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Dynamic Algorithms for Chordal and Interval Graphs
by-Louis W a lte r Ibarra
B.S.. Jo h n s Hopkins University. 1988 M.S., University of \ l r g i n i a . 1990
A D issertation S u b m it te d in P a rtia l Fulfillment of t h e
R e q u ire m en ts 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 e p a r tm e n t of C o m p u te r Science We accept this d iss e rta tio n as conform ing
to th e req u ired s ta n d a r d
Dr. Valerie King, Supervisor (D ept, of C o m p u t e r Science)
— Dr. J o h n Ellis. D e p a r tm e n ta l M e m b e r (D e p t, of C o m p u t e r Science)
. Michael rellows. D snA rtm ental Memh
D r. Michael rellows. D snA rtm ental M e m b e r (D ept, of C o m p u t e r Science)
Dr. J in g m ia n g . O u ts id e M e m b e r (D ept, of M a th e m a tic s )
Dr. Pavol Hell. E x t e r n a l/ E x a m in e r (School of C o m p u tin g Science. Sim on Fraser U niversity)
0 Louis W a lter Ibarra, 2001 University o f V ictoria
Al l rights reserved. This dissertation m a y not be reproduced in whole or in part, by photocopying or other means, without the permission o f the author.
A B S T R A C T
We present t h e first d y n a m ic a lg o r ith m t h a t m a in ta in s a clique t re e r e p r e s e n ta tio n of a chordal g r a p h a n d s u p p o rts th e following operations: (1) query w h e t h e r de le ting or inserting a n a r b i t r a r y edge preserves chordality. (2) delete or insert a n a r b i t r a r y edge, provided it preserves chordality. We give two im p le m e n ta tio n s . In th e first, e ach o p e ra tio n runs in 0 ( n ) tim e, w here n is the n u m b e r of vertices. In t h e second, a n insertion query runs in 0 ( lo g ^ n) t im e , an insertion in 0 { n ) tim e, a d e le tio n query in 0 ( n ) tim e, a n d a dele tio n in 0 ( n log n) time.
We also in tro d u c e t h e c liq u e -s e p a ra to r graph r e p re s e n ta tio n of a c h o rd a l g raph, which provides significantly m ore in fo rm a tio n a b o u t t h e g r a p h 's s t r u c t u r e t h a n th e well-known clique tree r e p r e s e n ta tio n . We present fu n d a m e n ta l p ro p e r tie s o^ the c lique-separator g r a p h a n d a d d itio n a l properties when th e in p u t g r a p h is interval. We th e n in tro d u c e th e t r a i n tre e r ep re s en ta tio n of interval g rap h s a n d use it to decide w h e th e r th e re is a c e rtain linear ord erin g of the g r a p h 's m a x im a l cliques. T h is yields a fully d y n a m ic a lg o rith m to recognize interval g rap h s in O ( n l o g n ) t i m e per edge insertion or deletion. T h e c liq u e -s e p a ra to r graph m a y lead to d y n a m ic a lg o r ith m s for every pro p er subclass of c hordal g rap h s, an d the tra in tre e m a y lead to fast d y n a m ic a lg o rith m s for problem s on interval graphs.
E xam iners:
Dr. Valerie King, S up e rv iso r (D e p t, o f C o m p u te r Science)
Ill
Dr. M ichael Fellows. D e p a r tm e n ta l M e m b e r (D e p t, of C o m p u t e r Science)
Dr. J in g H u a n g . O u ts id e \ I ^ b e r (D e pt, of .M athem atics)
Dr. Pavol Hell. E x te rn a l E xam iner (School of C o m p u tin g Science. Sim on Fraser University)
A b str a ct ii
C o n ten ts iv
L ist o f figures vii
1 In tro d u ctio n 1 1.1 D ynam ic g r a p h a l g o r i t h m s ... I 1.2 C hordal g r a p h s ... 3 1.3 Interval graphs ... 9 1.4 O t h e r classes ... 11 1.5 R e s u l t s ... 12
2 A d y n a m ic alg o rith m for chordal graphs 14 2.1 I n t r o d u c t i o n ... 14
2.2 Clique t r e e s ... 15
2.3 T h e d y n a m ic a lg o r ith m for chordal g r a p h s ... IS 2.3.1 D e l e t e - Q u e r y ... IS 2.3.2 D e l e t e ... 19
2.3.3 I n s e r t - Q u e r y ... 22
2.3.4 Insert ... 24
2.4 F irs t i m p l e m e n t a t i o n ... 26
2.4.1 C o n n e c te d c o m p o n e n t s ... 27
2.5 Second i m p l e m e n t a t i o n ... 27
2.5.1 C o m p u t in g th e w e i g h t s ... 28
2.6 C onclusions a n d open p r o b l e m s ... 29
3 A d y n a m ic a lg o rith m for interval graphs 31 3.1 I n t r o d u c t i o n ... 31
3.2 T h e c liq u e -se p a ra to r graph of a chordal g r a p h ... 33
3.2.1 T h e clique -se pa ra tor g r a p h ... 34
3.2.2 D e f i n i t i o n s ... 37
3.2.3 T h e M ain T h e o r e m ... 38
3.3 T h e c liq u e -s e p a ra to r graph of an interval g ra p h ... 45
3.3.1 Relevance of ^ ... 45 3.3.2 S t r u c t u r e and size ol Ç ... 46 3.4 T h e t r a in t r e e ... 50 3.4.1 P Q - t r e e s ... 50 3.4.2 T rain t r e e s ... 50 3.4.3 P r o p e rtie s of tra in t r e e s ... 54
3.4.4 P re v ie w of the tra in tree a l g o r i t h m ... 60
3.4.5 T h e O rd e rin g L e m m a ... 61 3.5 Deciding w h e th e r a g ra p h is interval ... 64 3.5.1 O verview ... 64 3.5.2 T h e t r a i n tree a l g o r i t h m ... 65 3.5.3 C o rrec tn ess a n d c o m p l e x i t y ... 71 3.5.4 C o m p u t in g a clique p a t h ... 81 3.6 U p d a t e s ... 82
3.6.1 Insert ... S2
3.6.2 Insert c o r r e c t n e s s ... 90
3.6.3 D e l e t e ... 92
3.6.4 Delete c o r r e c t n e s s ... 102
3.7 C o m p u t in g the cliq u e -se p a ra to r g r a p h ... 108
3.8 Conclusions a n d f u tu r e w o r k ... 110
List o f Figures
1.1...g r a p h ... 4
2.1 . \ chordal g r a p h G ... 16
2.2 clique t re e T of t h e grap h in Figure 1.1... 17
2.3 Deleting {u. ü} from A 'j... 19
2.4 Before dele tin g {u. e } ... 20
2..5 Deleting {u...t'}... 21
2.6 Clique tre e T ... 24
2.7 Inserting {u. I’} ... 2-i 2.8 .Adding n o d e z t o T ... 26 3.1 .A g r a p h ... 34 3.2 .A chordal g r a p h G ... 3-5 3.3 T h e c liq u e -s e p a ra to r g r a p h Ç of G. T h e su p e rs c rip ts in d ic a te th e se t's size... 35 3.4 G ... 39 3.5 Ç ... 41
3.6 Ç{a.'2) is not th e c liq u e -s e p a ra to r g ra p h of G[V’( ^ (< t,2 ))] ... 43
3.7 A b o x ... 46
3.8 A s e p a r a to r n o d e 5 w ith t h re e in te rn a l n e ig h b o rs... 48
3.9 A P Q - t r e e ... 50
3.10 An interval g rap h , its clique -se pa ra tor g ra p h , a n d its tra in tree. . . . .53
3.11 P ro o f of L e m m a 3.9... 55
3.12 P ro o f of L e m m a 3.10... 56
3.13 P ro o f of L e m m a 3.11... 58
3.14 T h e tra in tre e a lg o rith m a pplied to Figure 3.3... 66
3.15 Case P ... 69 3.16 Ccise Q ... 69 3.17 S e m i M e e t s i S . T ) ... 70 3.18 5 = k \ n /\2 an d 5 divides R'l. K o ... 83 3.19 Proof of L e m m a 3.20.4... 85 3.20 Insertion e x a m p le s ... 88 3.21 T h e I n s e r t a lg o r i th m ... 90 3.22 M axim al clique K ... 92 3.23 P roof of p a r t 2 ... 94 3.24 P roof of p a r t 3... 95 3.25 K is a leaf... 98 3.26 Deletion e x a m p le s ... 99 3.27 K is not a l e a f . ... 101 3.28 D eleting 5 ... 103 3.29 More e x a m p le s ... 104
Chapter 1
Introduction
1.1
D yn am ic graph algorithm s
A d y n a m i c graph algorithm m a in ta in s a solution to a graph problem as the g rap h undergoes a series of sm all changes, such as single edge deletions or insertions. For e very c hange, th e a lg o rith m u p d a te s th e solution faster t h a n re c o m p u tin g th e solution from sc ra tch , i.e.. w ith no previously c o m p u te d inform ation. For e x a m p le , t h e World W id e Web is a d y n a m ic g r a p h whose vertices an d edges represent netw ork nodes a n d links t h a t u n p re d ic ta b ly m ay gain or lose capability cis e q u ip m e n t fails, new e q u ip m e n t is in tro d u c e d , or t h e netw ork becomes congested. Typically, a d y n a m ic g r a p h a lg o rith m has a preprocessing step to c o m p u te a solution, w ith som e auxiliary info rm a tio n , for th e initial g rap h . For exa m p le, our d y n a m ic g ra p h a lg o rith m in C h a p t e r 3 builds a d a t a s t r u c t u r e for t h e initial g r a p h in O (n^) t im e a n d th e n u p d a te s it in O ( n l o g n ) tim e p e r edge insertion or deletion, whereas c o m p u tin g t h e solution from s c ra tch (w ith o u t previously c o m p u te d inform ation) requires 0 ( m -r n) tim e, w here m a n d n are respectively t h e n u m b e r of edges a n d vertices in t h e g raph.
