17,551 research outputs found
The Big Match with a clock and a bit of memory
The Big Match is a multi-stage two-player game. In each stage Player 1 hides one or two pebbles in his hand, and his opponent has to guess that number; Player 1 loses a point if Player 2 is correct, and otherwise he wins a point. As soon as Player 1 hides one pebble, the players cannot change their choices in any future stage.
Blackwell and Ferguson (1968) give an ε-optimal strategy for Player 1 that hides, in each stage, one pebble with a probability that depends on the entire past history. Any strategy that depends just on the clock or on a finite memory is worthless. The long-standing natural open problem has been whether every strategy that depends just on the clock and a finite memory is worthless. We prove that there is such a strategy that is ε-optimal. In fact, we show that just two states of memory are sufficient
Algorithms for 3D rigidity analysis and a first order percolation transition
A fast computer algorithm, the pebble game, has been used successfully to
study rigidity percolation on 2D elastic networks, as well as on a special
class of 3D networks, the bond-bending networks. Application of the pebble game
approach to general 3D networks has been hindered by the fact that the
underlying mathematical theory is, strictly speaking, invalid in this case. We
construct an approximate pebble game algorithm for general 3D networks, as well
as a slower but exact algorithm, the relaxation algorithm, that we use for
testing the new pebble game. Based on the results of these tests and additional
considerations, we argue that in the particular case of randomly diluted
central-force networks on BCC and FCC lattices, the pebble game is essentially
exact. Using the pebble game, we observe an extremely sharp jump in the largest
rigid cluster size in bond-diluted central-force networks in 3D, with the
percolating cluster appearing and taking up most of the network after a single
bond addition. This strongly suggests a first order rigidity percolation
transition, which is in contrast to the second order transitions found
previously for the 2D central-force and 3D bond-bending networks. While a first
order rigidity transition has been observed for Bethe lattices and networks
with ``chemical order'', this is the first time it has been seen for a regular
randomly diluted network. In the case of site dilution, the transition is also
first order for BCC, but results for FCC suggest a second order transition.
Even in bond-diluted lattices, while the transition appears massively first
order in the order parameter (the percolating cluster size), it is continuous
in the elastic moduli. This, and the apparent non-universality, make this phase
transition highly unusual.Comment: 28 pages, 19 figure
On the Relative Strength of Pebbling and Resolution
The last decade has seen a revival of interest in pebble games in the context
of proof complexity. Pebbling has proven a useful tool for studying
resolution-based proof systems when comparing the strength of different
subsystems, showing bounds on proof space, and establishing size-space
trade-offs. The typical approach has been to encode the pebble game played on a
graph as a CNF formula and then argue that proofs of this formula must inherit
(various aspects of) the pebbling properties of the underlying graph.
Unfortunately, the reductions used here are not tight. To simulate resolution
proofs by pebblings, the full strength of nondeterministic black-white pebbling
is needed, whereas resolution is only known to be able to simulate
deterministic black pebbling. To obtain strong results, one therefore needs to
find specific graph families which either have essentially the same properties
for black and black-white pebbling (not at all true in general) or which admit
simulations of black-white pebblings in resolution. This paper contributes to
both these approaches. First, we design a restricted form of black-white
pebbling that can be simulated in resolution and show that there are graph
families for which such restricted pebblings can be asymptotically better than
black pebblings. This proves that, perhaps somewhat unexpectedly, resolution
can strictly beat black-only pebbling, and in particular that the space lower
bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight.
Second, we present a versatile parametrized graph family with essentially the
same properties for black and black-white pebbling, which gives sharp
simultaneous trade-offs for black and black-white pebbling for various
parameter settings. Both of our contributions have been instrumental in
obtaining the time-space trade-off results for resolution-based proof systems
in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th
Annual IEEE Conference on Computational Complexity (CCC '10), June 201
On Characterizing the Data Movement Complexity of Computational DAGs for Parallel Execution
Technology trends are making the cost of data movement increasingly dominant,
both in terms of energy and time, over the cost of performing arithmetic
operations in computer systems. The fundamental ratio of aggregate data
movement bandwidth to the total computational power (also referred to the
machine balance parameter) in parallel computer systems is decreasing. It is
there- fore of considerable importance to characterize the inherent data
movement requirements of parallel algorithms, so that the minimal architectural
balance parameters required to support it on future systems can be well
understood. In this paper, we develop an extension of the well-known red-blue
pebble game to develop lower bounds on the data movement complexity for the
parallel execution of computational directed acyclic graphs (CDAGs) on parallel
systems. We model multi-node multi-core parallel systems, with the total
physical memory distributed across the nodes (that are connected through some
interconnection network) and in a multi-level shared cache hierarchy for
processors within a node. We also develop new techniques for lower bound
characterization of non-homogeneous CDAGs. We demonstrate the use of the
methodology by analyzing the CDAGs of several numerical algorithms, to develop
lower bounds on data movement for their parallel execution
On Characterizing the Data Access Complexity of Programs
Technology trends will cause data movement to account for the majority of
energy expenditure and execution time on emerging computers. Therefore,
computational complexity will no longer be a sufficient metric for comparing
algorithms, and a fundamental characterization of data access complexity will
be increasingly important. The problem of developing lower bounds for data
access complexity has been modeled using the formalism of Hong & Kung's
red/blue pebble game for computational directed acyclic graphs (CDAGs).
However, previously developed approaches to lower bounds analysis for the
red/blue pebble game are very limited in effectiveness when applied to CDAGs of
real programs, with computations comprised of multiple sub-computations with
differing DAG structure. We address this problem by developing an approach for
effectively composing lower bounds based on graph decomposition. We also
develop a static analysis algorithm to derive the asymptotic data-access lower
bounds of programs, as a function of the problem size and cache size
Spartan Daily February 20, 2012
Volume 138, Issue 12https://scholarworks.sjsu.edu/spartandaily/1011/thumbnail.jp
Spartan Daily February 20, 2012
Volume 138, Issue 12https://scholarworks.sjsu.edu/spartandaily/1011/thumbnail.jp
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