocedure in th e algorithm for g en eratin g ideals of a forest poset which is th e m ain result of th e first p art of this dissertation.

3.2

T h e E h rlich an d B E R A lg o rith m s

Now, w e in tro d u ce A lgorithm E by E hrlich [11]. In connection w ith our A lgorithm P for generatin g forest ideals, we describe A lgorithm E in term s of poset elem ents. Algorithm E works on n u n re la ted p o set elem ents (i.e., an antichain) A n . T h e algorithm lists all the possible subsets S of A n . W j assum e th e elem ents are labeled in increasing order from one to n and arran g ed accordingly w ith th e sm allest on yo u r left. T h e sm aller th e label is, the higher th e priority.

W e consider t h a t each elem ent of A n can ta k e eith er f r e e s ta tu s or s tu c k statu s. If th e s ta tu s of th e elem ent is free, th e elem ent m ay b e eith er rem oved from th e current subset S (th e elem ent is deletable) or added to S (th e elem ent is a u g m e n ta b le ); otherw ise, th e s ta tu s of the elem ent is stuck an d it can b e n eith er rem oved from S nor added to S . T he set A n is changeable if th ere exists an elem ent w hich is e ith e r d eletable or augm entable.

Initially, all th e elem ents are f r e e . T h e algorithm runs as long as th ere exists a free elem ent in S (ste p (4)). W e choose th e leftm ost free elem ent v from S (step (5)). T hen, all th e elem ents to th e left of v becom e free an d v itself becom es stuck (steps (6 ) and (7)). D epending on w h e th e r or n o t v is in S , we add v to or delete v from S (ste p (8)). After th e u p d a te d S is o u tp u t (step (9)), we re tu rn to step (4).

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( 1) forall v € S d o status(v) <— free; (2) S <— 0;

(3) O utput(A ');

(•1) w h ile S is changeable d o b e g in

(5) v <— min{v £ «? | v is changeable};

(fi) fo ra ll w < v d o status(w ) <— free ;

(7) status(v) <— siucfe; (^) i f n € 5 th e n S <— S — {n} e ls e S <— 5 U {«}; (9) O utput(A '); (10) end; F igure 3.1: A lgorithm E

3.3

T h e E x iste n c e o f a G ray C o d e o f L ow er Id ea ls o f a

F orest P o s e t

A pi oof of th e existence of a G ray code of lower ideals of a forest poset is given by induction on th e n u m b er of elem ents in th e poset. If a G ray code has th e p ro p e rty th a t th e first codeword and th e la st codeword difFer by a single elem ent, it is called a cyclic Gray code. N ecessary an d sufficient conditions for a G ray code of lower ideals of a forest poset to be cyclic are also given below.

F irst, we give tw o easy lem m as. We show th a t th e re is a H am iltonian p a th in J ( V ) if V is a forest poset. T his will follow as a consequence of m ore general results a b o u t th e ordinal and direct sum s of posets. A necessary condition for th e existence of a H am iltonian p a th in J ( V ) is th a t th e p a rity difference \d(V)\ < 1. L et us first d eterm ine d ( V ) if V is a forest poset.

L et i[V) denote th e nu m b er of elem ents in J ( V ) (th e n u m b er of ideals o f V ) . It is easy to show th a t i ( P + Q ) = i( V ) X i(Q ) and t h a t i(V ® Q) = - 1 + i( V ) + i(Q). Similarly, we have

d{V + Q) = d(V )d{Q )

d{V © Q) = • —1 + d ( V ) + d(Q ) if |V\ is even + 1 + d { V ) - d ( Q ) if |^>j is odd

F or forests we apply (3.2) w ith V = A \ . Since d ( A \ ) = 0, th e p a rity difference fm forests is 0 o r 1. However, th a t d ( V ) = 0 is no t a sufficient condition for J ( V ) to have a H am iltonian p~th. Such an exam ple is illu strated in F igure 3.2.

LEMMA 3 . 1 L et V and Q be non-em pty posets both o f whose ideal graphs hare Hamiltonian

i f \ J { V ) \ (or \ J { Q ) \) is even then J ( P + Q ) has a Hamiltonian cycle.