A d y n a m ic g r a p h a lg o rith m s u p p o rts two operations: query, a qu e stio n a b o u t th e s olution being m a in ta in e d , e.g.. 'W re vertices u. v connected?" or "Is th e g ra p h b ip a r
p ro b le m s on w eig h te d g rap h s also s u p p o rt u p d a te s t h a t change an edge weight. Some a lg o r ith m s s u p p o r t u p d a te s t h a t delete or insert a vertex, e.g.. [CH78. KS99. SP75]. .•\n i nsert ions- only or deletions-onlij d y n a m ic g r a p h a lg o rith m respectively allows o nly e dge insertions or only edge deletions. .\ f ull y d y n a m ic g r a p h a lg o rith m allows b o t h insertions a n d deletions. W ith som e problem s, m a in ta in in g a solution is easier w h e n only in se rtio n s or only deletions are allowed. For e x a m p le, a sim ple and fast in se rtions-only a lg o rith m can be readily o b ta in e d for c o n n e c tiv ity using disjoint-set u n ion [Tar75]. whereas fast fully d y n a m ic a lg o rith m s a re considerably m ore involved [HK99. H T98. HdT98].
Fully d y n a m i c a lg o rith m s have been developed for n u m ero u s problem s on u n d i re c te d g ra p h s , in cluding con n e ctiv ity [EGIN97. Fre85. HK 99. HT98. H dT 98j. bicon n e c tiv ity [EGIN97. HK95. Rau94]. 2-edge c o n n e c tiv ity [Fre97. HK99j. bipartireness [EG IX 97. H K 99. HK97]. m in im u m spa n n in g trees [CH78. EG IN 97. Fre85. HK97. H d T 9 8 . SP7Ô]. a n d p la n a rity [BT96. EG IS96. ILR93]. T h e r e are also a lgorithm s for p ro b le m s on plan e g rap h s [EGIS96. EIT'*'92. Fre85. HRS94. Rau94]. Som e of these re su lts a re d e s c rib e d in survey papers [EGI98. FK99].
Hell. S h a m ir , a n d S h a ra n [HSS99] developed a fully d y n a m ic a lg o rith m to rec o gnize p r o p e r in te rv al g rap h s t h a t m a in ta in s a re p r e s e n ta tio n of th e p ro p er interval g r a p h . T h e ir a lg o r ith m allows u p d a te s t h a t result in a p ro p er interval graph and s u p p o r t s queries to decide w h e th e r a n edge u p d a te is allowed. In this thesis, we will p res e n t a nalogous a lg o rith m s for chordal g rap h s a n d interval g rap h s, which are classes t h a t c o n ta in t h e p ro p e r interval g rap h class.
1.2
C hordal graphs
Let G = ( V’(G ), ^ ( G ) ) = (V. £ ) be a n u n d ire c te d g rap h w ith o u t loops or m u ltip le edges a n d let n = |L'| a n d m = | £ | . We w rite H Ç G \î H is a su b g ra p h of G and
H C G \{ H IS a pro p er s u b g ra p h of G. T h e s u b g ra p h of G induced by i Ç \ is
G[G] = ( £ . £ [ £ ] ) . w here E[U] = { { u . c } e E \ u . v e i ' } . Given 5 C V. let G - 5’
d e n o te G[V’ — 5 |. Let G — { u , v} d e n o te (V, E — {{-r.i/}}) a n d let G -f {x, y} d e n o te (V; E U { { x . y } } ) .
A g r a p h G is chordal (or triangulated) if every cycle of length 4 or m o re has a
chord, which is an edge joining two nonconsecutive vertices of th e cycle. Equivalently.
G is chordal if no in duced s u b g ra p h of G is a cycle of length g r e a te r t h a n 3. Since e very induced s u b g ra p h of a chordal g r a p h is chordal, some a u th o rs describe being ch o rd a l as a hereditary property, e.g.. [GolSOj. . \ g rap h G is perfect if for every induced s u b g ra p h H of G. t h e c h ro m a tic n u m b e r of H is equal to th e clique n u m b e r of H. A g r a p h G is th e intersection graph of a family of sets IF = { 5 i S\ } if T is the v e rte x set of G a n d {5,. 5^} is a n edge of G e x a ctly when i ^ j an d S’, n Sj = 0. E\ ery ch o rd a l g ra p h is perfect [GolSO] a n d every chordal g ra p h is the intersection g ra p h of a fam ily of subtrees of a tre e [Bun74. Gav74b. Wal72. Wal7S]. where th e s u b tre e s are view ed as sets. G olum bic [GolSO] a n d M cK ee a n d M cM orris [.\IM99] discuss chordal g ra p h s in th e c ontext of perfect g rap h s a n d intersection graphs, respectively.
T h e r e are linear t im e a lg o rith m s to recognize chordal graphs [RTL76. T\*S4] a n d to solve various problem s on chordal g rap h s t h a t are N P -c o m p le te on general graphs: C L IQ U E . C H R O M .\ T I C N U M B E R . I N D E P E N D E N T S E T . a n d P A R T I T I O N IN T O C L IQ U E S [Gav72|. C h o rd a l g rap h s have a p p lic atio n s in biology, d a ta b a s e s , sta tis tic s , facility location problem s, a n d especially in sparse m a t r ix c o m p u ta tio n [BP93. GolSO. MM99].
satisfying t h e p r o p e r ty such t h a t V C I . a n d ma xi ma l if th e r e is no L ' Q f satisfying th e p r o p e r ty such t h a t L ' D L'. A clique of G is a set of pairw ise a d ja c e n t vertices of
G. .An i ndependent set of G is a set of pairwise n o n a d ja c e n t vertices of G.
Let 5 C V’. If two vertices a re connected in G a n d not con n e cte d in G — S. th en 5 is a separator of G. If u. v G V are conn e cte d in G a n d not c o n n e cte d in G — S . th e n 5 is a uv-separator of G. If u . v G V a n d S is a m in im a l u e -s e p a ra to r, th en 5 is a m i n i m a l vertex separator o f G. If I j. l i C V a n d 5 is a u e -se p a ra to r of G for every u G V’l a n d e G l i . t h e n 5 separates V ÿ .li. In Figure 1.1, { u . x . y } . { v . x . y } are m a x im a l cliques, { x .y } . {y} are m inim al vertex s e p a ra to rs , an d { y } is a m inim al se p a ra to r.
Figure 1.1: .A graph.
We will discuss th e c h a ra c te riz a tio n s of chordal g ra p h s t h a t are th e most well known a n d m ost useful for o u r purposes. O th e r, less useful, c h a ra c te riz a tio n s of chordal g ra p h s include [Duc84. ShiS4. BCD'"'90. B.J96].
.A g r a p h G is chordal if a n d only if any of t h e following s t a t e m e n ts holds. 1. E v e ry m in im a l v e rte x s e p a r a to r of G is a clique [Dir61].
