Efficient parallel algorithms on distance-hereditary graphs

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Efficient Parallel Algorithms on Distance-Hereditary Graphs (Extended Abstract)

Sun-yuan Hsieh

*

Chin-Wen Ho

Dept. of Computer Science

&

Info. Eng.

National Taiwan University, Taiwan

e-mail: d3506013 @csie.ntu.edu.tw

Tsan-sheng Hsu

t

Ming-Tat KO

Dept. of Computer Science

&

Info. Eng.

National Central University, Taiwan

Institute of Information Science

Academia Sinica, Taiwan

Gen-Huey Chen

Dept. of Computer Science

&

Info. Eng.

National Taiwan University, Taiwan

In this papel; we present eficient parallel algorithms forjinding a minimum weighted connected dominating set, a minimum weighted Steiner tree for a distance-hereditary graph which take O(1og n ) time using O(n+m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique - of - a distance-hereditary graph in O(log2 n) time using O ( n

+

m) processors on a CREW PRAM.

1 Introduction

A graph is distance-hereditary if every two vertices have the same distance in every connected induced subgraph containing both (where the distance between two vertices is the length of a shortest path connecting them). Properties and optimization problems in distance-hereditary graphs have been extensively studied during the past two decades [ 2 , 7 , 8 , 9 , 111.

A dominating set D of a graph G = (V,

E)

is defined as D V such that every vertex in V

\

D is either in D or is adjacent to some vertex in D. A dominating set D is connected if the subgraph induced by D is connected. For a given graph G and a set K

C

V (of terminal ver- tices), a Steiner tree is a tree which spans all vertices of I<-. The connected dominating set problem CD (respectively, Steiner tree problem S7) asks for a minimum cardinality connected dominating set (respectively, Steiner tree). In this paper, we consider the following weighted version of the CD and ST on distance-hereditary graphs. Let w(w)

be a non-negative weight associated with each vertex w in

'Supported in part by Institute of Information Science, Pcademia Sinica, Teipei, Taiwan.

Supported in part by NSC Grant 86-22 13-E-001-012,

G. We want to find a connected dominating set D (respec- tively, Steiner tree T ) such that

xwED

w(w) (respectively,

CwET

w(w)) equals the smallest possible value. We show that the above problems can be solved in O(1ogn) time using O(n

+

m ) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively.

A graph C is a clique if there is an edge between every pair of vertices. We say that C is a clique of G if C is an induced clique of G. If each vertex is assigned anon-negative weight, a maximum weighted clique is a clique of G with the maximum total weight. In this paper, we also compute a maximum weighted clique of a distance-hereditary graph in O(log2 n ) time using O(n

+

m) processors on a CREW PRAM.

2 Preliminaries

This paper considers finite, simple loopless, undirected and connected graphs G = (V, E ) , where V and

E

are the vertex and edge set of G, respectively. Let n = IVI and m = (E l. The distance &(z, y) or d ( z , y) between two vertices z and y in G is the length of a shortest z-y path in G. Let v be a vertex of G. We denote the neighborhood

of w, consisting of all vertices adjacent to w , by N ( w ) , and the closed neighborhood of w , the set N ( w ) U {w}, by N[w]. Let S be a subset of V. We denote N ( S ) the open neighborhood of S , that is the set of vertices in G, exclusive of S, which are adjacent to any vertex in

S.

We also denote N [ S ] = N ( S ) U

S.

The subgraph induced by S , denoted by (S), consists of the vertices of S and edges (2, y) with 2, y E S and (z, y) E E .

The hanging of a connected graph G = (Vi E ) at a vertex U E

V ,

denoted as h,, is the collection of sets & ( U ) ,

Ll ( u ) ,..., Lt(u)(orsimplyLo,L1 ,..., Lt whennoambiguity

20

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arises), where t = maxUE v ( E G ( u , v) and Li(u) = {v E V : & ( U , W) = i} for0

5

i

5

t . For any vertex E Li and vertexsets C_ Li, 1 5 i

5

t,letN’(v) = N (v) n Li- land let N ’ ( S ) = N ( S )

n

Li-1. Any two vertices x , y E Li (1

5 i

5 t -

1) are said to be tied if x and y have a common neighbor in Li+l. A homogeneous set of a graph

G

is a set A of vertices such that every vertex in V ( G) - A

is adjacent to either all or none of the vertices of A. A graph is a cograph (also a P4-free graph) [5] if it is either a vertex, the complement of a cograph, or the union of two cographs (where by union of two graphs G1 =

(V,

,

E l ) and G2 =

(fi,

E2) we mean the graph G =

(V,

U

h,

El U

E2)). A cograph G has a parse tree representation called

cotree, denoted by TG, with (a) the leaves of TG are the vertices of G; (b) the internal nodes of TG are labelled 0 or 1 ; (c) 0 nodes and 1 nodes alternate along every path staring from the root; (d) two vertices x and y of the cograph are adjacent if and only if the least common ancestor of z and y in TG is a 1 node. In the following, we introduce properties of distance-hereditary graphs.

