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 CCCG 2014 1

Steiner tree problem Given an arbitrary undirected graph with weights G (lengths) on edges terminals Steiner or optional vertices Steiner tree for terminals is a subgraph connect- T ing all of them minimizing ‘s length T 2

Steiner tree problem (STP) 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 T Current best approximation    • ln( 4 ) 1 . 39 3

Online Steiner Tree Problem (OSTP) Terminals appear one at a time online; • future terminals are not known is formed step by step • T 1 Lower bound of • log k 2 Current best approximation of • log k is the number of terminals k 4

Steiner tree Graph 1 1 1 3 1 1 1 Cost 3 5

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 6

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 7

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 8

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 9

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 10

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 11

Online Steiner tree Graph 1 3 1 1 Terminals arrive in some order 12

Online Steiner tree Graph 1 3 1  1   Cost = 6 13

Online Steiner tree Optimal Steiner tree 1 1 1  Cost 3     Cost = 6 2 OPT 14

New Problem: Bursty Arrival of Terminals Bursty Steiner tree problem (BSTP) Instance: Graph and a set of terminals appearing • G S m  online in a sequence of groups (bursts), m k Question: Find a Steiner tree for in with minimum • S G length Captures known problems  Classic STP m 1 m  Online STP k 15

Bursty Steiner tree Graph 1 Group 1 3 1 1 Terminals arrive in groups 16

Bursty Steiner tree Graph 1 Group 2 Group 1 3 1 1 Terminals arrive in groups 17

Bursty Steiner tree Graph 1 Group 2 Group 1 3 1 Group 3 1 Terminals arrive in groups 18

Bursty Steiner tree Graph 1 3 1 1 Cost = 5 19

Bursty Online Optimal Steiner tree Steiner tree Steiner tree Cost =   Cost = Cost = 5 3 6 20

Contributions Bursty STP: Tight approximation bound of    min{log k , m } is the number of terminals k is the number of bursts m    Online STP: log k 21

Terminal Steiner tree problem (TSTP) All the terminals are the leaves of the Steiner • tree T TSTP is also NP-Hard like STP with best • approximation 2.17 Bursty TSTP contributions: log k m • Lower bound of min{ , } 4 4  • Upper bound of min{ 2 log , 3 } k m   is the current best approximation of TSTP 2 . 17 22

BSTP in directed graphs • Networks may contain links that are asymmetric in QoS they offer w ( u , v )   • Asymmetry max  { , } u v G w ( v , u ) is the weight of the edge w ( u , v ) ( u , v )  • We prove near-tight bounds for bounded   log k log k      O min{max{ , }, k , m }      log log log k   log k log k m            1 min{max{ , }, k , max{ , }} , 0       log log log log k 23

BSTP in undirected graphs   The upper bound proof O min{log k , m }    The lower bound proof min{log k , m } 24

  The upper bound proof O min{log k , m } Algorithm for O(m): when a new burst arrives B i 1. Compute a Steiner tree for T B i i 2. Find a closest to existing tree of v  B T i previous bursts and connect to v T 25

Group 2 Group 3 Group 1 Group 4 Optimal Steiner tree within each group 26

  The upper bound proof O min{log k , m }   O log k is from Online STP • • For , if we maintain Steiner trees, m ( m ) O one for each burst , we obtain  2 B m i approximation, is approx. of STP   1 . 39 27

  The upper bound proof O min{log k , m } • The factor of is because to join 2 individual trees to obtain , we may T T i need a path that has length as most OPT Q.E.D. 28

   The lower bound proof min{log k , m } 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] 29

   The lower bound proof min{log k , m } Consider a sequence of bursts B  m { B ,..., B } 0 m for graphs ,  0 G i i There exists a path between and for all v p v 1 0 the terminals in with length exactly 1 in OPT B Minimum tree sequence: for T  { T ,..., T } 0 m ,  w.r.t. such that B  G i i 0 { B ,..., B } T i 0 m must contain as a subgraph and connects T  1 i all the terminals in B i 30

G 0 1 STP cost = 1 31

G 1 1 2 1 1 2 STP cost = 1 32

G 2 1 4 1 2 1 4 1 1 1 4 2 1 4 STP cost = 1 33

G 3 1 8 1 1 8 4 1 1 8 1 2 4 1 1 8 1 8 1 1 4 2 1 8 1 1 4 8 1 STP cost = 1 8 34

   The lower bound proof min{log k , m }  • When for each , BSTP becomes B 1 i i  OSTP, hence applies to BSTP (log k ) • Therefore, we consider the case where     and prove ( m ) log m k 2  • Let and contains   i 1 B { B ,..., B }, i 1 B   i i m 1 i terminals beside and , contains v B v i 2  1 1 i 0 terminals, and so on 35

   The lower bound proof min{log k , m } • Now for consider and construct B G i i  T { T ,..., T }   1 i i m  • The length of tree is by any C A T ( ) 1 T i i algorithm A 1   1  as and the length • C A T ( ) 1 B 2 B  i 1  i i 2 of the edges in are half than G G  i 1 i  We can achieve this by choosing level nodes i 1 • that are not in 36 T i

G T 0 0 Group 0 1 cost = 1 cost = 1 37

G T 1 1 1 Group 1 2 1 1 2 1 1  cost = cost = 1 2 38

T G 2 2 1 4 1 2 1 Group 2 4 1 1 1 4 2 1 4 1 1 cost = 1   cost = 1 2 2 39

G T 3 1 3 8 1 1 8 4 1 1 8 1 2 4 1 1 Group 3 8 1 8 1 1 4 2 1 8 1 1 4 8 1 1 1 1 8    cost = cost = 1 1 2 2 2 40

   The lower bound proof min{log k , m } • The length of the shortest path from 1 each node in to a node in is B  T  j  i  i 1 i j j 2 i  • Moreover, level nodes are in  j  j i 1 2 terminals in B  i j  1 j 1     • C ( T ) C ( T ) 1    i j i j 1 A A 2 2 • This is as there are exactly  m ( m ) groups in Q.E.D. B 41

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 log k m min{ , } 4 4 k  Upper bound min{ 2 log , 3 } m 42

Conclusions 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. 43