scheduling time-constrained multicast messages in circuit-switched tree networks

Information Processing Letters 84 (2002) 109–116

www.elsevier.com/locate/ipl

Scheduling time-constrained multicast messages incircuit-switched tree networks

Oh-Heum Kwon

Pukyong National University, Electronics, Computer, and Communication Division, Daeyeon-dong 599-1,Nam-gu, 608-737 Pusan, South Korea

Received 11 July 2001; received in revised form 15 December 2001

Communicated by F.Y.L. Chin

Abstract

In this paper, we consider a kind of multicast scheduling problem in a tree network. Each multicast message is transmittedthrough a directed subtree within the tree network. The transmission time of each multicast message is assumed to be one unit.Two messages can be transmitted at the same time if their subtrees are edge-disjoint. Each message is constrained by a readytime and a deadline, and has a weight we can gain if it is scheduled within its deadline. The optimality criterion is the totalweight we gain. We assume that the degree of each subtree is bounded by a constantd and present an approximation algorithmof which the approximation ratio is at most 4d + 15. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Algorithms; Scheduling; Multicast routing; Approximation algorithms

1. Introduction

1.1. Problem definition

A multicast message is one that originates froma single source node and has multiple recipients.Each multicast message is delivered through a subtree,which spans the source and the destinations of themessage, within the network. At each branch node ofthe subtree, the message is replicated and transmittedsimultaneously toward the destinations.

In a store-and-forward network, packets can beblocked at an intermediate node within the path fromthe source to the destination. The packet transmission

E-mail address: [email protected] (O.-H. Kwon).

time depends heavily on the length of the path that themessage travels.

There are, however, other switching technologies,including circuit-switching, virtual cut-through, andwormhole routing, in which the message transmissiontime is not sensitive to the length of the path. In circuit-switched networks, once a message departs from itssource, it is never blocked at an intermediate node. Atthe branch nodes, the message is replicated and re-layed immediately upon the reception. In those tech-nologies, the transmission time is mainly determinedby the length of the message.

In this paper, we assume that every message hasthe same length and, therefore, the transmission timeof every message is equal to one unit time, regardlessof the location and the number of destinations.

0020-0190/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0020-0190(02)00226-0

110 O.-H. Kwon / Information Processing Letters 84 (2002) 109–116

We further assume that each link of the network hasa unit capacity, which means that, if two subtrees fordelivering two multicast messages, respectively, sharea common link, then they cannot be transmitted at thesame time.

In this paper, we consider tree networks. Treesare interesting since tree-like networks are often usedin the telecommunications industry. Even when thenetwork is not a tree, some multicast protocol usesa fixed spanning tree of the underlying network fortransmitting multicast messages [6].

Each link of the tree network is assumed to befull-duplex. We model the tree network by asymmetricdirected tree which is a directed graph obtained froman undirected tree by replacing each edge with a pairof opposite directed edges.

There arenmulticast messages{M1,M2, . . . ,Mn}.In tree networks, the route taken by each messageMiis uniquely determined and can be represented as adirected subtreeTi . The subtreeTi spans minimallythe source and destinations of the messageMi . We callTi themulticast tree ofMi .

Each multicast message is constrained by itsreadytime bi and itsdeadline di , and has aweight wi . Wegain the weightwi if the message is scheduled at atime instance between its ready time and its deadline.

A schedule consists of a subsetS of the multicasttrees. Each treeTi in S is associated with a scheduledtime instancesi where bi � si < di . A schedule isconsidered to be feasible if any pair of treesTi and

Tj in S with si = sj are edge-disjoint. Of course, twoopposite directed edges joining the same nodes areconsidered to be different edges. The objective of thescheduling algorithm is to find the setS of which thetotal weight is as large as possible.

This problem is NP-hard, and moreover, is difficultto approximate even when every message has the sameweight. Consider a star tree which consists of a singlecenter nodev0 and other nodesV = {v1, v2, . . . , vm}that are all adjacent tov0. The source of every messageis v0 and the destinations of messageMi is Vi ⊆ V ,1 � i � n. Suppose further that the ready times of allmessages are 0, the deadlines of all messages are 1,and the weights of all messages are 1. Fig. 1 shows anexample.