Proposition2.1

181

Suppose h, = (L0,Ll

,...,

Lt) is the hanging at U of a connected distance-hereditary graph. r f

x, y E Li (1

5

i

5

t ) are in the same component of

<

Li

>,

or x and y are tied, then N ’ ( x ) = N’(y).

Fact 1 A hanging of a connected distance-heredita graph can be computed in O(1ogn) time using O ( n

+

my processors on a CRCW PRAM.

Proo$ An implementation is described as follows. (a) We first find an arbitrary spanning tree S for G in O(1og n ) time using O ( ( m

+

n)a(m, n ) / logn) processors on a CRCW PRAM, where a is the inverse Ackermann function [4], and then transform it into a rooted tree with the root U in

O(1ogn) time using O ( n ) processors on an EREW PRAM [lo]. (b) For each vertex v(# U ) , we associate it with a

number num(v) which is the distance from v to U in S

(the computation of distances for those non-root vertices can be done in O(1og n ) time using O( n/ log n ) processors on an EREW PRAM [lo]). (c) For each vertex 2, let

(x , y) E E T ) if num( y) is the smallest value in { num(z)

I

t E N ( x )

i

.

This can he done in O(1ogn) time using O ( n

+

m) processors on an EREW PRAM by the sorting algorithm described in [3]. It can be verified that T =

(V, E ( T ) ) is a quasi-breadth-first-search tree which is a hanging at U . It has been shown that a quasi-breadth-first-

search tree is also a breadth first search tree on a distance- hereditary graph [6]. Note that Li is the set of vertices x E T , such that the unique path from x to U is of length

exactly i. Q.E.D

Proposition 2.2 [2] Suppose h, = (LO, L1

,

.

. .

,

L,) is a hanging of a connected distance-heredita graph at U .

For any two vertices x , y E Li, i

2

1, N ’ Z ) and N‘(y) are either disjoint, or one of the two sets is contained in the othel:

We call a family of subsets arboreal if any two elements of the family are either disjoint or comparable (by set inclu- sion). For example, { 1

,

2,3}, { 1,2}, (4)) is an arboreal family of subsets of

J!

1,2,3,4}. Given an arboreal family of distinct subsets, denoted by S I , S2,

...,

S,, we can define a partial order

5

between Si’s by S,

5

S,

e

S,

C

S,,

where 1

5

p , q

5

r . An element S is said to be minimal in the given family if S,

2

S,, for a i q

#

p and 1

5

q

5

r .

In [8], Hammer and Maffary defines an equivalence re- lation rabetween vertices of Li by x

s

i

y means

x

and y

are in the same connected component of La or IC and y are

tied. Let be defined on V ( G ) by x y means IC =i y

for some

i.

They make explicit the structural aspects of distance-hereditary graphs by deriving the following characteristics.

Proposition2.3 [8] Let U be a vertex of a connected

distance-hereditary graph G and let h, be the hanging at U of G. Let R I , R2,

...,

R, be the equivalence classes of

the relation sa. Then, ( a ) the graph obtained from G by shrinking each Rj into one vertex is a tree rooted at U (for

convenience, we call such a tree an equivalence-tree Th, .);

( b ) each Rj induces a P4-jree subgraph; (c) f o r each j , the family {NI( R k ) : N‘( R k ) Rj} is an arboreal family of

homogeneous subsets of Rj.

3 Minimum Weighted Steiner Tree

This section deals with the weighted Steiner tree proh- lem on a distance-hereditary graph with the terminal vertex set

Ii.

Given a hanging h,, where U is an arbitrary vertex

of IC, let Th, he the equivalence-tree defined in Proposition 2.3. Note that each node w in Th, represents an equiva- Ience class, denoted by R,. For each node v in a rooted tree T , let T ( v ) be the subtree rooted at v. We present our algorithm below.

