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Bottleneck Routing Games on Grids
Costas Busch Rajgopal Kannan Alfred Samman
Department of Computer Science Louisiana State University
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Bottleneck Routing Games on Grids Costas Busch Rajgopal Kannan - - PowerPoint PPT Presentation
Bottleneck Routing Games on Grids Costas Busch Rajgopal Kannan - - PowerPoint PPT Presentation
Bottleneck Routing Games on Grids Costas Busch Rajgopal Kannan Alfred Samman Department of Computer Science Louisiana State University 1 Talk Outline Introduction Basic Game Channel Game Extensions 2 2-d Grid: n n n nodes n
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2-d Grid:
Used in:
- Multiprocessor architectures
- Wireless mesh networks
- can be extended to d-dimensions
n n
n n
nodes
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Each player corresponds to a pair of source-destination Edge Congestion
3 ) ( 1 e C 2 ) (
2
e C
Bottleneck Congestion:
3 ) ( max
e C C
E e
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5
A player may selfishly choose an alternative path with better congestion
i i
C C 3 1
Player Congestion
i 3
i
C 1
i
C
Player Congestion: Maximum edge congestion along its path
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Routing is a collection of paths,
- ne path for each player
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Utility function for player :
i
i i
C p pc ) ( p
congestion
- f selected path
Social cost for routing :
C p SC ) (
p
bottleneck congestion
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We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality:
p
Price of Stability
) ( ) ( min
*
p SC p SC
p
Price of Anarchy
) ( ) ( max
*
p SC p SC
p
*
p
is optimal coordinated routing with smallest social cost
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Bends : number of dimension changes plus source and destination
6
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Price of Stability: Price of Anarchy:
) 1 ( O ) (n
even with constant bends
) 1 ( O
Basic congestion games on grids
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Better bounds with bends Price of anarchy:
n O log
Channel games:
Optimal solution uses at most bends
Path segments are separated according to length range
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There is a (non-game) routing algorithm with bends and approximation ratio
n O log
n O log
Optimal solution uses arbitrary number of bends Final price of anarchy:
n O
3
log
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Solution without channels: Split Games channels are implemented implicitly in space Similar poly-log price of anarchy bounds
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Some related work: Arbitrary Bottleneck games [INFOCOM’06], [TCS’09]: Price of Anarchy NP-hardness Price of Anarchy Definition Koutsoupias, Papadimitriou [STACS’99] Price of Anarchy for sum of congestion utilities [JACM’02]
1 O
| | E O
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Talk Outline Introduction Basic Game Channel Game Extensions
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] , , , , , [ ) (
2 1 N k
m m m m p M
number of players with congestion
k Ci
Stability is proven through a potential function defined over routing vectors:
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Player Congestion
3
i
C 1
i
C
In best response dynamics a player move improves lexicographically the routing vector
) ( ) ( p M p M
] ,..., , , 3 , 1 , [ ] ,..., , , , 2 , 2 [
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] , , , , , , , [ ) (
1 1 N k k k
m m m m m p M
Before greedy move
k Ci
] , , , , , , , [ ) (
1 1 N k k k
m m m m m p M
After greedy move
i i
C k k C
) ( ) ( p M p M
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Existence of Nash Equilibriums
Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium
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min
p
Price of Stability
Lowest order routing : ) ( ) (
* min
p SC p SC
- Is a Nash Equilibrium
- Achieves optimal social cost
1 ) ( ) ( Stability
- f
Price
* min
p SC p SC
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Price of Anarchy
Optimal solution Nash Equilibrium 1
*
C 2 / n C ) ( 2 /
*
n n C C Price of anarchy: High!
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Talk Outline Introduction Basic Game Channel Game Extensions
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Row: channels n log Channel holds path segments
- f length in range:
j
A ] 1 2 , 2 [
1 j j
A
1
A
2
A
3
A
] 1 , 1 [
] 3 , 2 [
] 7 , 4 [
] 15 , 8 [
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1
e
C 2
e
C different channels same channel Congestion occurs only with path segments in same channel
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Path of player
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Consider an arbitrary Nash Equilibrium p
i
i
C
maximum congestion in path
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must have a special edge with congestion Optimal path of player
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In optimal routing :
*
p
i
i
C 1
i
C C
) ( 1 1 1 *) ( p pc C C C p pc
i i i i
* *)
( C p SC
Since otherwise:
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C edges use that Players : Congestion
- f
Edges : E C E
In Nash Equilibrium social cost is:
C p SC ) (
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C
1 C 1 C
Special Edges in optimal paths of
First expansion
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C
1 C 1 C
1
1
1 1 1
edges use that Players : 1 least at Congestion
- f
Edges Special : E C E
First expansion
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C
1 C 1 C 2 C 2 C 2 C 2 C
1
1
Special Edges in optimal paths of
1
Second expansion
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C
1 C 1 C 2 C 2 C 2 C
1
1
2 C
2
2
2 2 2
edges use that Players : 2 least at Congestion
- f
Edges Special : E C E
Second expansion
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In a similar way we can define:
j j j
E j C E edges use that Players : least at Congestion
- f
Edges Special :
, , , , , , , ,
3 2 1 3 2 1
E E E E
We obtain expansion sequences:
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j j j
E j C E edges use that Players : 1 2 : r in far ly sufficient are edges and r channel some in majority the are ch whi least at Congestion
- f
Edges Special :
1
- r
Redefine expansion:
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* 1
| | | | aC E
j j
* 1
| | ) ( | | C a E j C E
j j
| | ) ( | |
j j
E j C
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* 1
| | ) ( | | C a E j C E
j j
If then
) log (
*
n C C
| | | |
1 j j
E k E
2
| | n E
Contradiction constant k
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) log (
*
n C O C
Therefore:
Price of anarchy:
) log ( ) log (
*
n O n O C C
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Optimal solution Nash Equilibrium 1
*
C ) ( ) (
2
n C ) ( ) (
2 *
n C C Price of anarchy: Tightness of Price of Anarchy
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Talk Outline Introduction Basic Game Channel Game Extensions
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A
1
A
2
A
3
A
Split game
A
1
A
2
A
3
A
Price of anarchy:
) log (
2 n
O
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d-dimensional grid
Price of anarchy:
n d O log
Channel game
n d O
2 2 log