Optimal Oblivious Routing in Hole-Free Networks Costas Busch - PowerPoint PPT Presentation

Optimal Oblivious Routing in Hole-Free Networks Costas Busch Louisiana State University Malik Magdon-Ismail Rensselaer Polytechnic Institute 1

Routing: choose paths from sources to destinations v 3 u 1 u v 2 2 u 3 v 1 2

Edge congestion Node congestion C C edge node maximum number of maximum number of paths that use any node paths that use any edge 3

Length of chosen path Stretch= Length of shortest path 12   stretch 1 . 5 shortest path 8 u v chosen path 4

Oblivious Routing Each packet path choice is independent of other packet path choices 5

Path choices: 1  q , , q k Probability of choosing a path: Pr[ i q ] k   Pr[ ] 1 q i  i 1 q 1 q 2 q 3 q 4 q 4 q 5 6

Benefits of oblivious routing: • Distributed • Needs no global coordination • Appropriate for dynamic packet arrivals 7

Hole-free network 8

Our contribution in this work: Oblivious routing in hole-free networks Constant stretch Small congestion  stretch O ( 1 ) node  * C O ( C log n ) node 9

Holes 10

Related Work Valiant [SICOMP’82]: First oblivious routing algorithms for permutations on butterfly and hypercube butterfly butterfly (reversed) 11

Maggs, Meyer auf der Heide, Voecking, Westermann [FOCS’97]:     d-dimensional Grid: * C O d C log n edge edge Lower bound   * log C n     edge C for oblivious routing:   edge d   12

Arbitrary Graphs (existential result):   Racke [FOCS’02]:  * 3 C O C log n edge edge    Racke [STOC’08]: * C O C log n edge edge Constructive Results: Azar et al. [STOC03] Harrelson et al. [SPAA03] Bienkowski et al. [SPAA03] 13

General Approach: Hierarchical clustering 14

General Approach: Hierarchical clustering 15

At the lowest level every node is a cluster 16

source destination 17

Pick random node 18

Pick random node 19

Pick random node 20

Pick random node 21

Pick random node 22

Pick random node 23

Pick random node 24

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Problem: Big stretch Adjacent nodes may follow long paths 26

An Impossibility Result Stretch and congestion cannot be minimized simultaneously in arbitrary graphs 27

Example graph: ( n ) Each path has length  n paths n nodes Length 1 Destination Source of of all packets n packets 28

1 Stretch = n Edge congestion = packets in one path n 29

n Stretch = Edge congestion = 1 1 packet per path 30

Result for Grids: Busch, Magdon- Ismail, Xi [TC’08]     * C O d C log n edge edge  2 stretch ( ) O d For d=2, a similar result given by C. Scheideler 31

Special graphs embedded in the 2-dimensional plane: Busch, Magdon-Ismail, Xi [SPAA 2005]: Constant stretch Small congestion  stretch O ( 1 ) node  * C O ( C log n ) node    * C O ( C log n ) edge edge degree 32

Embeddings in wide, closed-curved areas 33

Graph models appropriate for various wireless network topologies Transmission radius 34

Basic Idea source destination 35

Pick a random intermediate node 36

Construct path through intermediate node 37

However, algorithm does not extend to arbitrary closed shapes 38

Our contribution in this work: Oblivious routing in hole-free networks 39

Approach: route within square areas node   * C O ( C log n ) stretch O ( 1 ) node 40

grid n  n 41

simple area in grid (hole-free area) 42

Hole-free network 43

Canonical square decomposition 44

Canonical square decomposition 45

Canonical square decomposition 46

Canonical square decomposition 47

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Shortest path v u 50

Canonical square sequence v u 51

A random path in canonical squares v u 52

Path has constant stretch v u 53

Random 2-bend paths or 1-bend paths in square sequence 54