Algorithm WCD

Input: A distance-hereditary graph G = (V, E ) and a set I<

c

V composed of terminal vertices.

Output: A minimum weighted Steiner tree T on It”.

Step 1. Let U be an arbitrary vertex in the target set

IS-

and let U =

Ii.

Determine a hanging h, and construct the equivalence-tree T h ,

.

Step 2. For each node w in Th,, if all of the following conditions (a) U X f V ( T h u ( W ) ) { R X

n

K}

#

0,

(b) N I ( & ) contains no vertex in K , and (c) N’( R,) contains no proper subset N’(R,), such that the node v in Th, satisfies that { R,

n

Ii}

#

0;

hold, then select a minimum weighted vertex y E N’(R,) and add it to U .

Step 3. Find a spanning tree T for

<

U

>, which is the

desired output.

Theorem 3.1 Algorithm WCD generates a minimum

weighted Steiner tree for the distance-hereditarv nraDh in

U X E V ( T h u ( u ) )

O(lu0g n> time using O ( n

+

m ) processors on’

a“

CRCW PRAM.

Proofi For each vertex

w(#

U ) E

It‘,

it can be verified that

there is a path, connecting w and U , in

<

U

>

after Step

2. Thus < U

>

is a connection of K. We then show that

< U

>

has the minimum weight. Observe that for each node w in Th, satisfying conditions (a), (b) and (c) in step 2 of algorithm W C D , a minimum weighted connection of I< must contain a vertex of N’(R,). Since the algorithm selects a vertex with the minimum weight from w , the weight of U is clearly minimum. After executing Step 3, the algorithm finds an arbitrary spanning tree T for < U

>.

By definition, T is a minimum weighted Steiner tree.

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We now analyse the time-processor complexities. In Step 1, the implementation of a hanging is described in Fact 1. We note that the computation of T h , can be done by finding the equivqlence classes with respect to h,. It takes O(1ogn) time using O ( n

+

m ) processors on a CRCW PRAM. In step 2, associate each vertex v with t a g ( v ) such that t a g ( v ) = 1 for v E K and t a g ( v ) = 0 otherwise. Using the above information, the Euler-tour technique [lo] can be applied to check the condition (a) which takes O(1og n ) time with O ( n / log n ) processors on an EREW PRAM. Condition (b) can be checked in O(1og n ) time using O ( n / log n ) processors on an EREW PRAM by slightly modifying optimal parallel prefix-sum computation [lo]. A method to check condition (c) can be done similar to the implementation of the algorithm in Section 3 for finding a connected dominating set. In Step 3, finding a spanning tree can be done in O(1og n ) time using O ( ( m + n ) a ( m , n)/ l og n ) processorsonaCRCWPRAM

[41. Q.E.D.

4 Maximum weighted Clique

It is easy to see that any clique of a distance-hereditary graph G belongs to Li U Li+l for some 0

5 i

5

t

-

1. Let R denote an equivalence class of G with respect to h,

.

Observation 1 A maximum weighted cliaue of a distance- hereditary graph is a maximum“weight.cd clique of some

<

R U N’(R)

>.

We call < R U NI(

R)

> a

candidate subgraph of a maxi- mum weighted clique. For abbreviation, we also call it is a candidate subgraph. Since there are at most n equivalence classes for G, we have the following observation.

Observation 2 There are O ( n ) candidate subgraphs f o r a distance-hereditary graph.

For two graphs

G I

and G2, the join of GI and G2, denoted by G1 @ G2, is the graph consists of G1 U G2 and all edges joing V(G1) and V(G2) [12]. From structure characteristics of G, every candidate subgraph

<

RU N’(R)

>=<

R

>

@

<

N’(R)

>.

If C1 and C2 are maximum weighted cliques of R and N’(R) re- spectively, then C1 @ C2 is a maximum weighted clique of

<

R U N’(R)

>.

By Observation 2 and 1 , a maximum

weighted clique of G is a clique with the maximum weight among O ( n ) candidate subgraphs.

It is assured by Proposition 2.3 that

<

R

>

and

<

N’(R)

> are cographs. In

[I], Abrahamson, et. al. present an algorithm to find a maximum weighted clique of a cograph G: Given a cotree T of G a maximum weighted clique can be found by applying the tree contraction technique to T which takes O(1ogn) time using O ( n / logn) processors on an EREW PRAM. Hence, if cotrees TI and

T2 for < R > and < N ’ ( R ) > are given, a maximum

weighted clique of the candidate subgraph < R U N’( R )

>

can be computed by applying the algorithm in [ l ] to TI and T2. But it is necessary to construct cotrees for each R and N ’ ( R ) , the above algorithm takes a large number of processors to achieve the desired computation. Here we provide a different way (described latter) which requires fewer processors than the above one.

