2007/5/29 presenter : wei-ni chu, chuan-ju wang, wei-yang chen

Algorithms for the optimum communication spanning tree problem

Prabha SharmaAnn Oper Res (2006) 143:203-209

2007/5/29

Presenter : Wei-Ni Chu, Chuan-Ju Wang, Wei-Yang Chen

Outline

• Introduction• Notation• Algorithm OCST I• Example• Algorithm OCST II )(arc

ijr

Introduction

• Optimum Communication Spanning Tree Problem (OCSTP)– Given a graph .– A set of requirements , which represent the volume

of communication between the nodes and .– A set of distances for all .– The cost of communication for the tree T

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ijr

i j

ijd E)j,i(

Vi Vj

Tij)j,i(dr

2

1)T(C

Introduction (cont.)

• Johnson, Lenstra and Rinooy Kan(1978) have shown the OCSTP even with all is NP-hard.

• Hu(1974) considered the OCSTP on a complete graph.– Optimal Requirement Spanning Tree– Optimal Distance Spanning tree

• He proved that if the satisfy a generalized triangle inequality, the optimum tree will be a star tree.

1rij

ijd

Introduction (cont.)

• Concept of a near optimum tree– A tree is said to be near optimum if its communication

cost is better than the cost of all trees which differ from it in one arc only.

• Two algorithms for constructing near optimum trees are given.– Algorithm OCSTP I is a pseudo-polynomial algorithm.– Algorithm OCSTP II construct a near optimum tree in

. )mn(O 3

Notation

Algorithm OCST I

• Begin by keeping all equal to the minimum .• By Hu’s (1974) result, the optimal

communication spanning tree in this case is a cut tree and can be constructed in time.

• Increase the value of each tree arc one at a time and maintain a near optimal tree.

• When all arcs have attained their true values we stop.

ijd

)n(O 4

ijd

ijd

ijd

Algorithm OCST I (cont.)


• Algorithm OCST I is an algorithm– Step1

– Step3 Computation for each may be repeated at most times. For each computation for ,

has to be computed for each

and for each and tree with minimum cost has to be found.

Total computations for steps3 are , where

is equal to the largest of the values.

)Mn(O 4

)n(O 4

)j,i(arc

ijd )j,i(arc

))t,s()q,p(T(C )X,X()t,s(

ji )t,s(cycle)q,p(

)Mn(O 4

Mij

d


• We define

• We know that each iteration

• It means

.v)T(C)D(vDvDv)T(C

DD

ijijDijijijijijT)j,i(

ijijD

ijij

new

ij

ijnewij

.v)T(C))t,s()k,l(T(C

v/))T(C))t,s()k,l(T(C(

ijij

ijij

).T(C))t,s()k,l(T(C newijij DD


• Because and

,

• is near optimum with .

)T(C))t,s()k,l(T(C newijij DD

))t,s()q,p(T(Cmin))t,s()k,l(T(Cijij DD

)).t,s()q,p(T(C)T(C newij

newij DD

T new

ijD


• Problem– Is always ?– Why the computation for each may be

repeated at most times?– In each iteration, can be removed?

ij 0

)j,i(arc

ijd

)j,i(arc

Example

• Consider our problem on a complete graph – Three nodes A,B,C

– , ,

– , ,

– We use sum-requirement to compute the communication cost of the tree.

3K

1

23

34

2

A

BC

4ABd 2BCd 3ACd1)( Ar 2)( Br 3)( Cr

Example (cont.)

• Under this case, optimal solution can be easily found.

1

23

3

2

A

BC

Example (cont.)

• Apply OCST I• Step 1:

– Construct a tree with for all ijij dD min VVji ),(

1

23

22

2

A

BC

Example (cont.)

– Apply Hu’s Optimal Requirement Spanning Tree, and we will get an optimal tree T.

• • • •

• Step 2: – Assign

1

23

2

2

A

BC

3,2 ACAC dD

2,2 BCBC dD

30)( TC

4,2 ABAB dD

4 ABAB dD

Example (cont.)

• Step 3:– In the first iteration, – And compute cost for all possible adjacent trees

),(),( CAji

1

23

4

2

A

BC

1

23

4

A

BC

2

46)( 1 TC 50)( 2 TC

Example (cont.)

– Choose– – – ,go to step 3 – We found that no , such that .– We are done.

• Step4:– Output T as a near optimal tree.

),(),(1 klhgTT

7

16

7

3046/)}()),(),(({

ACAC vTCklhgTC

7

162 ACACAC DD

337

162 ACACAC DdD

Tji ),(

1

23

3

2

A

BC

ijij dD

Algorithm OCST II

• Generalized triangle inequality– Let be the distances associated with three

sides of any triangle formed by three nodes in the n-node network (n>4).

– Let .– If there exist a positive not larger than such

that for all triangles in the network, we say that the distances of the network satisfy the generalized triangle inequality.

candb,a

cba t 2n2

2n

ctba

Algorithm OCST II (cont.)

• Hu (1974) proved that if the distances satisfy the generalized triangle inequality and the

are all equal, the OCST is a star tree.• Begin by keeping all equal to the smallest

value.• One at a time is increased and critical values

such that for all in . • The process has to be repeated till all non-tree

arcs attain their true values.

ijd

s'rij

ijr

ijr

ijr

l

ik

k

ij}{ ij

r ],[ 1k

ij

k

ij


• Validity of OCST II


• Validity of OCST II– The critical value is chosen such that

– Also are adjacent trees. Thus, it follows that is a near optimum tree for

– By lemma, it follows that for the , if , then can be increased to its true value without disturbing the near optimality of

k

ij

kT 1 k ij

k ij ij ,λλR

kTp,q arcpqpq rR pqR

kT

1 and k-k T T


• Algorithm OCST II is an algorithm

– For each arc not in the star tree, there are at

most (n-1) critical values, since for each critical value

, the path is one arc less than the

number of arcs in– To compute all the critical values ,where h < n

– There are m-(n-1) non-tree arcs in the beginning, and

step 3 may have to be performed for all these arcs.– Total complexity of the algorithm is

)mn(O 3

ji,

k ijλ k

ij Tp

1kij Tp

h

k k ijλ

1)O(n 3

)mn(O 3

2007/5/29 presenter : wei-ni chu, chuan-ju wang, wei-yang chen

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