A Note on Online Steiner Tree Problems Gokarna Sharma and Costas - - PowerPoint PPT Presentation

A Note on Online Steiner Tree Problems Gokarna Sharma and Costas Busch

Division of Computer Science and Eng. Louisiana State University

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CCCG 2014

Steiner tree problem

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Steiner tree for terminals is a subgraph connect- ing all of them minimizing ‘s length terminals Steiner or optional vertices Given an arbitrary undirected graph with weights (lengths) on edges

G T T

Steiner tree problem (STP)

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Steiner trees have applications in

• VLSI design
• Computational biology
• Network and group communication

Classic Steiner Tree Problem (STP)

• All the terminals are known in advance
• NP-hard problem
• Approximation solutions are given comparing length of

with the length of the optimal tree OPT

• Current best approximation

39 . 1 ) 4 ln(  

T

Online Steiner Tree Problem (OSTP)

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• Terminals appear one at a time online;

future terminals are not known

• is formed step by step
• Lower bound of
• Current best approximation of

is the number of terminals

k log 2 1 k log k T

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Cost 3 Steiner tree

1 1 1 1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree Terminals arrive in some order

1 1 1 3

Graph

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Online Steiner tree

1 1 1 3

Graph Cost =

  6 

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Online Steiner tree Cost =

OPT    2 6  

Cost 3 Optimal Steiner tree

1 1 1

New Problem: Bursty Arrival of Terminals

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Bursty Steiner tree problem (BSTP)

• Instance: Graph and a set of terminals appearing
• nline in a sequence of groups (bursts),
• Question: Find a Steiner tree for in with minimum

length

Captures known problems

m S S G G k m  1  m

Classic STP

k m 

Online STP

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Bursty Steiner tree Terminals arrive in groups

1 1 1 3

Graph Group 1

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Bursty Steiner tree Terminals arrive in groups

1 1 1 3

Graph Group 2 Group 1

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Bursty Steiner tree Terminals arrive in groups

1 1 1 3

Graph Group 2 Group 1 Group 3

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Bursty Steiner tree

1 1 1 3

Graph Cost = 5

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Bursty Steiner tree Cost = 5 Cost = Optimal Steiner tree Online Steiner tree Cost =

  6 3

Contributions

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Bursty STP: Tight approximation bound of

 

} , min{log m k  k

is the number of terminals

m

is the number of bursts Online STP:

 

k log 

Terminal Steiner tree problem (TSTP)

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• All the terminals are the leaves of the Steiner

tree

• TSTP is also NP-Hard like STP with best

approximation 2.17

T

Bursty TSTP contributions:

• Lower bound of
• Upper bound of

} 4 , 4 log min{ m k } 3 , log 2 min{ m k 

17 . 2  

is the current best approximation of TSTP

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BSTP in directed graphs

        } , }, log log log , log log min{max{ m k k k k O    

• Networks may contain links that are

asymmetric in QoS they offer

• Asymmetry

is the weight of the edge

• We prove near-tight bounds for bounded

) , ( ) , ( max

} , {

u v w v u w

G v u 

 

) , ( v u w

) , ( v u

, }} log , max{ , }, log log log , log log min{max{

1

         



      



m k k k k



BSTP in undirected graphs

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The upper bound proof

 

} , min{log m k O

The lower bound proof

 

} , min{log m k 

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The upper bound proof

 

} , min{log m k O

Algorithm for O(m):

when a new burst arrives

• 1. Compute a Steiner tree for
• 2. Find a closest to existing tree of

previous bursts and connect to

i

B

i

T

i

B v 

i

B T v T

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Group 4 Group 2 Group 3 Group 1 Optimal Steiner tree within each group

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The upper bound proof

 

} , min{log m k O

• is from Online STP
• For , if we maintain Steiner trees,
• ne for each burst , we obtain

approximation, is approx. of STP

) (m O m

 

k O log

i

B m  2 39 . 1  

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The upper bound proof

 

} , min{log m k O

• The factor of is because to join

individual trees to obtain , we may need a path that has length as most OPT Q.E.D.

2

i

T T

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The lower bound proof

 

} , min{log m k 

Idea: create a sequence of terminal burst

instances based on a sequence of graphs and apply adversarial argument

Variation of OSTP lower bound [Imase and Waxman 91]

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The lower bound proof

There exists a path between and for all the terminals in with length exactly 1 in OPT

 

} , min{log m k 

Consider a sequence of bursts for graphs

m } ,..., {

m

B B B  ,  i Gi p v

1

v B

Minimum tree sequence: for w.r.t. such that must contain as a subgraph and connects all the terminals in

} ,..., {

m

T T T  ,  i Gi } ,..., {

m

B B B 

i

T

1  i

T

i

B

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G 1

STP cost = 1

32

1

G 1

2 1 2 1

STP cost = 1

33

2

G 1

2 1 2 1 4 1 4 1 4 1 4 1

STP cost = 1

34

3

G 1

2 1 2 1 4 1 4 1 4 1 4 1

8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1

STP cost = 1

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The lower bound proof

 

} , min{log m k 

• When for each , BSTP becomes

OSTP, hence applies to BSTP

• Therefore, we consider the case where

and prove

• Let and contains

terminals beside and , contains terminals, and so on

1 }, ,..., {

1

 

 

i B B B

m i i

1 

i

B i

) (logk 

 

k m log  ) (m 

i

B

1

2 

i

v

1

v

1  i

B

i

2

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The lower bound proof

 

} , min{log m k 

• Now for consider and construct
• The length of tree is by any

algorithm

• as and the length
• f the edges in are half than
• We can achieve this by choosing level

nodes that are not in

i

B

i

G } ,..., {

1  



m i i

T T T 1 ) ( 

i A T

C

i

T A 2 1 1 ) (

1

 

 i A T

C

i i

B B 2

1   1  i

G

i

G 1  i

i

T

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T G 1

cost = 1 Group 0 cost = 1

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1

T

1

G 1

2 1 2 1

cost = 1 Group 1 cost =

2 1 1

39

2

G 1

2 1 2 1 4 1 4 1 4 1 4 1

2

T

Group 2 cost = 1 cost =

2 1 2 1 1  

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3

G 1

2 1 2 1 4 1 4 1 4 1 4 1

8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1

cost = 1

3

T

Group 3 cost =

2 1 2 1 2 1 1   

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The lower bound proof

 

} , min{log m k 

• The length of the shortest path from

each node in to a node in is

• Moreover, level nodes are in

terminals in

• This is as there are exactly

groups in Q.E.D.

) (m  2 1 1 2 1 ) ( ) (

1

    

  

j T C T C

j i A j i A j i

B 

1

2

  j i j i

B 

1   j i

T

j i

2 1 j i  m B

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BTSTP in complete graphs

• Results follow the proof structure of

BSTP, but in complete graphs

• And graph sequence construction and

adversarial argument are more involved

• Bounds are tight with in a constant

factor Lower bound Upper bound

} 4 , 4 log min{ m k } 3 , log 2 min{ m k 

Conclusions

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Tight and near-tight results for online Steiner tree problem variations These variations provide trade-offs to existing solutions Open problems:

• Provide similar trade-offs for other

Steiner tree variations, e.g. node-weighted, group Steiner, etc.