For any node v in a tree T , let l e a f ( v ) be the ieaves of T (v ) (recall that T ( v ) is the subtree of T rooted at

v). Let SR =

{SI

there is an equivalence class R’ with N’(R’) = S and S R } . By the algorithm in [l], given a cotree T all maximum weighted cliques of cographs induced by deaf (v), where v is an internal node of T , can be found simultaneously in O(1og n ) time using O ( n / log n ) processors on an EREW PRAM (see [ 11 for details). Based on this result, we concentrate our effort to construct a spe- cial cotree TR for

<

R

> such that for each

S E SR there exists a node v in TR with Z e a f ( v ) = S and T R ( Y ) is a cotree of

<

S

>.

For convenience, we call TR a canonical cotree.

Lemma 4.1 I f a canonical cotree of

< R >

is given, all maximum weighted cliques of

<

S

>,

where S

E

SE, can be computed in O(1og IRI) time using O(lRI/ log IRI) processors on an EREW PRAM.

In the following, we show how to construct the canonical cotree for

<

R

>. By Proposition 2.3,

SR is the arboreal family of homogeneous subsets of R. Let S’R = SE U { R } . According to the partial order

5

defined in Section 2 , we

first construct an auxiliary tree structure AR as follows. We create IS’I nodes where each node represents an element (a set) in S’R. Let S, denote the set represented by the node

v in A R , and p a r ( v ) (respectively, child(v)) be the parent (respectively, children) of v. For any two created nodes a and @, there is a directed edge from a to ,B if S ,

5

Sp and no other S, satisfies Sa

5

S,

5

Sp.

Lemma 4.2 The auxiliary tree structure AR can be con- structed in O(1ogp) time using O ( p

+

q ) processors on an EREW PRAM, where p =

I

RI and q =

I

E (< R

>

)

I.

Proof: Suppose R = { V I , 212,

...,

vp} and SR

=

{SI, S 2 , ..., S,.}. Let

Si

= N’(R;) where Ri is an equiva- lence class. For ease of description, we assume the vertices are represented by their indices. So R = { 1 , 2 , . . . , p } .

We construct AR as follows. Let U be any vertex of Ri and Si = {tl,t2, . . . , t k } , where ti E {1,2,

...,

p } . Note that N’ U ) =

Si.

We assign ISi

I

processors to the edges

( u , t l ) ,

r

u,t2) ,..., (u,tk)forcreatingtupIes(tl,i),(t;?,i) ,...,

( t k ,

i).

Thus there are totally

[Si

I

created tuples that

are stored in an array B.

We first sort those tuples according to their first indices. Then B is divided into p blocks (recall that /RI = p)

B1, B2,

...,

B,, where

Bi

records those tuples (i, a)’s for all a such that i E Sa. We then sort the tuples in each

Bi

according to the cardinalities of those sets indicated by second indices. Suppose (i, a l ) , (i,

4,

..., (i, a k ) are the

sorted elements in

Bi

and let ai be the created node representing Sa*. Set par(ai) = ai+l, 1

5

i

5

k

-

1 and let p a r ( a k ) be the node representing R.

After executing the above algorithm, AR can be con-

structed. The correctness follows directly from the property that if S,

5

S,,then: (i)

ISPI

<

I

S

,

l

and (ii) 2 E

s,

implies 2 E

S,.

Using the sorting algorithm described in

[ 3 ] , the implementation can be done in O(1ogp) time using O ( p

+

q ) processors on an EREW PRAM. Q.E.D Lemma4.3 A canonical cotree TR can be generated in O(log2p) time using ~ (

+

pq ) processors on a CREW

PRAM,

wherep = ]RI and q = IE(< R

>)I.

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Pro08 Given the auxiliarly tree structure A R , we generate a tree TR in the following way. Let us consider the cograph

<

S,

> for each node

U E AR. We first shrink each

<

S,

>

in

<

S,

>,

where

x

E child(u), into a vertex

s called shrinked vertex. Since S, is a homogeneous set of

< S,

>,

the resulting graph, denoted by H , is still a cograph. Applying the algorithm in [6] to H , a cotree

TH

can be constructed for H in O(log2 p ) time using O ( p

+

q )

processors on a CREW PRAM. Note that each shrinked vertex is a leaf in T H . There are totally ~ S ' R

1

constructed cotrees. For each leaf of the generated cotrees, if it is a

shrinked vertex s,, we replace it with the "corresponding" cotree T,. It takes a constant time using O ( p ) processors.