2. G has a perfect e lim in a tio n ordering [FG65].
3. G hcis a clique tre e [Bun74. G av74b. Wal72. Wal78j. C h a ra cteriza tio n 1
Using this c h a ra c te r iz a tio n , we can give a n a lg o rith m t h a t decides w h e th e r a graph is chordal: for e very p a ir u, u of vertices, find every m in i m a l u c -se p a ra to r an d test
w h e th e r it is a clique. Since th ere a re 0 ( n ’ ) vertex pairs a n d each test requires 0 ( r r ) tim e , not including t h e t im e t o find t h e s e p ara to rs, this is a very inefficient a lg o rith m when c o m p a re d to th e linear tim e a lgorithm s we will s u b seq u e n tly describe. T his c h a ra c te riz a tio n is im p o r ta n t , however, for th e clique -se pa ra tor g ra p h we define in C h a p t e r 3.
C h a ra cteriza tio n 2
.\ ve rte x v is simplicial if Adj(L') is a clique, where Adj{ v] is th e set of neighbors of
V. For e x a m p le, the c o m p lete g ra p h on n vertices has n simplicial vertices an d a p a th
has e x a c tly 2 simplicial vertices. .A. perfect elimination ordering ( P E O l of G is an ord erin g of its vertices ( (.q. ... /•„) such t h a t for each c,. t h e set A d j { i \ ) n {c. | i <
j } is a clique. Equivalently, each c, is a sim plicial vertex in th e s u b g ra p h induced by
{ c , Un}. For exa m p le, th e g ra p h in Figure 1.1 has perfect e lim in a tio n orderings [ a . c. X. y . ~) an d (z. u. x . y . v ) . which shows t h a t t h e P E O m ay not be unique.
Every chordal grap h has a sim plicial vertex [Dir6I]. This fact leads to a short proof [FG65] t h a t a g r a p h G is chordal if a n d only if G has a PE O : (=>) Let c be any sim plicial vertex of G. Since chordality is h ereditary, th e n G — v is chordal. By i n d u ctio n . G — c has a P E O . which when a p p e n d e d to v yields a P E O of G. (<= ) Let C be a cycle of length 4 or more. Let c, be t h e v e rte x of C w ith sm allest index in t h e P E O . Since i\ is sim plicial in t h e su b g ra p h in duced by { c ,... t \ } . th e n c's two neighbors in C are con n e cte d by an edge, which is a chord of C.
Fulkerson a n d Gross [FG65] gave th e first a lg o rith m to c o m p u te a P E O : r e p e a t edly find a n d delete an y sim plicial vertex until no vertices rem ain. T h e a lg o r ith m 's correctness follows from t h e proof t h a t a g r a p h is chordal if a n d only if it has a P E O . Since a sim plicial v ertex can be found in 0 { n ^ ) tim e , this a lg o rith m runs in 0 ( n ‘‘ ) tim e. T h e r e is an a lg o rith m to recognize chordal graphs in • ] tim e using fast m a t r i x m u ltip lic a tio n [Gav74a]. b u t it is supe rse d e d by a lg o rith m s t h a t c o m p u te a
a lg o rith m b a sed on a g r a p h search called Lexicographic BFS (L E X -B F S ):
N u m b e r t h e vertices from n to 1 in decreasing order, as follows. .\ v e rte x 's label is t h e list of its n u m b e r e d neighbors in decreasing order. Initially all labels a re e m pty. N u m b e r t h e v e rte x w ith th e largest label in lexicographic order, w ith ties b ro k en arb itra rily , an d rep e a t.
Every ord erin g p r o d u ce d by L E X -B F S could be p roduced by B r e a d th - F irs t Search. Rose. T a rja n . a n d Lueker show t h a t if a g ra p h is chordal, th e n every ord erin g produced by L E X -B F S is a P E O . T h e y im p le m e n t L EX -B FS in linear tim e a n d give a linear tim e a lg o rith m to test w h e th e r a given vertex ordering is a P E O . T h u s, this a lg o rith m recognizes chordal graphs in 0 ( m -|- n) tim e. T arjan a n d Y annakakis [T ^’S4] later gave a n o th e r linear tim e a lg o rith m using a sim pler g ra p h search called M a x im u m C a r d in a lity Search (M C S):
N u m b e r t h e vertices from n to 1 in decreasing order, as follows. N u m b e r th e v e rte x w ith t h e m o st n u m b e r e d neighbors, with ties broken a rb itra rily , a n d r ep e a t.
T a rja n a n d Y annakakis show t h a t if a grap h is chordal, th e n every ord erin g pro du c e d by M CS is a P E O . L sing a h e a p . M CS runs in 0 (ttilog n-|-n log n ) tim e. T arjan
a n d Y annakakis im p le m e n t M CS to ru n in 0 ( m -h n) tim e a n d th ey use t h e a lg o rith m from L E X -B F S to test w h e th e r a given v ertex ordering is a P E O . T h u s, this a lg o rith m also recognizes chordal g r a p h s in 0 ( m -|- n) tim e.
M CS can pro d u ce orderings t h a t c a n n o t be p roduced by L E X -B F S a n d vice versa, a n d th e r e a re P E O s t h a t a re c a n n o t be pro d u ce d by MCS or L E X -B F S . Shier [ShiS4] gives a n a lg o rith m t h a t c a n p r o d u ce a n y P E O . alth o u g h it does not ru n in linear tim e. P a n d a [Pan96] shows t h a t D e p t h - F ir s t Search guided by M CS ( th e n e x t v e rte x t o visit
is th e ne ig h b o r of th e c u rre n t vertex w ith th e m o st n u m b e re d neighbors ) c o m p u te s a P E O o f a c h o rd a l g r a p h in linear tim e , as does B r e a d th - F i r s t Search guided by MCS.
-A. P E O is req u ired for t h e linear t im e a lg o rith m s for C L IQ U E . CHRO.M.ATIC N U M B E R . I N D E P E N D E N T S E T . a n d P .A R T IT IO N IN T O C L IQ U E S on chordal g ra p h s [Gav72]. .A P E O is also im p o r ta n t for several a lg o rith m s t h a t recognize interval g r a p h s , which we will discuss in th e next section.
C h a ra cteriza tio n 3
B u n e m a n [Bun74]. Gavril [Gav74b]. a n d W a lte r [Wal72. Wal78j in d e p e n d e n tly discovered t h a t a g ra p h is chordal if a n d only if it is th e intersection g rap h of a family of s u b tre e s of a tree. In p a rtic u la r, th e y showed t h a t a g ra p h G is chordal if a n d only if G has a clique tree, which is a tree T on t h e m a x im a l cliques of G w ith th e clique
i ntersection property, for any two m a x im a l cliques K . l \ ' . th e set /v H k'' is co n ta in ed
in every m a x im a l clique on th e A'-A ' p a th in T.
Let KIg be th e set of m a x im a l cliques of G a n d let I'l be th e set of m a x im a l cliques of G c o n ta in in g v e rte x v. It is s tra ig h tfo rw a rd to show th a t th e clique in te r section p r o p e r ty is equivalent to th e induced subtree property: for any v e rte x r . the su b g ra p h o f T in d u c e d by ICc{e!) is a tree. Let be th e s u b tr e e of T in d u ce d by
ICg(^')- Since ( u . e} is an edge of G if a n d only if u. v a re b o th co n ta in ed in a m ax im a l
clique of G. t h e n {u. e} is an edge of G if a n d only if T.j, a n d T^. intersect. T h u s. G is t h e in te rse c tio n g r a p h of t h e fam ily of s u b tre e s {T^. | v € V’}.
T h e weighted clique intersection graph Wg o f G is th e weighted g r a p h on th e m a x i m a l cliques o f G where {A'. A''} is a n edge if a n d only if A' i~l K ' == 0. a n d each edge { K . A''} has weight | A' fl A''|. Wg is c o n n e c te d if a n d only if G is connected. B e r n s te in a n d G o o d m a n [BG81] showed t h a t for a n y c onne cte d graph G. a tre e on
Kg is a m a x im u m -w e ig h t s p a n n in g tre e of Wg if a n d only if it has th e induced su b tre e p ro p erty .
• r is a clique tre e of G.
• T has th e clique inte rse c tion property.
• T has t h e induced s u b tr e e property.
• r is a m a x im u m -w e ig h t s p a n n in g tre e of W'c (if G is connected).
Ho a n d Lee [HLS9] a n d L u n d q u ist [Lun90] in d e p e n d e n tly discovered th e following p r o p e r ty of clique trees, which we will use extensively, especially in th e interval graph alg o rith m s in C h a p t e r 3.