In this situation, two multicast treesTi andTj inter-sect if and only ifVi ∩Vj �= ∅. Therefore, the schedul-ing problem is identical to finding the maximum car-dinality subsetS ⊆ {V1,V2, . . . , Vn} such that no twosetsVi andVj in S intersect. This problem is known tobe theset packing problem [5] and cannot be approx-imated with an approximation ratio ofn1−ε for anyε > 0 unless NP= ZPP [1,4].

In this paper, we restrict our concern to the caseswhere the degree (the in-degree plus the out-degree)of each multicast tree is no larger than a constantd .This restriction is natural in the sense that, in practice,there exists a limit on the number of messages that anode can transmit at the same time.

Fig. 1. The star tree.

O.-H. Kwon / Information Processing Letters 84 (2002) 109–116 111

This bounded-degree version of the schedulingproblem also embeds, as a special case, a restrictedversion of the set packing problem in which thecardinality of each set is bounded byd . This problemis called thed-set packing problem. Thed-set packingproblem is known to be APX-complete [4], and thebest known approximation ratio isd/2 + ε, for anyε > 0, for unweighted cases [7], and 2(d + 1)/3 forweighted cases [3], respectively.

1.2. Results

The approximation ratio of our scheduling algo-rithm depends on that of thed-set packing prob-lem. Our algorithm achieves an approximation ratio of6α+11 for weighted cases and 3α′ +3 for unweightedcases, whereα andα′ are the best known approxima-tion ratios for the weighted and the unweightedd-setpacking problem, respectively.

As a byproduct, our algorithm can be applied to theunicast message scheduling problem, in which eachmessage has only one destination, and achieves a ratioof 11 and 4 for the weighted and the unweighted cases,respectively.

1.3. Related works

There have been extensive studies on the unicastmessage scheduling in tree networks, however, mostof them considered the makespan problem. See [2] andthe references therein.

2. Independent set of multicast trees

Let T be the symmetric directed tree representingthe underlying tree network. Since it’s symmetric, if adirected edge〈w,v〉 exists, then〈v,w〉 also exists. Fora directed edgee= 〈v,w〉, we mean bye the oppositedirected edge〈w,v〉, that is,e= 〈w,v〉.

We choose at random a noder of T as itsroot node.The level of each nodev in T , denoted byl(v), isdefined to be the number of edges in the unique pathbetweenr andv.

We are givenn multicast messagesMi , 1� i � n,and each messageMi corresponds to a multicast treeTi , 1� i � n. Theroot of multicast tree Ti is definedto be the node ofTi that is nearest to the rootr of tree

Fig. 2. The root of a multicast tree.

T . We denote the root ofTi by ri . Note that the root ofthe multicast tree is not the source of the message, ingeneral.

Fig. 2 shows an example of the multicast tree and itsroot node. In this figure, we represent two symmetricdirected edges of the underlying tree by one flat line.The thick arrowed lines represent a multicast treeTi .The source of the message isv1, the destinations are{v3, v4, v7}, and the root ofTi is v8.

For any nodev in Ti , let Eout(v, Ti) be the setedges outgoing fromv in Ti , and let ein(v, Ti) bethe edge incoming tov in Ti , if any. LetE(v,Ti) =Eout(v, Ti) ∪ {ein(v, Ti)}. In Fig. 2, Eout(v7, Ti) ={〈v7, v3〉, 〈v7, v4〉} andein(v7, Ti) = 〈v8, v7〉. The de-gree,|E(v,Ti)|, of each nodev of each multicast treeTi , 1� i � n, is assumed to be at mostd .

The weight of a multicast treeTi , denoted byw(Ti),means the weightwi of the corresponding messageMi . For a set of multicast treesT , let w(T ) denotethe total weight of the trees inT , that is,

w(T )=∑

Ti∈Tw(Ti).

Two multicast treesTi and Tj are said to beindependent if they do not intersect. In this section, weconsider the problem of finding themaximum weightindependent set which is defined to be the set of the


pairwise independent multicast trees of which the totalweight is as large as possible.

2.1. Independent set within a group

In this subsection, we consider a subproblem of theindependent set problem. Consider a setT of multicasttrees that are all rooted at a common node, that is, theroots of the multicast trees inT are the same. LetrTbe their common root node.

