Oblivious Routing in Wireless networks. Costas Busch

Transcription

1 Obliious Routing in Wireless networks Costas Busch Rensselaer Polytechnic Institute Joint work with: Malik Magdon-Ismail and Jing Xi 1

2 Outline of Presentation Introduction Network Model Obliious Algorithm Analysis Discussion 2

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

4 Edge congestion Node congestion C edge C node maximum number of paths that use any edge maximum number of paths that use any node 4

5 Stretch= Length of chosen path Length of shortest path stretch u shortest path chosen path 5

6 Obliious Routing Each packet path choice is independent of other packet path choices 6

7 Path choices: q, 1,q k Probability of choosing a path: Pr[ q i ] k Pr[ q i ] 1 i 1 q 3 q 2 q 1 q 4 q 4 q 5 7

8 Benefits of obliious routing: Distributed Needs no global coordination Appropriate for dynamic packet arrials 8

9 Related Work Valiant [SICOMP 82]: First obliious routing algorithms for permutations on butterfly and hypercube butterfly butterfly (reersed) 9

10 Maggs, Meyer auf der Heide, Voecking, Westermann [FOCS 97]: d-dimensional Grid: C O d C * logn edge edge Lower bound for obliious routing: C edge C * edge d logn 10

11 Racke [FOCS 02]: Arbitrary Graphs: C O C * log 3 n edge edge existential result Azar et al. [STOC03] Harrelson et al. [SPAA03] Bienkowski et al. [SPAA03] constructie 11

12 Approach: Hierarchical clustering 12

13 13

14 At the lowest leel eery node is a cluster 14

15 source destination 15

16 Pick random node 16

17 Pick random node 17

18 Pick random node 18

19 Pick random node 19

20 Pick random node 20

21 Pick random node 21

22 Pick random node 22

23 23

24 Problem: Big stretch Adjacent nodes may follow long paths 24

25 An Impossibility Result Stretch and congestion cannot be minimized simultaneously in arbitrary graphs 25

26 Example graph: Each path has length ( n ) n paths n nodes Length 1 n Source of packets Destination of all packets 26

27 Stretch = 1 Edge congestion = n n packets in one path 27

28 Stretch = n Edge congestion = 1 1 packet per path 28

29 Contribution Busch, Magdon-Ismail, Xi [SPAA 2005]: Obliious algorithm for special graphs embedded in the 2-dimensional plane Constant stretch stretch O (1) Small congestion C node O ( C * node logn ) C edge O ( C * edge logn ) degree 29

30 Embeddings in wide, closed-cured areas 30

31 Our algorithm is appropriate for arious wireless network topologies Transmission radius 31

32 Basic Idea source destination 32

33 Pick a random intermediate node 33

34 Construct path through intermediate node 34

35 Preious results for Grids: Busch, Magdon-Ismail, Xi [IPDPS 05] C edge O d C * edge logn 2 Stretch = O ( d ) For d=2, a similar result gien by C. Scheideler 35

36 Outline of Presentation Introduction Network Model Obliious Algorithm Analysis Discussion 36

37 Network G Surrounding area A 37

38 Perpendicular bisector x,y space point x x,y y A space point 38

39 ( x, y ) s x, y x,y space point x y A space s point 39

40 Area wideness: min ( x, y ) x, ya A 40

41 Coerage Radius : R maximum distance from a space point to the closest node A graph node R x space point 41

42 For all pair of nodes there exist, : dist G ( u, u, ) Euclidian distance: u u, A Shortest path length: dist G ( u, ) dist G ( u, ) u,

43 Consequences of dist G ( u, u, ) Max Euclidian distance between adjacent nodes u, 1 u edge (max transmission radius in wireless networks) 43

44 Consequences of dist G ( u, u, ) Min Euclidian Distance between any pair of nodes: u u, 1 2 O ( r ) nodes r 44

45 Good Network embeddings: Small,,R and large Suppose they are constants 45

46 Outline of Presentation Introduction Network Model Obliious Algorithm Analysis Discussion 46

47 Eery pair of nodes is assigned a default path u default path A z default path w Examples: Shortest paths Geographic routing paths (GPSR) 47

48 The algorithm s source t A destination 48

49 Perpendicular bisector s t A 49

50 Pick random space point y s t A y 50

51 Find closest node to point y s t A R y w 51

52 Connect intermediate node to source and destination w s t A default path w default path 52

53 Outline of Presentation Introduction Network Model Obliious Algorithm Analysis Discussion 53

54 Consider an arbitrary set of packets:,, 1 N Suppose the obliious algorithm gies paths: P p,, 1 p N 54

55 We will show: stretch O 1 C node O C * node logn optimal congestion 55

56 Theorem: stretch O 1 Proof: Consider an arbitrary path and show that: p P stretch ( p) O 1 56

57 57 s A t default path default path w y q 1 q 2 p ), ( ) ( ) ( ), ( ) ( ) ( 2 1 t s dist q length q length t s dist p length p stretch G G