T h eo rem 1.1 Let G be a connected chordal graph with clique tree T.
1. .4 set S is a m i n i m a l vertex separator o f G i f a nd only i f S = A H A ' f o r s ome
{A'. A"} ^ E ( T ) .
2. f f S = A n A ' f o r {A'. A '} € E { T ) . then S separates K — S and A ' — .S'.
Fulkerson a n d Gross [FG65] showed t h a t a chordal g ra p h w ith n vertices has at m o st n m a x im a l cliques. By T h e o r e m 1.1.1. a chordal g r a p h w ith n vertices has at m ost n —1 m in im a l ve rte x s e p ara to rs. .Also, if 5 satisfies th e prem ise of T h e o r e m 1.1.2. th e n 5 is a m in im a l u e -s e p a ra to r for an y u Q K — S a n d v € A ' — 5' because every v ertex of 5 is a d ja c e n t to every vertex of A’ — 5 a n d A ' — 5 . which m eans 5 is a m in im a l vertex se p a ra to r.
B u n e m a n . G avril, a n d W a lte r showed how to c o n s tr u c t a clique tree of a chordal g ra p h in p olynom ial tim e , for e x a m p le . G a v ril's a lg o rith m runs in 0(n'*) tim e. Subse quently, o th ers developed linear t im e a lg o rith m s to c o m p u te a clique tre e of a chordal g r a p h , as well as its m a x im a i cliques a n d m in im a l v e rte x s e p ara to rs. T a rja n a n d A'an- nakakis [TY84] im plicitly gave t h e first linear t im e a lg o r ith m in t h e c o n te x t of acyclic
h y p e rg ra p h s for relational d a ta b a s e s . Lewis. P e y to n , a n d P o th e n [LPP89] gave the first explicit linear t im e a lg o rith m in t h e c o n te x t of sparse m a tr ix c o m p u ta tio n . Blair a n d P e y to n [BP93] gave a sim p ler a lg o rith m in a g ra p h th eo ry con te x t. Each of these linear t im e a lg o rith m s uses MCS.
T h e t e x t by G o lu m b ic [GolSO] describes th e P E O a n d t h e clique tre e , but does not include m ore recent clique tre e results like T h e o r e m 1.1. T h e excellent p rim e r by Blair a n d P e y to n [BP93] em ph a size s t h e clique tre e an d its a pplications in sparse m a tr ix c o m p u ta tio n . .\ lt h o u g h b o th c h a ra c te riz a tio n s have been very useful in developing a lg o rith m s for chordal graphs, b o th m ay not be e qually useful in developing d y n a m ic a lg o rith m s for chordal grap h s. Since th e P E O is c o m p u te d with a g ra p h search such as Lex-BFS or M CS, a single edge u p d a te could yield a very different P E O . T here are pro p e rtie s t h a t im ply a given ordering is a P E O . for e x a m p le P r o p e rty P in [T\'S-lj. b u t again, a single edge u p d a te could yield a very different ordering. T hus, the P E O a p p e a rs to be difficult to m a in ta in u n d e r edge u p d a te s . In c o n tra s t, in C h a p te r 2. we prese n t a sim ple a lg o rith m t h a t m a in ta in s a chordal g r a p h 's clique tre e in 0 [ n ) tim e p e r u p d a te .
1.3
Interval graphs
-A. g r a p h is interval if it is th e intersection g r a p h of a set of intervals on th e real line. T h e r e a re as m a n y c h a ra c te riz a tio n s of interval graphs as th e re are of chordal graphs. We will briefly discuss t h e m ost well known c h a ra c te riz a tio n s , for which we require th re e definitions. .An asteroidal triple is a set of th re e vertices such t h a t between any two of t h e vertices, th e r e is a p a t h t h a t does not c ontain a neighbor of t h e third vertex. .An asteroidal trip le is a n in d e p e n d e n t set. T h e c o m p le m e n t of a g r a p h G = (V. E) is t h e g r a p h G = ( V . Ë ) . where {u. u} E Ê if a n d only if {u. c} ^ E. .A grap h is
t h a t if (u. v), (u. w) a re d ire c te d edges, th e n (u, w) is a directed edge.
g r a p h G is interval if a n d only if a n y of t h e following s ta t e m e n ts holds.
1. G is chordal a n d G has no a s te ro id a l trip le [LB62].
2. G is chordal a n d G is a c o m p a r a b ility g ra p h [GH64].
3. T h e r e is a linear ordering of t h e m a x im a l cliques of G such t h a t t h e m a x im a l cliques c ontaining an y given v e rte x are consecutive in th e ordering [GH64].
T h e linear ordering in th e th ir d c h a ra c te r iz a tio n is a clique tree of G t h a t is a p a th , which is a clique path of G. It follows t h a t a grap h is interval if a n d only if it is th e inte rse c tion graph of a family of s u b p a th s of a p a th . Therefore, th e interval graph class is a subclass of t h e chordal g r a p h class. It is a proper subclass, as shown by t h e n e t. which is th e g r a p h form ed by a tt a c h i n g an edge to each corner of a triangle to form a g r a p h w ith 6 vertices. Since not every clique tree of an interval g r a p h is a clique p a th , a clique tre e a lg o rith m m a y not p r o d u ce a clique p a th when a p p lie d to an interval graph.
Interval graphs often m odel pro b le m s involving a linear order. C onsequently, interval g rap h s have even m ore a p p lic a tio n s t h a n chordal graphs, including a p p li c a tio n s in archeology, biology, genetics, psychology, an d scheduling [GolSO. .\lM99j. F u r th e r m o r e , m ost well known .\ P - c o m p le t e problem s can be solved in polynom ial t im e on interval graphs, including H .W I IL T O N IA N C I R C U I T . DO.\IIN.A.TI.N‘G S E T . a n d G R . \ P H IS O M O R P H IS M [JohSô]. B o d la e n d er [BodS9] showed t h a t .ACHRO M A T IC N U M B E R is N P -c o m p le te for interval g rap h s, which is one of t h e few known N P -c o m p le ten e ss results for interval g ra p h s [JohSô].
T h e r e are n um erous algorithm s to recognize interval graphs, including th e fol lowing linear t im e a lg o rith m s. B o o th a n d Lueker [BL76] gave t h e first (linear tim e) a lg o r ith m a n d in tro d u c e d t h e P Q - tr e e . w hich repre sents a set of p e rm u ta tio n s . T h e ir
u
a lg o rith m c o m p u te s t h e m a x im a l cliques a n d th en uses a c o m p lic a te d set of tem p la te s to c o n s tru c t a P Q - t r e e t h a t rep re sen ts t h e set o f valid orderings of t h e m a x im a l cliques. K o rte an d M ohring [KM89] gave an a lg o rith m t h a t c o n s tru c ts a n M P Q -tre e . which is a sim p ler version of t h e P Q -tre e . T his c o n stru c tio n uses a L E X -B F S ordering and d e p e n d s on L E X -B F S pro p ertie s t h a t MCS does not have. H abib. Paul, and V'iennol [HMPVOO] gave an a lg o r ith m t h a t c o m p u te s a clique tree a n d t h e n uses a LEX -B FS ord erin g to m a n i p u la t e t h e clique tre e into a clique p a th . Subsequently, o th ers have developed a lg o rith m s t h a t do not c o m p u te a linear ord erin g of t h e m a x im a l cliques: Hsu a n d M a [HM99] gave a n a lg o r ith m t h a t uses a s u b s titu tio n d e com position c o m p u te d w ith L ex-B FS. a n d CorneiL O lariu. a n d S te w a rt [COS98] gave a pa rticu la rly sim ple a lg o rith m t h a t uses four sweeps of Lex-BFS. w ith ties broken in a p a rticu la r way.
