bottleneck routing games on grids
DESCRIPTION
Bottleneck Routing Games on Grids. Costas Busch Rajgopal Kannan Alfred Samman Department of Computer Science Louisiana State University. Talk Outline. Introduction. Basic Game. Channel Game. Extensions. 2-d Grid: . nodes. Used in: Multiprocessor architectures - PowerPoint PPT PresentationTRANSCRIPT
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Bottleneck Routing Games on Grids
Costas BuschRajgopal KannanAlfred Samman
Department of Computer ScienceLouisiana State University
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Talk Outline
Introduction
Basic Game
Channel Game
Extensions
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2-d Grid:
Used in:• Multiprocessor architectures• Wireless mesh networks• can be extended to d-dimensions
n
n
nnnodes
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Each player corresponds to a pair of source-destination
EdgeCongestion
3)( 1 eC
2)( 2 eC
Bottleneck Congestion: 3)(max
eCCEe
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A player may selfishly choose an alternativepath with better congestion
ii CC 31
PlayerCongestion
i
3iC
1iC
Player Congestion: Maximum edge congestion along its path
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Routing is a collection of paths, one path for each player
Utility function for player :i
ii Cppc )(
p
congestionof selected path
Social cost for routing :
CpSC )(p
bottleneck congestion
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We are interested in Nash Equilibriumswhere every player is locally optimal
Metrics of equilibrium quality:
p
Price of Stability
)()(min *pSCpSC
p
Price of Anarchy
)()(max *pSCpSC
p
*p is optimal coordinated routing with smallest social cost
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Bends :number of dimension changes plus source and destination
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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: nO log
Channel games:
Optimal solution uses at most bends
Path segments are separated accordingto length range
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There is a (non-game) routing algorithmwith bends and approximation ratio
nO log nO log
Optimal solution uses arbitrary number of bends
Final price of anarchy: nO 3log
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Solution without channels: Split Gameschannels 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 AnarchyNP-hardness
Price of Anarchy DefinitionKoutsoupias, Papadimitriou [STACS’99]
Price of Anarchy for sum of congestion utilities [JACM’02]
1O
|| EO
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Talk Outline
Introduction
Basic Game
Channel Game
Extensions
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],,,,,[)( 21 Nk mmmmpM
number of players with congestion kCi
Stability is proven through a potential functiondefined over routing vectors:
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PlayerCongestion
3iC
1iC
In best response dynamics a player move improves lexicographically the routing vector
)()( pMpM ]0,...,0,0,3,1,0[]0,...,0,0,0,2,2[
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],,,,,,,[)( 11 Nkkk mmmmmpM
Before greedy move kCi
],,,,,,,[)( 11 Nkkk mmmmmpM
After greedy move
ii CkkC
)()( pMpM
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Existence of Nash Equilibriums
Greedy moves give lower order routings
Eventually a local minimum for every playeris reached which is a Nash Equilibrium
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minp
Price of Stability
Lowest order routing :
)()( *min pSCpSC
• Is a Nash Equilibrium
• Achieves optimal social cost
1)()(Stability of Price *
min pSCpSC
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Price of AnarchyOptimal solution Nash Equilibrium
1* C 2/nC
)(2/* nnCC
Price of anarchy: High!
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Talk Outline
Introduction
Basic Game
Channel Game
Extensions
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Row:
channelsnlog
Channel holds path segments of length in range:
jA]12,2[ 1 jj
0A1A2A3A
]1,1[]3,2[
]7,4[]15,8[
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1eC
2eC
different channels
same channel
Congestion occurs only with path segmentsin same channel
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Path of player
Consider an arbitrary Nash Equilibriump
i
iCmaximum congestionin path
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must have a special edge with congestion
Optimal path of player
In optimal routing :*p
i
iC1 iCC
)(111*)( ppcCCCppc iiii
**)( CpSC
Since otherwise:
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C
00
0
edges use that Players: Congestion of Edges :ECE
In Nash Equilibrium social cost is: CpSC )(
0 0
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C 1C1C
0 0
Special Edges in optimal paths of 0
First expansion
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C 1C1C
0 01 1
11
1
edges use that Players:1least at Congestion of Edges Special :
ECE
First expansion
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C 1C1C 2C 2C2C2C
0 01 1
Special Edges in optimal paths of 1
Second expansion
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C 1C1C 2C 2C2C
0 01 1
2C
2 2
22
2
edges use that Players:2least at Congestion of Edges Special :
ECE
Second expansion
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In a similar way we can define:
jj
j
E
jCE
edges use that Players:
least at Congestion of Edges Special :
,,,,
,,,,
3210
3210
EEEE
We obtain expansion sequences:
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jj
j
E
jCE
edges use that Players:
12 :rin far ly sufficient are edges and
r channel somein majority thearech whi
least at Congestion of Edges Special :
1-r
Redefine expansion:
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*1
||||
aCE jj
*1
||)(||
CaE
jCE jj
||
)(|| jj
EjC
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*1
||)(||
CaE
jCE jj
If then )log( * nCC
|||| 1 jj EkE
2|| nE Contradiction
constant k
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)log( * nCOC Therefore:
Price of anarchy:)log()log(* nOnO
CC
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Optimal solutionNash Equilibrium1* C)()( 2 nC
)()( 2* nCCPrice of anarchy:
Tightness of Price of Anarchy
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Talk Outline
Introduction
Basic Game
Channel Game
Extensions
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0A
1A
2A3A
Split game
0A
1A
2A3A
Price of anarchy: )log( 2 nO
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d-dimensional grid
Price of anarchy:
nd
O logChannel game
nd
O 22 log
Price of anarchy:Split game