Bottleneck Routing Games on Grids. Costas Busch Rajgopal Kannan Alfred Samman Department of Computer Science Louisiana State University

Transcription

1 Bottleneck Routing Games on Grids ostas Busch Rajgopal Kannan Alfred Samman Department of omputer Science Louisiana State University 1

2 Talk Outline Introduction Basic Game hannel Game xtensions 2

3 2-d Grid: n n n nodes n Used in: Multiprocessor architectures Wireless mesh networks can be extended to d-dimensions 3

4 ach player corresponds to a pair of source-destination dge ongestion ( e 1 ) 3 ( e 2 ) 2 Bottleneck ongestion: max ( e) 3 e 4

5 A player may selfishly choose an alternative path with better congestion Player ongestion i 3 i i i i Player ongestion: Maximum edge congestion along its path 5

6 Routing p is a collection of paths one path for each player Utility function for player : i pc ( p) i i congestion of selected path Social cost for routing : p S ( p) bottleneck congestion 6

7 We are interested in Nash quilibriums where every player is locally optimal p Metrics of equilibrium quality: Price of Stability S( p) min p S( p * ) Price of Anarchy S( p) max p S( p * ) * p is optimal coordinated routing with smallest social cost

8 Bends : number of dimension changes plus source and destination 6 8

9 Basic congestion games on grids Price of Stability: O(1) Price of Anarchy: (n) even with constant bends O(1) 9

10 Better bounds with bends hannel games: Path segments are separated according to length range Price of anarchy: O log n Optimal solution uses at most bends 10

11 There is a (non-game) routing algorithm with Olog n approximation ratio bends and Olog n Optimal solution uses arbitrary number of bends Final price of anarchy: O log 3 n 11

12 Solution without channels: Split Games channels are implemented implicitly in space Similar poly-log price of anarchy bounds 12

13 Some related work: Price of Anarchy Definition Koutsoupias Papadimitriou [STAS 99] Price of Anarchy O 1 for sum of congestion utilities [JAM 02] Arbitrary Bottleneck games [INFOOM 06] [TS 09]: Price of Anarchy NP-hardness O 13

14 Talk Outline Introduction Basic Game hannel Game xtensions 14

15 Stability is proven through a potential function defined over routing vectors: M( p) [ m m m m 1 2 k N ] number of players with congestion i k 15

16 In best response dynamics a player move improves lexicographically the routing vector Player ongestion 3 i i 1 [ ] [ ] M ( p) M ( p) 16

17 17 ] [ ) ( 1 1 N k k k m m m m m p M Before greedy move i k ] [ ) ( 1 1 N k k k m m m m m p M After greedy move i i k k ) ( ) ( p M p M

18 xistence of Nash quilibriums Greedy moves give lower order routings ventually a local minimum for every player is reached which is a Nash quilibrium 18

19 Price of Stability p min Lowest order routing : Is a Nash quilibrium Achieves optimal social cost S( p min ) S( p * ) S( p ) Price of Stability min S( p * ) 1 19

20 Price of Anarchy Optimal solution Nash quilibrium * 1 n / 2 Price of anarchy: n / 2 ( n) * High! 20

21 Talk Outline Introduction Basic Game hannel Game xtensions 21

22 log n channels hannel A j of length in range: holds path segments [2 j 2 j1 1] Row: A 3 A 2 A A 1 0 [23] [11] [47] [815] 22

23 different channels e 1 same channel 2 e ongestion occurs only with path segments in same channel 23

24 onsider an arbitrary Nash quilibrium p Path of player i i maximum congestion in path 24

25 In optimal routing p * : S ( p * ) * Optimal path of player must have a special edge with congestion i i 1 i Since otherwise: pc i ( p *) pc ( p) i i i 25

26 In Nash quilibrium social cost is: S( p) : dges of ongestion : Players that use edges 0 26

27 First expansion Special dges in optimal paths of 0 27

28 First expansion :Special dges of ongestion at least 1 1 : Players that use edges 1 28

29 Second expansion Special dges in optimal paths of 1 29

30 Second expansion :Special dges of ongestion at least : Players that use edges

31 31 In a similar way we can define: j j j j Players that use edges : at least ongestion of dges :Special We obtain expansion sequences:

32 Redefine expansion: j :Special dges of ongestion at least j which are the majority in some channel r and edges are sufficiently far in r : 2 r-1 1 j : Players that use edges j 32

33 33 * 1 a j j * 1 ) ( a j j j ) ( j j j

34 ( j j 1 j) * a If ( * log n) then k j1 j constant k n 2 ontradiction 34

35 Therefore: O( * log n) Price of anarchy: O( log n) O( log n) * 35

36 Tightness of Price of Anarchy Nash quilibrium ( n 2 ) ( ) Optimal solution * 1 Price of anarchy: ( n 2 ) ( ) * 36

37 Talk Outline Introduction Basic Game hannel Game xtensions 37

38 Split game A 0 A 1 A 2 A 3 A 0 A 1 A 2 A 3 2 n Price of anarchy: O( log ) 38

39 d-dimensional grid hannel game Price of anarchy: O d log n Split game Price of anarchy: O d 2 log 2 n 39