1.4
O ther classes
We describe one subclass o f interval graphs. . \ g ra p h is proper interval (or unit
interval) if it is t h e inte rsec tion g ra p h of a set of intervals on th e real line such
t h a t no interval is properly c o n ta in e d in a n o th e r. T h e subclass is proper, as shown by t h e claw
(/v'1.3).
which is t h e g r a p h consisting of one ve rte x a d ja c e n t to three n o n a d ja c e n t vertices. T h e r e a re linear t im e a lgorithm s to recognize pro p er interval g ra p h s derived from th e B o o th a n d Lueker a lg o rith m a n d th e K o rte a n d M ohring a lg o rith m . T h e r e a re also sim p ler linear t im e a lg o rith m s t h a t do not use P Q -tre e s or M P Q - tre e s , in clu d in g a lg o rith m s by Corneil. K im . X a t a r a ja n . O la riu . a n d Sprague [CKX'*‘95], Deng, Hell, a n d H u a n g [DHH96]. a n d de Figueiredo. M eidanis. a n d de Mello [dFM dM 95]. Hell. S h a m ir , a n d S h a ra n [HSS99] gave a fully d y n a m ic algorithm to recognize a n d rep re sen t p r o p e r interval graphs in O ( l o g n ) t im e per edge insertion or deletion.W e d e scrib e one m ore subclass of ch o rd a l graphs. graph is .•split if its v ertex set can. be p a rtitio n e d into a clique I\ a n d a n in d e p e n d e n t set / . with no re stric tio n on edges be tw e en vertices in K a n d vertices in / . g r a p h G is split if a n d only if
G a n d its c o m p lem en t G are ch o rd a l [FH77]. Split g ra p h s a n d interval graphs are
in c o m p a ra b le classes.
J o h n s o n [JohS5] discusses th e s e a n d o th e r g r a p h classes w ith "broad a lg o rith m ic significance", including undirected path graphs, directed path graphs, an d strongly
chordal graphs. T h e G olum bic [GolSO] t e x t an d t h e M cKee and McMorris m ono
g r a p h [MM99] discuss m any of t h e s a m e classes in th e c ontext of perfect graphs a n d in te rse c tio n graphs, respectively.
1.5
R esu lts
In th is thesis, we present th e first d y n a m ic a lg o rith m s t h a t recognize chordal graphs a n d interval graphs. B oth a lgorithm s m a i n ta i n a g r a p h re p re sen ta tio n w ith 0{ n ) size. In C h a p t e r 2. we present th e d y n a m ic a lg o r ith m t h a t recognizes a chordal g rap h by m a i n ta i n in g its clique tree. T h e a lg o r ith m allows u p d a te s t h a t result in a chordal g r a p h a n d s u p p o rts queries to decide w h e th e r an edge u p d a te is allowed. In a very s im p le im p le m e n ta tio n , each o p e ra tio n runs in 0 ( n ) tim e , a n d in a m ore c o m p lic a ted im p l e m e n ta t io n , one o p e ra tio n ru n s in O ( lo g ^ n ) tim e , a n o th e r in O ( n l o g n ) tim e , a n d t h e o th e r s in 0 ( n ) tim e.
In C h a p t e r 3. we in tro d u c e t h e clique-separator graph r ep re s en ta tio n of a chordal g r a p h , which is a g ra p h on its m a x i m a l cliques a n d m in im a l vertex se p ara to rs. We prove f u n d a m e n ta l properties of t h e c liq u e -s e p a ra to r g r a p h a n d a d d itio n a l pro p ertie s w hen t h e i n p u t g rap h is inter\"al. W e t h e n in tro d u c e t h e train tree re p re s e n ta tio n of in te rv a l g rap h s a n d use it to d e cide w h e th e r t h e r e is a clique p a t h on th e g r a p h 's m a x i m a l cliques. This yields t h e d y n a m ic a lg o r ith m to recognize interval g rap h s in
13 log n) tim e per edge in se rtio n or deletion. T h e a lg o rith m allows u p d a te s t h a t resu lt in an interval g r a p h a n d s u p p o rts queries to decide w h e th e r an edge u p d a t e is allowed. T h e c liq u e -s e p a ra to r g r a p h a n d t h e t r a in tree are rep re s e n ta tio n s t h a t provide significantly m o re in fo rm a tio n a b o u t th e g r a p h 's s tr u c tu r e th a n t h e clique t re e repre senta tion.
A dynam ic algorithm for chordal
graphs
2.1
In tro d u ctio n
We present th e first d y n a m ic alg o rith m t h a t m a in ta in s a clique tre e rep re s en ta tio n of a c hordal g r a p h G a n d s u p p o rts th e following o pe ra tions:
• D c l e t e - Q u e r y i u . v) r e tu rn s "yes" if G — {u. c} is c hordal a n d "no" otherwise.
• D e l e t e ( u . v ) d eletes {u, u} from G. provided G — { u .c } is chordal.
• I n s e r t - Q u e r y ( u . v ) r e tu r n s "yes" if G '+ { u . v } is c hordal a n d "no" otherwise.
• / n s e r t { u . v] in se rts [u . i’} into G. provided G + {u. c} is chordal.
T h e a lg o r ith m also m a in ta in s solutions to th e proble m s C L I Q U E a n d C H R O - M.A.TIC N U M B E R . We give two im p le m e n ta tio n s . In t h e first, each o p e ra tio n runs in 0 { n ) tim e . T h is i m p le m e n ta tio n is very sim ple a n d uses no c o m p le x d a t a s t r u c tu re s. In t h e second. D e l e t e - Q u e r y runs in 0 ( n ) tim e . D e l e t e in O ( n l o g n ) . I n s e r t -
Q u e r y in G ( l o g ^ n ) tim e , a n d I n s e r t in 0 ( n ) tim e. B o th im p le m e n ta tio n s im prove
15
t h e 0 { m + n) tim e o b ta in e d by ru n n in g a s ta tic a lgorithm (e.g. iRTL76]) for each o p e ra tio n . .A.11 of the bo u n d s in this p a p e r are worst-case, unless specified otherw ise.
O ptionally, the a lg o rith m can m a i n ta i n t h e connected c om pone nts of G. It s u p p o rts th e q u e ry " .\r e vertices u . v c o n n e cte d ?" in 0 ( 1 ) tim e w ith 0 { n ) t im e per u p d a te . If this query is not needed, its E T - tre e d a ta s tr u c tu r e [HK99] is easily o m i t ted. T h e c onnected c o m p o n e n ts of a n a r b i t r a r y grap h can be m a in ta in e d w ith 0{ 1 ) t im e per qu e ry a n d 0 ( y / n ) t im e p e r u p d a te using topology trees [FreSo] c om bined w ith spa rsih ca tio n [EGIN97]. b u t t h e resulting d a t a s tr u c tu r e is m uch m ore c o m p li c a te d t h a n E T - trees.
O u r d y n a m ic a lg o rith m has been used to im prove a result for m inim al filled graphs. Informally, a filled g r a p h of G is G w ith edges a d d e d to m ak e it chordal. See [BHTOl] or [RTL76] for formal definitions. G iven an a r b itr a r y graph G and a filled g rap h G ^ o f G. th e problem is to rem ove fill edges from G’"*" to o b ta in a m inim al filled g ra p h of
G t h a t is also a su b g ra p h of G'*’. T h e a lg o rith m in [BHTOl] solves this pro b le m in 0 { f { m + f ) ) tim e, where m = \ E{ G) \ a n d / = |£ ’(G''^)| — m . which is th e n u m b e r of
fill edges in G"*". Using th e first im p le m e n ta tio n of our algorithm . Heggernes and Telle have im proved th e ru n n in g tim e to 0 ( n f + m) tim e [Hegj. M inim al filled g rap h s are d esirable in sparse m a tr ix c o m p u t a ti o n , as well as o th e r areas of c o m p u t e r science. Since f is 0 { n ) for m a n y p rac tic a l m a tr ic e s [BHTOl]. 0 { n f + m) c o m p a res favorably w ith th e best known r u n n in g t im e for c o m p u tin g any m in im a l filled g r a p h o f G. which is 0 { n m ) [RTL76].
2.2
C lique trees
By T h e o r e m 1.1.1. a clique tre e of G has nodes a n d edges th a t c orrespond to th e m a x im a l cliques a n d m in im a l v e rte x se p ara to rs of G. respectively. Figure 2.1 shows a c h o rd a l g ra p h G a n d F ig u re 2.2 shows a clique tre e of G. T h e set 5 = A = {c. d]
is a m in im a l u u -se p a ra to r for every u t Ax — 5 = {a. 6} a n d c € I\y — S = {g}. Also, rep lacin g th e edge { K y , A’u,,} w ith {A’x. A',x} gives a different clique tre e for G. w hich shows t h a t a g r a p h ’s clique tre e m ay not b e un iq u e. We will refer to th e vertices of
G an d th e nodes o f T an d will usually use s . t . u . v as v ertex nam es a n d œ . x . y . : as
node n am es. In th is c h a p te r only, node z E T c o rresp o n d s to m ax im a l clique A \ of
G an d a n edge { x . y } E T has w eight w { x . y ) .
b
a
c
I 9
F igure 2.1: .A. chordal g ra p h G.