We introduce an additional notion about the inde-pendence between two multicast trees. Two multicasttreesTi andTj in T are said to bealmost-independentif E(ri , Ti)∩E(rj , Tj )= ∅. In other words,Ti andTjmay share an edge, but do not share any edge that isincident torT .

See Fig. 3, in which three multicast trees (black,shaded, and dotted one) are pairwise intersecting,however, no pair of them intersect at the edge incidentto their common root. Therefore, they are not indepen-dent, but, almost-independent to each other.

The maximum weight almost-independent set ofT is defined to be the subset ofT that consists ofpairwise almost-independent trees and has the totalweight as large as possible.

Since |E(ri, Ti)| � d for any multicast treeTi ,the problem of finding the maximum weight almost-independent set is identical to the weightedd-setpacking problem, which can be approximated withinα = 2(d + 1)/3 [3]. Therefore, we can find, in poly-nomial time, an approximated almost-independentset I ′

appr(T ) of which the total weight is at least3/2(d + 1) times of the maximum weight almost-independent setI ′

opt(T ).

Fig. 3. Almost-independent trees.

In general, the almost-independent set is not an in-dependent set. However, we can find an independentsubset of it. To do that, letG be a graph defined as fol-lows: each node ofG corresponds to a multicast treein I ′

appr(T ) and two nodes ofG are connected by anedge if their corresponding trees are not independent.In Lemma 2.1 and the following paragraph, we willshow thatG is a 3-colorable graph, which implies thatthere exists an independent subset of which the cardi-nality is at least|I ′

appr(T )|/3.

Lemma 2.1. Any connected component of the graphG has at most one cycle.

Proof. We will convert the graphG into a directedgraph as follows: Consider any edge(ui, uj ) in G andlet Ti andTj be the trees that correspond toui anduj ,respectively. SinceTi andTj are members ofI ′

appr(T ),they do not share any edge that is incident to their com-mon rootrT , but they share some other edges. This ispossible only with eitherein(ri , Ti) ∈ Eout(rj , Tj ) orein(rj , Tj ) ∈ Eout(ri , Ti). We assume, without loss ofgenerality, thatein(ri , Ti) ∈ Eout(rj , Tj ). We assign adirection fromui to uj to the edge(ui , uj ) of G. Inthis way, we assign directions to each edges ofG.

Since the trees inI ′appr(T ) are almost-independent,

there could not exist two treesTj and Tk , j �= k,such thatein(ri , Ti) ∈ Eout(rj , Tj ) and ein(ri , Ti) ∈Eout(rk, Tk), which means that the out-degree of eachnode of the directed version ofG is at most one.

Suppose for contradiction that two cyclesC1 andC2 exist in any connected component ofG. The onlyway of assigning directions to the edges within a cycle,satisfying the out-degree one requirement, is makingthe cycle a directed cycle, that is, a cycle of which theedges have the same direction along the cycle.

There are several ways of connecting two directedcycles. Fig. 4 shows two representative examples ofconnecting two directed cycles. Fig. 4(a) shows thecase where two cycles share no nodes, while Fig. 4(b)shows the case where two cycles share some nodes.It is easy to verify that, in any case, at least one nodemust have an out-degree of two, which completes theproof. ✷

A connected graph which contains only one cyclecan be colored using three colors. By removing anarbitrary edge within the cycle, the graph becomes a


(a)

(b)

Fig. 4. Examples of connecting two cycles.

tree. A tree can be colored using two colors. Aftercoloring the tree, we recover the removed edge. If thenodes joined by the recovered edge have the samecolor, we change the color of one of them into the thirdcolor. Then we have a 3-coloring of the graph. A graphof which the connected components are all 3-colorableis also 3-colorable.

Now thatG is a 3-colorable graph, we can alwayschoose an independent setIappr(T ) of which the totalweight is at least13w(I

′appr(T )). In fact, Fig. 3 implies

that this value,13, is the best we can achieve.

Lemma 2.2. The “ independent set within a group”problem can be approximated with an approximationratio of 3α = 2(d + 1).