We now show that TR is the canonical cotree of

< R

>.

Clearly, TR has the following properties: (1) the leaves of TR are all vertices of R, (2) each internal nodes are either 0 node or 1 node, and (3) if two vertices

x

and y of

<

R > are adjacent, the least common ancestor of

x

and y in TR is a 1 node (this property can be verified from the fact that SR is a family of homogeneous sets of R). Moreover, from the construction of TR, for each

S

E SR there exists a node

U in

TR

such that l e a f ( u ) = S. Hence T' is a canonical

cotree of < R

>.

Q.E.D

Remark: Note TR is not the standard form of the cotree since it violates the definition (c) (refer to Section 2.1) that 0 nodes and 1 nodes alternate along every path staring from the root. For an internal node U of a cotree

with children 1-11, 1-12,

...,

I l k , let G ( u ) be the cograph in-

duced by l e a f ( u ) and G p , be the cograph induced by l e a f ( p ; ) . From the definition (d), if U is a 1 (res ec-

tively, 0) node, G ( u ) = G(p1) f3 G(1-12) @

. . .

f3 G & E )

(respectively, G(u) = G(p1) U G(p2) U.. .U G ( p k ) ) . Ac- cording to the above observation, the definition (c) can be relaxed (note that this kind of cotree representation is no longer unique).

Theorem 4.4 The maximum weighted clique problem f o r distance-hereditary graphs can be solved in O(log2 n ) time using O ( n

+

m ) processors on a CREW PRAM.

Proofi It takes O(1ogn) time using O ( n f m) CRCW processors to determine a hanging and compute the equivalence classes by Fact 1 and Lemma 5.2. By Observation

2, we have O ( n ) candidate subgraphs, i.e., O ( n ) candidate cliques. According to Lemma 4.3, we compute canonical cotrees for those equivalence classes of G in O(logn2) time using O ( n

+

m ) processors on a CREW PRAM. By applying the algorithm in [ 11 to canonical cotrees, O ( n ) candidate cliques can be found in O(1ogn) time using O ( n / log n ) processors on an EREW PRAM. Thus we can find the one with the largest total1 weight in O(1og n ) time using O(n/ log n ) processors [IO]. Q E D

5 Minimum Weighted Connected Dominat-

Given a hanging h,, let RI, R2, ..., R,. be those equivalence classes of the relation -I+] (see Proposition 2.3). Let

S = { S C LII there exists an R; with N ' ( R i ) = S}. For convenience, we call S the upper-neighborhood system of

Lt+1. By Proposition 2.3, S is arboreal. So the partial order

5

described in Section 2 can be defined on

S.

Let

U

=

{S

E SI there exists no SI E S, S ;I! SI} , i.e.,

U

ing Set

23

is the set composed of those minimal elements of

5.

We call U the minimal upper-neighboorhoods of Ll+l. For

each

5'

E

U ,

we select a vertex x, E S with the minimum weight. Let

D I

= {rsl S E

U}.

Lemma5.1

[ I l l

Suppose G = (V, E ) is a connected distance-hereditary graph with a non-negative weightfunc- tion

w

on vertices. Let U be a vertex of minimum weight and

h , = (Lo, L1,

. . . ,

L,) be the hanging at U . Then, one of the

following three sets of vertices is a minimum weighted con- nected dominating set of G: ( I ) U;,: DI ; ( 2 ) U:=: DI U { U } ;

(3) some vertex of G .

Lemma5.2 Given a hanging h,, D I , f o r all 1

5

1

5

t

-

1, can be computed in O(1ogn) time using O ( n

+

m )

processors on a CRCW PRAM.

Proot We first compute the auxilary tree structure AR for each R L I (see Lemma 4.2). Let l e a f ( A R ) denote the leaves of AR. So U={SlS E leaf(&), R

C

L I } (recall that

U

is the minimal upper-neighborhood of Ll+l).

Clearly, DI can be determined from

U.

Q.E.D. Theorem 5.3 A minimum weighted connected dominating set for distance-hereditary graphs can be found in O( log n ) time using O ( n

+

m ) processors on a CRCW PRAM.

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Efficient parallel algorithms on distance-hereditary graphs