M ost discussions o f ch o rd a l g rap h s, such as [B P93. L PPS9]. assu m e a connected g ra p h , since d isc o n n e c te d g ra p h s are ty p ic a lly h a n d le d by a p p ly in g th e re su lts to each c o n n e cte d c o m p o n e n t. To m a in ta in th e g r a p h ’s clique tre e in a unified way in our d y n a m ic a lg o rith m , we e x te n d th e definitions as follows. Let G be a disco n n ected ch o rd al g ra p h a n d let T [ . T2 Tk be clique tre e s of G”s co n n e cte d c o m p o n en ts.
k > I. Let X a n d y be nodes of different r , ’s. so th a t A'x a n d I\y are c o n ta in ed in
different co m p o n e n ts of G. Since A'x H A'y = 0 a n d { z .y } is not an edge of l l ’c . we call { x . y } a d u m m y edge w ith w eight w { x . y ) = 0. W e jo in T i . T j Tt w ith a r b itr a r y d u m m y edges to form a tre e T. w hich satisfies th e in d u ced s u b tre e an d clique in te rse c tio n p ro p e rtie s. We e x te n d W q by a d d in g all d u m m y edges to it. so th a t r is a m axi m um -w eight sp a n n in g tre e of W q . T h u s. T satisfies all th re e clique tre e p ro p e rtie s. W e will use clique tree to refer to T a n d strict clique tree to refer to th e Ti's. w hich a re th e su b tre e s of T co rresp o n d in g to th e c o n n e cte d c o m p o n e n ts of G.
2
K.W
2.3
T h e d yn am ic algorith m for chordal graphs
T h ro u g h o u t, G d en o tes th e c u rre n t ch o rd al g rap h and T d en o tes th e c u rre n t clique tre e of G.
2.3.1 D elete-Q uery
We show how to decide w h e th e r G — {u. c} is chordal for {u. c} 6 E.
L em m a 2.1 ([RTL76]) Let G be a chordal graph with edge { u .c } . Then either
G — {u. I’} (S chordal or G — {u . u} has a chordless cycle o f length -/•
L em m a 2.2 ([G JSW 84]) .4 graph H has no chordless cycle o f length 4 i f and only if f o r all distinct vertices s . t with { s .f} ^ E ( H ) . H + { s . t } has exactly one m a xi m al clique containing { s . f }.
By L em m as 2 an d 3. G — { u .c } is ch o rd al if and only if for all d istin c t vertices s. t w ith { a .f} ^ E { G — {u. u}). G — {u. I'} 4- { s . t } has e x a ctly one m ax im a l clique c o n ta in in g {s. (}. T h u s, a necessary co n d itio n for G — { u .c } to be ch o rd al is: G has e x a c tly one m ax im a l c liq u e c o n ta in in g { u .c } . In fac t, th is is also a sufficient co n d itio n . .Although sufficiency does not follow im m ed ia te ly from L em m a 3. its p ro o f is very sim ila r to L em m a 3 ’s p ro o f in [G JSW 84].
T h eo rem 2.3 Let G be a chordal graph with edge {u. c}. Then G — {u. c} zs chordal i f a nd only i f G has exactly one m ax i m a l clique containing u and v.
P r o o f (=>) By L em m as 2 a n d 3. (<=) S uppose G — { u .c } is not ch o rd al. By L em m a 2. G — {u. u} htis a chordless cycle of len g th 4. say ( u . s . v . t ) . Since {u. c} is a n edge of G a n d { s .f} is n o t an edge o f G. th e n { u .u .s } a n d { u .c .f } a re tw o cliques of G th a t cajin o t be c o n ta in e d in th e sam e m ax im a l clique o f G. T h u s. [ u . c} is co n ta in ed in a t least tw o d is tin c t m a x im a l cliques of G. □
19
D e l e t e - Q u e r y ( u . c)
For ev ery n ode x of T . te s t w h e th e r { u .e } Ç A’x- If th ere is e x a c ily one such node, r e tu r n "yes", an d oth erw ise, re tu rn "no".
E n d Del ete-Query
2.3.2
D elete
We n ex t show how to u p d a te T for G — { u .c } , given th a t T has a un iq u e node x such th a t { u . c } Ç A'^. Let A'“ = A’x — {c}. A’_^' = A'x — {«}. In G — { u .c } . every clique A 'y,y x is m ax im al b u t A'x has sp lit in to th e cliques A'". A'^. w hich m ay not be m a x im a l. (See F igure 2.3.)
u
VF igure 2.3; D eleting {u. c} from A'x.
We d ecid e w h e th e r A'“ .A'^’ a re m ax im a l in G — { u .c } as follows. P a rtitio n th e set -V(x) of x 's neighbors into
A'u = {y € -V(x) I u € A'y} = {z E N { x ) I V e K ; } .Vw = { w € :V(x) I U. l> ^ A'u,}
(See F ig u re 2.4.) Let k = |A'x|. Since T is a clique tre e , th e n for a n y y E .V (x).
w { x . y ) < k - I.
iV„
N„
F ig u re 2.4: Before d e le tin g {u. r }.
/\ “ is not m ax im a l in G — {u. u} if an d o nly if 3 y e T . y ^ x . A'“ C k'y
if an d only if 3 y € .\’u. /v'“ C Ky (by th e clique in te rse c tio n p ro p erty ) if an d only if 3 y € .\'u. A'^ fl A'y = A'“
if an d only if 3 y G Au. w{x. y) = k — I
S im ilarly. A'j: is not m ax im a l if and o nly if 3c 6 u:{x. :) = k — I. D e l et e{ u. v)
1. F ind th e un iq u e node z of T such th a t {u. c} Ç A'^. If th e re is m ore th a n one such node, reject th e d e le tio n . O th erw ise, for every y E .\ (x ). test w h e th e r a € A'y or V € A'y a n d w h e th e r w ( x . y ) = k - I.
2. (W e m odify T as if A'“ a n d A'^ w ere m ax im a l an d th e n m odify T ag ain if th ey a re n o t. It is stra ig h tfo rw a rd to re w rite th e o p e ra tio n so th a t T is m odified once.)
R eplace node x w ith new nodes x i a n d x , rep resen tin g A'“ a n d Aj). respectively, a n d a d d edge { x i.x o } w ith u ;(x i. X]) = k — '2. In th e following, each new e d g e ’s weight is th e (rep laced ) old ed g e 's w eight. If (/ E .V^. rep lace { x .y } w ith { x [.y } . For each z E -V^., rep lace { x .- } w ith { x j .c } . For each w E -V,r- rep lace {x. w ith { x i.tc } (or { x ;, w}). O b serve th a t T has th e in d u ced su b tre e pro p erty . (See F ig u re 2.5.)
3. If A'“ a n d A'J a re b o th m eixim al in G — ( u . i ’}. sto p . If A'“ is not m ax im al b e cau se A’“ C A'y, for som e y, E A'^. c o n tra c t { x i.y ,} a n d rep lace x i w ith yi. T his
21
k-2
N.
wF ig u re 2.5: D eleting { u .r } .
m a in ta in s th e in d u ce d s u b tre e p ro p erty , even if “ C for m o re th a n one y,- E . \ \ . S im ilarly, if is n o t m a x im a l becau se A'^' C A\, for som e r, E . \ \ . c o n tra c t { x j . r , } a n d rep la ce w ith z,. T h u s, x has been replaced w ith 0. 1. or 2 nodes.
In e v e ry case, th e edge a d d e d betw een x t a n d x , is not c o n tra c te d an d th is edge's e n d p o in ts (w hich m ay have changed) c o n ta in u a n d v. We will su b seq u e n tly use this o b se rv a tio n .
E n d Delete
De l e t e { u . v) u p d a te s T so th a t th e re is a b ije c tio n betw een th e nodes of T and
th e m a x im a l cliques o f G — {u. v}. F u rth e rm o re . T has th e induced s u b tre e p ro p e rty an d th e w eight of a n y edge { y . z } E T is |Ay Pi A \|. H ence. T is a clique tre e for
G — { u . e }.