Proof. Obvious from the fact that

w(Iappr(T )

)� 1

3w

(I ′appr(T )

)� 1

3αw

(I ′opt(T )

)

� 1

3αw

(Iopt(T )

) = 1

2(d + 1)w

(Iopt(T )

),

where Iopt(T ) is the maximum weight independentset. ✷2.2. Independent set problem

Now consider the problem of finding the maximumweight independent set of the entire multicast trees.We first partition the multicast trees into groupsG1,

G2, . . . ,Gm so that all trees in each groupGi , 1 �i � m, have the same root noderGi . The groupsare ordered non-increasingly by the levels of theircommon roots, that is,l(rGi ) � l(rGi+1), 1 � i �m− 1. The ties are broken arbitrary. The trees withineach group are also ordered arbitrary. The algorithmINDEPENDENT-SET is as follows:

Algorithm INDEPENDENT-SET.

Let I = ∅; // which is the independent set //for i = 1 tom do

for each treeTj in Gi doletX⊆ I be the set of multicast trees

that intersectTj ;if w(X) > 1

2w(Tj )

then removeTj fromGi ;[Pruning phase]

LetG′i be the set of remaining trees inGi ;

Find the independent setIappr(G′i ) by

the described algorithm;[Independent set within a group phase]

Remove all trees inI that intersect any treein Iappr(G

′i );

[Purging phase]Let I = I ∪ Iappr(G

′i );

End of Algorithm.

It is obvious that the algorithm INDEPENDENT-SET constructs an independent set.

Consider a treeTj ∈Gi that is not selected by thealgorithm as a member of the final independent set.We have three cases. First, if treeTj is removed inthe pruning phase, we say thatTj is pruned. Second,if it is neither pruned nor selected as a member ofIappr(G

′i ) in the “Independent set within a group”

phase, we say thatTj is rejected within a group. Third,if Tj was selected as a member ofI at the iterationof consideringGi and removed later in the purgingphase, we say that it ispurged.

Let Iappr be the final independent set produced byour algorithm, and letI ′

appr be the set of trees thatare neither pruned nor rejected within the group inour algorithm. Obviously,Iappr ⊆ I ′

appr and a tree inI ′appr− Iappr is the one that is purged. LetIopt denote

the maximum weight independent set.

Lemma 2.3. w(Iappr)�w(I ′appr)/2.


Proof. We construct a directed graphF of whichthe nodes correspond to the trees inI ′

appr. Consider apurged treeTi ∈ I ′

appr and suppose thatTi is purged atthe iteration of considering groupGk. SinceIappr(G

′k)

is an independent set, there is a unique treeTj ∈Iappr(G

′k) that causes the purging ofTi . We add, inF ,

a directed edge from the node corresponding toTi tothe node corresponding toTj .

The out-degree of each node ofF is at most one.Moreover, there could not be a cycle inF . Therefore,the graphF consists of disjoint trees. In other words,F is a forest. The out-degree zero nodes inF are calledthe local roots. The multicast trees corresponding tothe local roots are the members ofIappr.

In the followings, the weight of a node inF meansthe weight of the multicast tree that corresponds to thatnode ofF . The proposition in this lemma is that thetotal weight of the local roots inF is at least half of thetotal weight of all nodes inF . We claim that the weightof any nodev in F is at least half of the total weightof all nodes (includingv itself) of the subtree rooted atv. We use mathematical induction on theheight of v,which is defined to be the maximum distance fromvto one of its descendent leaf nodes.

If the height ofv is zero (that is,v is a leaf node),the claim holds obviously. Assuming that the claimholds for any node with the height less thani, considerany nodev with the heighti. Let u1, u2, . . . , ul bethe children ofv. Sincev purgesu1, u2, . . . , ul , thenw(v) � 2

∑1�j�l w(uj ). The total weight of the

subtree rooted atuj is at most 2w(uj ) by the inductionhypothesis. Therefore, the total weight of the subtreerooted atv is at most

∑1�j�l 2w(uj ) + w(v) �

2w(v), which completes the proof.✷Consider the setIopt− I ′

appr. A tree inIopt− I ′appr is

the one that is either pruned or rejected within a groupby our algorithm. LetA ⊆ Iopt − I ′

appr be the set ofpruned trees, and letB ⊆ Iopt− I ′

apprbe the set of treesthat are rejected within a group. Note thatA ∩ B = ∅andA∪B = Iopt − I ′

appr.