PROOF: Let H be a H am ilto n ian p a th in J { V ) and I I ' be a H am iltonian p ath in J (Q ). T h e n H x H ' is a sp an n in g su b g ra p h of J ( V ) x J ( Q ) . Hence, by (2 .1) it is again isom orphic to a sp anning su b g rap h of J ( V + Q ). Since I I and I I ' are p ath s, I I X I I ' is an | / / | by \II'\ grid graph. T h e ideal 0 o f J ( V + Q) corresponds to th e corner vertex (0 ,0 ) of II x I I '. It is easy to c o n stru c t a H am iltonian p a th in a grid grap h , sta rtin g a t a corner. Furtherm ore, unless b o th \H\ and \H'\ are o d d , a H am iltonian cycle can be found easily in II x II '. □ paths starting at 0. T h en J ( V + Q) has a Hamiltonian path starting at 0. Furthermore,

J(P)

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Le m m a 3 . 2 Let V be a poset f o r which J { V ) has a Hamiltonian path starting at 0 and ending at S ( V ) , and let Q be a poset fo r which J ( Q ) has a Hamiltonian path starting at 0. T hen there is a Ham iltonian path in J { V © Q) starting at 0.

PROOF: Lot a i , a 2, .. .,a iv be a H am iltonian p a th in J ( V ) where a i = 0 an d ajv = S ( V ) . Lot b \ , 62, .. . , 6a / he a H am iltonian p a th in J ( Q ) where 6/ = 0. T h en th e p a th

a i , a 2, . . . ,ayv U b i , a ^ U b2, . . . ,a /v U 6a/

is a H am iltonian p a th in J ( V © Q) sta rtin g a t 0. □

N o te th a t the condition o f Lem m a 3.2 th a t th e H am iltonian p a th in J { V ) ends a t SlfP) cannot bo relaxed since S ( V ) is a cut v ertex of J ( V @ Q) (if V and Q are b o th non-em pty).

As a consequence of L em m as 3.1 an d 3.2 we have th e following theorem . T h e p ro o f is im m ediate by in duction on th e num ber o f nodes in th e forest, n o tin g th a t an y forest poset T can be constructed by re p eated applications o f eith er h i + T2 or T\ 0 T2 where Tj for .7 = 1 ,2 are disjoint subtrees o f T such t h a t i ( T j ) < i(!F) for j = 1,2.

Th e o r e m 3 . 1 I f V is a forest poset, then there is a H amiltonian path in J ( V ) .

We can also characterize tho se forests posets whose ideal graphs h ave H am iltonian cycles.

Th e o r e m 3 . 2 Let IF be a forest poset consisting o f trees T \ , T 2, • • •, T*. There is a Hamil­

tonian cycle in J(IF) i f and only i f k > 2 and there is at least one tree T{ for which | J(T,-)| is even.

P r o o f : If k = 1, th e n 0 is a p e n d a n t v e rte x in J(IF). If all ] J(T,-)| are o d d th en i ( T ) is odd, an d since J(IF) is b ip a rtite it cannot have a H am iltonian cycle. O n th e o th er han d , if k > 2 and there is a t least one tree T t- for w hich |7 (2 i)| is even, th en by Lem m a 3.1

3.4

A lg o r ith m P

A lgorithm P generates all lower ideals of a forest poset w ithout rep etitio n . It determ ines successive ideals by traversing th e previous ideal in effect in preorder. T h e tran sitio n from one ideal to th e n e x t takes 0 (n) tim e in th e w orst case, where n is th e num ber of elem ents of P . E xperim entally, however, th e algorithm has been found to exhibit on th e average co n stan t tran sitio n tim e from one ideal to th e next. We offer th e conjecture (see C h ap ter 7) t h a t A lgorithm P has co n stan t average tra n sitio n tim e, b u t th e problem ap p ears to be difficult.

T h eorem 3.1 is constructive. N ote, however, th a t the algorithm corresponding to die proof of T h eo rem 3.1 m ay yield different H am iltonian p ath s in J ( P ) for a given forest poset V depending on how V is decom posed in to sm aller tree posets. In this section we will describe a n algorithm th a t finds such a p a rtic u la r H am iltonian p a th system atically. T he m ain idea is to use a decom position based on a depth-first search. In C h a p te r 1 we will show how to im prove this algorithm so th a t it is loopless.

In C hap ters 3 an d 4 we deal w ith a fixed forest poset T w ith n elem ents. T h e elem ents of a forest poset are referred to as nodes (W e will reserve the term v e rtic e s for discussion of th e ideal g ra p h ). T h e p a re n t of a node v is denoted by par(v). We extend th e notions of s t a t u s , and changeable defined on an elem ent in an antichain to those defined on an elem ent of a poset. E ach node v has a sta tu s , status(v), which takes on one of tw o values, free o r stuck.

DEFINITION: L et F b e a rooted subforest o f a forest poset T , and v a node for which status(v) = free.

• If v F , th e n v is augmentable if v is a ro o t o r par(v) is in F.

• If v E F th e n , v is deletable if none of th e children of v is in F an d the s ta tu s of each child is stu c k .

A node th a t is aug m en tab le or deletable is said to be changeable. If there is a change­ able node in F , th e n F is changeable; if u is n o t in F , we say th a t v is null w ith respect