W e c a n rea d ily d ecide w h e th e r { u .c } is a cut edge o f G. i.e.. w h e th e r d ele tin g { u .i ’} d isc o n n e c ts a c o n n e cte d co m p o n en t of G . as follows. If {u. c} is a cu t edge, it m u st b e a m a x im a l clique. If {u. c} is not a cu t edge, th e n G has a cycle co n ta in in g
{u. (.'} a n d th e sh o rte s t such cycle has le n g th 3 or else G is not ch o rd al, w hich im plies {u . I’} is n o t a m a x im a l clique. T h u s, [u, c} is a c u t edge of G if an d only if {u. c} is
2.3.3 Insert-Query
VVe show how to d ecid e w h e th e r G '+ { u . i’} is chordal for {u. c} ^ E . W e will im p lic itly use th e fact th a t if a n d a re th e su b tre e s of T induced by AJc(“ ) a n d ICcie). resp ectiv ely , th e n a n d T^, do not in te rse c t (because no clique co n ta in s {u. c}). T h e o r e m 2 .4 Let G be a chordal graph without edge { u .r } . Then G -f {u .t-} chordal i f a nd only i f G has a clique tree T with u G A'^. c G h'y f o r s o me {x. y} G E { T ) .
P r o o f (=>) Since G' = G + { u .c } is ch o rd a l, th en G' has a clique tre e T'. By th e o b se rv a tio n a t th e e n d of D e l e t e ( u . v). if { u .u } is d e le ted from G ' . th e a lg o rith m p ro d u ce s a clique tre e T of G w ith u G A'^. e G A'y for som e { x . y } G E ( T ) . ( •=) If { j . y } is a d u m m y edge of T . th e n u a n d v a re in different co n n ected c o m p o n e n ts of
G a n d G + [u. i,'} is c e rta in ly ch o rd a l. O th e rw ise , u an d v are in th e sa m e c o n n e cte d co m p o n e n t G". w hich has a s tr ic t clique tre e T". We ap p ly T h e o rem I to G" a n d
T " . Let I = K i n A'y 7^ 0. S ince [u. c} is n o t an edge, th en u 0 A'y. c G E f a n d th u s
u G A'r — / . u G A'y — / . (See F ig u re 2.7.) B y T heorem I. / i s a u c -s e p a ra to r.
To show th a t G + {u. c} is c h o rd a l, it sufBces to show th a t G" + ( u . c} is ch o rd al. Let C be any cycle in G" + [ a . e} w ith le n g th 4 or m ore such th a t C co n ta in s {(/. r} . Let P = C — ( u . {.’}. so t h a t P is a u - v p a th of length 3 or m ore. Since / i s a ur- s e p a ra to r. P m u st c o n ta in a v e rte x s G / . T h e n e ith e r ( s . u} or {.s. c} is a c h o rd of C . H ence. G" + {u. c} is ch o rd a l. □
Suppose G has a clique tre e T t h a t d o es not satisfy th e c o n d itio n in T h e o re m 5. T h e following th e o re m show s t h a t if G htis a n o th e r clique tre e T ' th a t does satisfy th e co n d itio n , th e n T ' differs fro m T by o n e edge.
T h e o r e m 2 .5 Let G be a chordal graph without edge ( u . l '} . Let T be a clique tree
23
There exists a clique tree T ' o f G with u t K i ' . c € k ’y' and {x '. (/'} € E { T ' ) i f and only i f the m i n i m u m weight edge e on the x - y path in T satisfies w(e) = œ ( x . y ) .
P r o o f (4=) S ince T " = T — e + { x . y } a n d T have th e sam e w eight, th e n T " is a clique tree o f G. M oreover, u Ç. K f . v € Ay an d { x .y } € E ( T " ) . (=>) We first consider T. Let Tu a n d T^ b e th e su b tre e s of T respectively in d u ced by A2c(u) a n d fCci c). T h en x . x ' 6 Tu a n d y . y ' 6 Ty. (See F igure 2.6.) By a ssu m p tio n , we have ( x '.y '} %' T. F u rth e rm o re , x . y a re on th e x '- y ' p a th P' in T. w hich im plies th e x - y p a th P in T is c o n ta in e d in P ' . By th e clique in tersectio n p ro p erty . AT' H A'y/ Ç AT H A’y an d so u'(x. y) > w ( x ' . y ' ) . Sim ilarly, for any edge f E P. w { f ) > w ( x . y } > w i x ' . y ' ) .
We now c o n sid er T'. Let V}. W be th e n ode sets of th e tre e s of T ' — { x '.y '} . w here
x' 6 V’l a n d y ' € Vi. C o n sid er th e cut [V'l.Vi] of VV'g. Since P' -f { x '.y '} is a cycle c o n ta in in g an edge crossing [Vq. Vi] (nam ely, { x '.y '} ) . it m ust c o n ta in an edge e E. P' crossing [V}. Vi]. S ince { x '.y '} € T'. th e n e ^ T'. T h en wi e) < u’( x '.y ') . or else
T ' — { x '.y '} -i- e is a tre e w ith g re a te r w eight th a n T'. a c o n tra d ic tio n .
We claim th a t e P' — P. Every edge in P ' — P is c o n ta in ed in e ith e r I ’l P j
or V'(Tu). M oreover, x ' E Tu a n d y' E T^ im plies V'(Tu) Ç V} a n d V l P j Ç \ j . B ut e crosses [Vq.Vi]. w hich m eans e ^ P' — P. T h erefo re, e E P- T h en /r(e) > tc (x .y ) > w ( x ' . y ' ) . Since u’(e) < w { x ' . y ' ) . th en a ’(e) = w { x . y ) = ic ( x '.y ') . T h u s, e is a m in im u m w eight edge on P and «.’(e) = w ( x . y ) . □
T h e o re m 6 holds even if {u. e} jo in s different co n n e cte d c o m p o n e n ts of G. In th is case, { x .y } is a d u m m y edge jo in in g different s tric t clique tre e s . T h e n th e m in im u m w eight ed g e e on th e x - y p a th in T m u st also be a d u m m y edge an d so
if{e) = w ( x . y ) = 0. T h e n T — e -r { x .y } is a clique tre e satisfy in g th e co n d itio n in
T
Ta or Ta
F igure 2.6: C lique tre e T.
l n s e r t - Q u e r y ( u . v)
F in d th e closest nodes x . y ^ T such th a t u € A 'r -<•’ € A’y. If {-r.y} € T . re tu rn "yes". O therw ise, find th e m in im u m w eight edge e on th e x - y p a th in T . If
w[e) = w { x . y ) , re tu rn "yes", and if w{e) > t c ( x . y ) . re tu rn "no". (W e c a n n o t have iv(e) < u ' ( x , y ) because T is a clique tree .)
E n d I nse rt -Qu er y
2.3.4
Insert
W e n ex t show how to u p d a te T for G 4- { u .f } . given th a t u € A’- . e € A’y a n d { i . y} 6 E { T ) . Let / = A 'r H A'y. T h e n A ' = / U { u. e} is a clique in G 4- {u. c}. (See
F ig u re 2.7.)
W e claim th a t A is m a x im a l in G 4- {u. y}. O th erw ise, th e re is a v e rte x of V — A a d ja c e n t to ev ery v ertex in A , w hich m ea n s u a n d u are connected in G — I. B ut by
25
F ig u re 2.7: In se rtin g { u .u } .
T h e o rem I, / is a u n -se p a ra to r of G. a c o n tra d ic tio n . T h u s. K is a m a x im a l clique
in G -f { u .n } . Since K is n o t a clique in G. we m u st add a new node z to T w ith
K ; = K . F u rth e rm o re , if K,^ C for som e u' € T . th e n K,^ is not m ax im a l in G -i- { u . v } a n d we m u st rem ove n ode w from T.
F irs t, we co n sid e r w h e th e r h \ , is m ax im a l in G -f {u. n} for som e iv ^ x . y. S uppose A'u.. C A’;. S ince AT, does n o t c o n ta in th e edge {u. n}. e ith e r AT C f U {u} Ç A - or AT C / U {n} Ç A’y. a c o n tra d ic tio n . T h u s, A’^, is a m ax im a l clique in G -j- {u. n} for every w ^ x . y .
Second, we c o n sid er w h e th e r AT. Ay are m ax im a l in G -f { u .n } . Since r <t A’^. th e n A x C A’; if a n d only if A'x = / U {u} if and only if |A ’x| = | f | -i- 1. Sim ilarly.
A'y C A '; if a n d o nly if A'y = / U {c} if an d only if |A 'y| = | / | -I- 1. T h u s, we decide
w h e th e r A'x or A'y is m a x im a l in G -t-{u. u } by co m p arin g |A 'r |. | A 'y |. and | / | = œ i x . y ) . I n s e r t { u . v)
1. F in d th e closest nodes x . y € T such th a t u € A'x-c € A'y. If { x .y } t T. go to S te p 2. O th e rw ise , find th e m in im u m weight edge e on th e x - y p a th in T. If u;(e) = w { x . y ) . rep lace e w ith { x .y } in T. an d if u’(e) > w ( x . y ) . reject th e in sertio n .