Lemma 2.4. w(A)� 4w(I ′appr).

Proof. We construct a bipartite graphH which con-sists of two node setsVA andVI ′

appr. Each node inVA

corresponds to a tree inA and each node inVI ′appr

cor-responds to a tree inI ′

appr.

Suppose thatu ∈ VA is the node that correspondsto Ti ∈ A and X is the set of multicast trees thatcauses the pruning ofTi . Note thatX ⊆ I ′

appr andw(X) > w(Ti)/2. Let v1, v2, . . . , v|X| ∈ VI ′

apprbe the

nodes that correspond to the trees inX. We add|X|edges(u, vj ),1 � j � |X|, into H . The edge(u, vj )

is assigned a weight ofw(Ti)w(Tj )

w(X), whereTj is the

tree corresponding tovj . Note that the total weight ofthe edges incident tou is equal tow(Ti), and also thatthe weight of the edge(u, vj ) is

w(Ti)w(Tj )

w(X)� 2w(Tj ).

We claim that the degree of any node inVI ′appr

is at most two. Suppose for contradiction that nodev ∈ VI ′

appris adjacent to three nodesu1, u2, and u3

in VA. Assume thatv corresponds to a multicast treeTj ∈ I ′

appr. The root ofTj is rj . Letp(rj ) be the parentnode ofrj in T .

Since three multicast trees corresponding tou1, u2,and u3, respectively, are considered later thanTj inour algorithm, their roots must have the level less thanthat ofrj . The only way for a tree having a root higherthan that ofTj to intersectTj is that the tree containseither〈rj ,p(rj )〉 or 〈p(rj ), rj 〉.

Since each of three trees contains at least oneamong two edges〈rj ,p(rj )〉 and〈p(rj ), rj 〉, at leasttwo of them intersect, which contradicts the fact thatall three are members ofA ⊆ Iopt. Consequently, theclaim has been proven.

Remember that, for any nodev ∈ VI ′appr

correspond-ing to Tj , the weight of any edge incident tov is nomore than 2w(Tj ). By combining this with the factthat the degree ofv is at most two, we conclude thatw(Tj ) is at least a quarter of the total weight of theedges incident to it. Moreover, the total weight of theedges ofH is equal tow(A). Consequently,w(A)is at most four times the total weight of the trees inI ′appr. ✷

Lemma 2.5. w(B)� 3αw(I ′appr).

Proof. Consider a groupGi and letBi = B ∩ Gi .Let G′

i be the set of trees inGi that are not pruned.Note thatBi ⊆G′

i . Our algorithm has tried to choosethe maximum weighted independent set from the trees


in G′i and found an independent setIappr(G

′i ) such

that

w(Iappr(G

′i )

)� 1

3αw

(Iopt(G

′i )

),

whereIopt(G′i ) is the optimal weighted independent

set ofG′i . SinceBi is an independent set withinG′

i ,we have

w(Bi)�w(Iopt(G

′i )

)� 3αw

(Iappr(G

′i )

).

Consequently, we have

w(B) =∑

1�i�mw(Bi)�

∑

1�i�m3αw

(Iappr(G

′i )

)

� 3αw(I ′appr). ✷

Sincew(Iopt − I ′appr) = w(A) + w(B), we have

w(Iopt − I ′appr)� (3α+ 4)w(I ′

appr).

Theorem 2.6. w(Iopt) � (6α + 10)w(Iappr) = (4d +14)w(Iappr).

Proof. Since

w(Iopt − Iappr) � w(Iopt − I ′appr)+w(I ′

appr− Iappr)

� (3α+ 4)w(I ′appr)+w(Iappr)

� (6α+ 9)w(Iappr),

we have

w(Iopt) � (6α + 10)w(Iappr)

= (4d + 14)w(Iappr). ✷

3. Scheduling algorithm

A multicast treeTi is said to berelevant to a timeinstancet if bi � t < di . Our scheduling algorithmis greedy in the sense that, for each time instancet ,it tries to find the maximum weight independent setamong the trees that are relevant tot and schedulethem at time instancet .

Algorithm SCHEDULE.