58 length ( q1) length ( q2) stretch ( p) dist ( s, t ) G when default paths are shortest paths stretch ( p) dist G ( s, w dist ) dist G ( s, t ) G ( w, t ) we show this is constant 58

59 59 R t s R y s w s w s dist G,,, ), ( s A t w y Default path (shortest) w s w s dist G, ), ( R R t s w t dist G, ), ( Similarly:

60 dist G ( s, t ) s, t s t A dist G ( s, t s, t ) 60

61 61 t s R t s t s dist t w dist w s dist p stretch G G G,, 2 ), ( ), ( ), ( ) ( For constants:,,r 1 ) ( O p stretch End of Proof

62 Theorem: Expected case: C O C * logn denotes C node * C node Proof: Consider some arbitrary node and estimate congestion on 62

63 Deiation of default paths: maximum distance from geodesic deiation max q i deiation ( q i ) u deiation ( q 1 ) A deiation z ( q 2 ) geodesic w 63

64 Consider some path from s to t s t 64

65 the use of depends on the choice of space point y s one choice t R y w 65

66 w y R another choice s t 66

67 If you choose node w in the cone the respectie path may use s t deiation w 67

68 If you choose node w outside the cone the respectie path does not use s t deiation w 68

69 Segment of space points affecting s 1 R t deiation R y w R 69

70 Probability of using node : Pr[ ] 1 2 s 2 1 R t A deiation (Q ) R y w R 70

71 It can be shown that: constant k Pr[ ] R s, t deiation s, 71

72 s, s, t R deiation ( Q) for simplicity assume: s, s, t s,t deiation R s s, s,t t R deiation 72

73 73 t s s,, s deiation t s R k,, ] Pr[ 1 s deiation R k, ] Pr[ 1,R,deiation : constants s k, ] Pr[ 2

74 Diide area A into concentric circles r i 2 i A A 2 A 0 A 1 r 1 r0 A 3 r 2 r 3 74

75 Max Euclidian distance between any two nodes = n u 1 u 2 u 3 u i, u i 1 1 A Longest path has at most n nodes u n 1 u n 75

76 r i 2 i Maximum ring radius A 2 A 0 A 1 r 0 r 1 A n log r 2 r n log n 76

77 N i C i = number of packets that can affect = number of paths that use We will bound Ring A i r i 77

78 Pr[ ] k 2 s, k 2 r i 1 s s, t w A i 1 r i 1 A i r i 78

79 Expected congestion: E [ C i ] N i Pr[ ] k 2 r N i 1 i A i 1 r i 1 A i r i 79

80 80 1 ] [ i i i r N O C E 1 * i i r N C ) ( ] [ * C O C E i We hae proen we proe next i r i r

81 r s, s, t i 1 we showed earlier s s s, r i 1, t r i 1 t A i 1 r i 1 A i r i 81

82 Similarly, each packet that affects traerses distance at least r i 1 r i 1 r i 1 r i 1 r i 1 r i 1 r i 1 A i 1 r i 1 r i 1 r i 1 A i r i r i 1 82

83 dist G ( s, t s, t ) dist G ( s, t ) ri 1 s, t r i 1 r i 1 r i 1 r i 1 r i 1 Area r i 1 r i 1 X r i 1 A i 1 r i 1 r i 1 r i 1 A i r i r i 1 A i 1 r i 1 83

84 Total number of nodes used N i r i 1 r i 1 r i 1 r i 1 r i 1 Area r i 1 r i 1 X r i 1 A i 1 r i 1 r i 1 r i 1 A i r i r i 1 A i 1 r i 1 84

85 Aerage node utilization N #nodes in i r i 1 area X r i 1 r i 1 r i 1 r i 1 Area r i 1 r i 1 X r i 1 A i 1 r i 1 r i 1 r i 1 A i r i r i 1 A i 1 r i 1 85

86 #nodes in area = X 2 ( 1 ) O r i Area X A i 1 r i 1 86

87 ) ( i i i i i r N r O r N Aerage node utilization aerage node utilization * C 1 * i i r N C

88 88 1 ] [ i i i r N O C E 1 * i i r N C ) ( ] [ * C O C E i We hae proen: i r i r

89 89 Considering all the rings: ) log ( log ] [ ) ( * * log 0 n C O n C O C E C E n i i End of Proof

90 Recap We presented a simple obliious algorithm which has: Constant stretch Small congestion stretch O 1 C node O ( C * node logn ) C edge O ( C * edge logn ) when the parameters of the Euclidian embedding are constants 90

91 Outline of Presentation Introduction Network Model Obliious Algorithm Analysis Discussion 91

92 Holes 92

93 Arbitrary closed shapes there is no 93