2. (W e m odify T aa if AT a n d A'y were m ax im a l a n d th e n m odify T again if th ey are n o t.)
R eplace edge { x .y } in T w ith new node z rep re sen tin g AT = / U {u. c} a n d add edges { x .z } ,{ y ,z } , each w ith w eight \I\ -h 1. (See F ig u re 2.8.) D e te rm in e w h e th e r A'x, A'y axe m a x im a l in G -f { u .u } by co m p arin g |A'x|. |A'y|. a n d u '( x .y ). If A'x an d
K y are b o th m ax im a l, sto p . O th e rw ise , if /v’x is not m a x im a l, c o n tra c t { r .z } and
re p la c e x w ith c: if A'y is not m ax im a l, c o n tra c t { y , z} a n d rep lace y w ith z. T h u s, x a n d y have b een rep la ce d w ith 1. 2, or 3 nodes.
E n d Insert
X z y
•--- • --- •
I / I + i 1 / 1 + /
F igure 2.8: .\d d in g node z to T.
I n s e r t { u , v) u p d a te s T so th a t th e re is a b ije c tio n betw een th e nodes of T and
th e m a x im a l cliques of G + {u. u}. F u rth e rm o re . T has th e in d u ced su b tre e p ro p e rty a n d th e correct edge w eights. H ence. T is a clique tre e for G + {u. i-}.
N ote th a t ( u . c} jo in s different c o n n e cte d c o m p o n e n ts of G if a n d only if { x . y } is a d u m m y edge of T . i.e.. w { x . y ) = 0. In th is case. I \ \ = [u. r} .
2.4
F irst im p lem en ta tio n
In th is sim p le im p le m e n ta tio n , we re p re se n t each m a x im a l clique w ith a c h a ra c te ristic v e c to r, i.e.. an a rra y of n b its. T h e n m e m b e rsh ip te s tin g requires 0 ( 1 ) tim e.
T h e preprocessing ste p ru n s in 0 ( m -|-n ) tim e , as follows. C o m p u te th e m ax im a l cliques of G a n d b u ild a clique tre e T o f G [B P 9 3 |. Label each edge {x. (/} E T w ith its w eight w ( x . y ) a n d each node x € T w ith a p o in te r to th e c h a ra c te ris tic vecto r for A 'x.
VVe im p lem en t D e l e t e - Q u e r y a n d I n s e r t - Q u e r y by search in g T . Since T has at m o st n nodes, each o p e ra tio n ru n s in 0 {n) tim e .
VVe im p lem e n t D e l e t e by se arc h in g T to find n o d e x a n d th e n e x a m in in g x 's neigh b o rs. Since th e re are 0 ( n ) changes to T . D e l e t e ru n s in 0 { n ) tim e . VVe im p lem en t
I n s e r t by search in g T to find x a n d y. VVe m u st th e n c o m p u te z 's c h a ra c te ristic
a n d y's vectors a n d a d d u . u to it. O th erw ise, we copy x or y 's vecto r an d a d d th e a p p ro p ria te vertices {u or v or b o th ) to it. Since each v ecto r has len g th n. I n s e r t requires 0 (n) tim e.
2.4.1 Connected com ponents
W e can rea d ily m a in ta in th e c o n n e cte d co m p o n e n ts of G' by using a degree-d E T -tre e [HK99] to rep resen t a sp a n n in g tre e for each c o m p o n e n t, w here d = n '.O < e < 1. D eciding w h e th e r tw o v ertices are in th e sam e E T -tre e requires G ( l ) tim e , so each " .\r e vertices u . c c o n n e c te d ? ’’ q u e ry runs in 0 ( 1 ) tim e. T h e d e scrip tio n s of De lete a n d In se rt show how to decide w h e th e r d e le tin g an edge d isco n n ects a co m p o n en t a n d w h e th e r in se rtin g an edge co n n ects two c o m p o n en ts. Since s p littin g or jo in in g th e E T -trees requires 0 ( n " ) tim e , each u p d a te runs in 0 { n ) tim e.
T h e only difficulty occurs w hen an edge { u .u } to be d eleted is co n ta in ed in its c o m p o n e n t’s sp a n n in g tre e T^p b u t it is not a c u t edge of its c o m p o n e n t. T h e n d eletin g
{u. I’} sp lits Tap in to th e tree s Tj),. resp ectiv ely c o n ta in in g u. c an d we m ust find a
replacement edge, w hich is an edge o f th e c o m p o n en t th a t co nnects Tj), an d T’jp. Let
A’x be th e u n iq u e m a x im a l clique c o n ta in in g {u, i } before th e d eletio n . X ote j A’x! > - becau se ( u . e} is n o t a c u t edge. S ince any v e rte x f € A’x — {u. r} is a d ja c e n t to b o th
u an d V. th e n t is in th e sa m e co m p o n e n t as u. v an d so t 6 T,p. If t € Tj),. th en {/. r} is a rep la ce m e n t edge, a n d i f f € rjj,, th e n (f, u} is a rep la ce m e n t edge. We jo in Tj), a n d r,p w ith th e re p la c e m e n t edge, re sto rin g th e c o m p o n e n t’s sp a n n in g tree.
2.5
Second im p lem en ta tio n
In th is im p le m e n ta tio n , we rep re sen t each m ax im a l clique w ith a c h a ra c te ris tic vecto r a n d th e clique tre e T w ith a S le a to r-T a rja n d y n a m ic tre e [ST83]. For ev ery v ertex
[ n s e r t - Q u e r y { u . v) as follows.
Let u G a n d u G AL. R eroot th e d y n a m ic tre e for T a t w. Use b in ary search on th e p a th in T from z to th e root to find th e closest nodes x . y E. T such th a t
u G A'x. L’ E A'y. If { x . y } G T . re tu rn "yes". O th erw ise, find th e m in im u m weight
edge e on th e x - y p a th in T by rero o tin g at x a n d finding th e m in im u m w eight edge from y to th e ro o t. If w(e) = w { x . y ) . re tu rn "y e s": if iv{e) > w ( x . y ) . re tu rn "no".
E ach of th e following d y n a m ic tre e o p e ra tio n s requires O (lo g n ) tim e: rerooting th e tre e , finding th e ith node on a p a th to th e ro o t, finding th e m in im u m weight edge on a p a th to th e ro o t, s p littin g th e tre e by d e le tin g an edge or jo in in g two tre e s w ith an edge. T h u s. I n s e r t - Q u e r y requires 0 ( lo g ‘ n] tim e . Since Del ete m akes
0 { n ) changes to T . it now requ ires 0 [ n log n) tim e. T h e o p e ra tio n s De Je te - Que r y and I n s e r t still req u ire 0 ( n ) tim e . T herefore, we have reduced th e tim e for I n s e r t - Q u e r y
to 0 (lo g " n) a t th e e x p e n se of increasing th e tim e for D e l et e to 0 { n log ii).
2.5.1
C om puting the weights
In th e first im p le m e n ta tio n of I n s e r t - Q u e r y . we c o m p u te d w { x . y ) = |A ' x U A 'y} in
0 ( n ) tim e by c o m p a rin g th e c h a ra c te ristic vectors for A 'x a n d A 'y . In th e second im p le m e n ta tio n , we c o m p u te a ' ( x . y ) in 0 (1) tim e by m a in ta in in g a n .< n m a trix lU w ith fU (x .y ) = w ( x . y ) . T h e preprocessing tim e is d o m in a te d by th e O (n^) tim e to b u ild W . We u p d a te W in 0 { n ) tim e a fte r I n s e r t or De l et e, as follows.
.After I n s e r t ( u . v ) a d d s node : to T an d c re a te s z's c h a ra c te ris tic vector. A L =
I U { u .u } . we m u st c o m p u te W '(u’.z ) for all œ E T: th e o th e r e n tries in lU are
u n ch an g ed . W e have f U I x .z ) = lU (y .z ) = | / | + 1. (See F ig u re 2.8.) Let w ^ x. y. z. W e show how to c o m p u te W '(tu.z) from W { w . x ) a n d lU(u.’.y ) .
S uppose A'x is not m a x im a l in G '+ { u .i;} . T h e n A'x = / U { u } a n d h \ = A'xU {u}. T h e refo re, if v E A'u,, th e n W { w . z ) = W '(u t.x ) + 1. a n d o th erw ise. lU (u -.z ) =