Let T be the set of all trees;t = min1�i�n bi ;while T �= ∅ do

Let Y ⊆ T be the set of trees that are relevant tot ;

Select an independent setS(t) amongYusing the INDEPENDENT-SET algorithm;

Schedule the selected trees at timet ;Let T = T − S(t);Incrementt to the minimumt ′ such that

there is, inT , at least one that is relevant tot ′;End of Algorithm.

Let Sappr be the set of multicast trees that arescheduled by our algorithm and letSopt be the setof trees that are scheduled by the optimal schedulingalgorithm. LetSopt(t) be the set of trees which arescheduled at time instancet in the optimal schedule,and letSappr(t) be the set of trees that are scheduledat time t in the schedule generated by our algorithm.Consider a treeTi ∈ Sopt(t)−Sappr. SinceTi is relevantto time slott and it is not scheduled by our algorithm,our algorithm must have tried to schedule the treeTiat time slott , but rejected it. Therefore, we concludethatw(Sopt(t)−Sappr)� (6α+ 10)w(Sappr(t)), whichimplies thatw(Sopt)� (6α + 11)w(Sappr).

Since at least one tree is scheduled at each iterationof the algorithm, the time complexity of the algorithmis obviously polynomial.

Theorem 3.1. There is a polynomial time algorithmfor scheduling multicast trees with an approximationratio of 6α+ 11, or, equivalently, 4d + 15.

4. Restricted problems

In this section, we discuss some restricted problemsincluding the unweighted version and the unicastversion of our problem.

4.1. The unweighted version

Applying our algorithm for unweighted trees, notrees will be purged, which implies thatw(I ′

appr) =w(Iappr) instead of Lemma 2.3. Instead, any tree willbe pruned if it intersects with any other tree that hasalready been selected. Moreover, the unweightedd-setpacking problem can be solved withind/2+ ε for anyε > 0. Since no more than two trees inIopt − Iappr canbe pruned by any tree inIappr, the number of prunedtrees,|A|, is less than or equal to 2|Iappr| (instead of


Lemma 2.4). Therefore,w(Iopt) � (3α′ + 3)w(Iappr),whereα′ = d/2 + ε. This impliesw(Sopt) � (3α′ +4)w(Sappr).

Lemma 4.1. The unweighted multicast schedulingproblem can be approximated within 3d/2+ 4+ ε forany ε > 0.

4.2. The unicast version

For unicast messages, the tree becomes a di-rected path. For directed paths, both weighted and un-weighted “independent set within a group” problemscan be solved optimally within polynomial time sinceit is identical to the 2-set packing problem [5,8].

We construct a bipartite graphH ′ similar toH inLemma 2.4. The node set ofH ′ consists ofIopt− I ′

appr(instead ofA) andI ′

appr. A treeTi ∈ Iopt − I ′appr must

have been either pruned or rejected within a group. IfTi was pruned by a setX of previously selected trees,we make the same edges betweenTi and those inX asdone in Lemma 2.4. Consider the case whereTi ∈Gjwas rejected within a group. LetY ⊆ Gj be the setof trees that are selected within the groupGj and letY ′ ⊆ Y be the set of trees that intersectTi . Obviously,w(Y ′) � w(Ti). Otherwise, we can construct a betterindependent set by replacingY ′ by Ti . Then we addedges connectingTi and each member ofY ′. An edge(Ti , Tk), whereTk ∈ Y ′, is associated with weight

w(Ti)w(Tk)

w(Y ′) �w(Tk).

Consider any treeTi in I ′appr. There can be at most

two trees inIopt − I ′appr that are connected to it by an

edge withinH ′; the one that intersects withTi at theincoming edge of the root ofTi and the other that inter-sects withTi at the outgoing edge of the root ofTi . The

weight of each edge incident toTi withinH ′ is at most2w(Ti). Therefore, we can conclude thatw(Iopt −I ′appr) � 4w(I ′

appr), which implies that the weightedunicast problem can be approximated within 11.

Lemma 4.2. The weighted unicast scheduling problemcan be approximated within 11.

For unweighted cases, it is that|Iopt − I ′appr| �

2|I ′appr| and I ′

appr = Iappr. Therefore, the unweightedproblem can be approximated within 4.

Lemma 4.3. The unweighted unicast scheduling prob-lem can be approximated within 4.

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scheduling time-constrained multicast messages in circuit-switched tree networks

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