Compact Routing Schemes Mikkel Thorup Uri Zwick AT&T Labs – Research Tel Aviv University.
Uri Zwick Tel Aviv University
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Transcript of Uri Zwick Tel Aviv University
Uri ZwickTel Aviv University
Simple Stochastic GamesMean Payoff Games
Parity Games
CSR 2008Moscow, Russia
Mean Payoff Games
Simple Stochastic Games
Parity Games
Randomized subexponential
algorithm for SSG
Deterministic subexponential
algorithm for PG
Mean Payoff Games
Simple Stochastic Games
Parity Games
R
R
R
R
A simple Simple Stochastic Game
Simple Stochastic game (SSGs) Reachability version [Condon (1992)]
Objective: MAX/min the probability of getting to the MAX-sink
Two Players: MAX and min
MAX minRAND
R
MAX-sink
min-sink
Simple Stochastic games (SSGs)Strategies
A general strategy may be randomized and history dependent
A positional strategy is deterministicand history independent
Positional strategy for MAX: choice of an outgoing edge from each MAX vertex
Simple Stochastic games (SSGs)Values
Both players have positional optimal strategies
Every vertex i in the game has a value vi
positional general
positional general
There are strategies that are optimal for every starting position
Simple Stochastic game (SSGs) [Condon (1992)]
The outdegrees of all non-sinks are 2
Terminating binary games
Easy reduction from general gamesto terminating binary games
All probabilities are ½.
The game terminates with prob. 1
“Solving” terminating binary SSGs
The values vi of the vertices of a game are the unique solution of the following equations:
Corollary: Decision version in NP co-NP
The values are rational numbersrequiring only a linear number of bits
Value iteration (for binary SSGs)
Iterate the operator:
Converges to the unique solution
But, may require an exponentialnumber of iterations just to get close
Simple Stochastic game (SSGs) Payoff version [Shapley (1953)]
MAX minRAND
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Limiting average version
Discounted version
Markov Decision Processes (MDPs)
Values and optimal strategies of a MDP can be found by solving an LP
Theorem: [Epenoux (1964)]
MAX minRAND
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SSG NP co-NP – Another proof
Deciding whether the value of a game isat least (at most) v is in NP co-NP
To show that value v ,guess an optimal strategy for MAX
Find an optimal counter-strategy for min by solving the resulting MDP.
Is the problem in P ?
Mean Payoff Games (MPGs)[Ehrenfeucht, Mycielski (1979)]
Non-terminating version
Discounted version
MPGsPayoffSSGs
Pseudo-polynomial algorithm (PZ’96)
MAX minRAND
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ReachabilitySSGs
Mean Payoff Games (MPGs)[Ehrenfeucht, Mycielski (1979)]
Value(σ,) – average of cycle formed
Again, both players have optimal positional strategies.
Selecting the second largest element with only four storage locations [PZ’96]
Parity Games (PGs) A simple example
2
1 4 1
3 2
EVEN wins if largest priorityseen infinitely often is even
Priorities
Parity Games (PGs)
EVEN
3
ODD
8
EVEN wins if largest priorityseen infinitely often is even
Equivalent to many interesting problemsin automata and verification:
Non-emptyness of -tree automata
modal -calculus model checking
Parity Games (PGs)
EVEN
3
ODD
8
Replace priority k by payoff (n)k
Mean Payoff Games (MPGs)
Move payoffs to outgoing edges
[Stirling (1993)] [Puri (1995)]
Switches
…i
Value vector of strategy σ of MAX with respect to the optimal counter
strategy of min
Strategy/Policy Iteration
Start with some strategy σ (of MAX)
While there are improving switches, perform some of them
As each step is strictly improving and as there is a finite number of strategies, the algorithm
must end with an optimal strategy
SSG PLS (Polynomial Local Search)
Strategy/Policy IterationComplexity?
Performing only one switch at a time may lead to exponentially many improvements,even for MDPs [Condon (1992)]
What happens if we perform all profitable switches [Hoffman-Karp (1966)]
???
Not known to be polynomialBest upper bound: O(2n/n) [Mansour-Singh (1999)]
No non-linear examplesBest lower bounds: 2n-O(1) [Madani (2002)]
A randomized subexponential algorithm for simple stochastic games
Start with an arbitrary strategy for MAX
Choose a random vertex iVMAX
Find the optimal strategy ’ for MAX in the gamein which the only outgoing edge of i is (i,(i))
If switching ’ at i is not profitable, then ’ is optimal
Otherwise, let (’)i and repeat
A randomized subexponentialalgorithm for binary SSGs
[Ludwig (1995)][Kalai (1992)] [Matousek-Sharir-Welzl (1992)]
A randomized subexponentialalgorithm for binary SSGs
[Ludwig (1995)][Kalai (1992)] [Matousek-Sharir-Welzl (1992)]
There is a hidden order of MAX vertices under which the optimal strategy returned by
the first recursive call correctly fixes the strategy of MAX at vertices 1,2,…,i
All correct !Would never be switched !
MAX vertices
The hidden order
ui(σ) - the maximum sum of values of a strategy of MAX that agrees with σ on i
The hidden order
Order the vertices such that
Positions 1,..,iwere switched
and would neverbe switched again
SSGs are LP-type problems[Björklund-Sandberg-Vorobyov (2002)]
[Halman (2002)]
General (non-binary) SSGs can be solved in time
AUSO – Acyclic Unique Sink Orientations
Parity Games (PGs) A simple example
2
1 4 1
3 2
EVEN wins if largest priorityseen infinitely often is even
Priorities
Exponential algorithm for PGs[McNaughton (1993)] [Zielonka (1998)]
Vertices of highest priority
(even)
Vertices from whichEVEN can force the
game to enter A
Firstrecursive
call
Lemma: (i)
(ii)
Exponential algorithm for PGs[McNaughton (1993)] [Zielonka (1998)]
Second recursive
call
In the worst case, both recursive calls are on games of size n1
Deterministic subexponential alg for PGs Jurdzinski, Paterson, Z (2006)
Second recursive
call
Dominion
Idea: Look for small
dominions!
Dominion: A (small) set from which one of the players can win without the play ever leaving this set
Dominions of size s can be found
in O(ns) time
Open problems
● Polynomial algorithms?● Is the Policy Improvement algorithm
polynomial?● Faster subexponential algorithms
for parity games? ● Deterministic subexponential algorithms
for MPGs and SSGs?● Faster pseudo-polynomial algorithms
for MPGs?