32 research outputs found
Pseudo-random graphs and bit probe schemes with one-sided error
We study probabilistic bit-probe schemes for the membership problem. Given a
set A of at most n elements from the universe of size m we organize such a
structure that queries of type "Is x in A?" can be answered very quickly.
H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe
scheme based on expanders. Their scheme needs space of bits, and
requires to read only one randomly chosen bit from the memory to answer a
query. The answer is correct with high probability with two-sided errors. In
this paper we show that for the same problem there exists a bit-probe scheme
with one-sided error that needs space of O(n\log^2 m+\poly(\log m)) bits. The
difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh
is that we consider a bit-probe scheme with an auxiliary word. This means that
in our scheme the memory is split into two parts of different size: the main
storage of bits and a short word of bits that is
pre-computed once for the stored set A and `cached'. To answer a query "Is x in
A?" we allow to read the whole cached word and only one bit from the main
storage. For some reasonable values of parameters our space bound is better
than what can be achieved by any scheme without cached data.Comment: 19 page
Upper Tail Estimates with Combinatorial Proofs
We study generalisations of a simple, combinatorial proof of a Chernoff bound
similar to the one by Impagliazzo and Kabanets (RANDOM, 2010).
In particular, we prove a randomized version of the hitting property of
expander random walks and apply it to obtain a concentration bound for expander
random walks which is essentially optimal for small deviations and a large
number of steps. At the same time, we present a simpler proof that still yields
a "right" bound settling a question asked by Impagliazzo and Kabanets.
Next, we obtain a simple upper tail bound for polynomials with input
variables in which are not necessarily independent, but obey a certain
condition inspired by Impagliazzo and Kabanets. The resulting bound is used by
Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number
of calls in a black-box construction of a pseudorandom generator from a one-way
function.
We then show that the same technique yields the upper tail bound for the
number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph,
matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math,
2002).Comment: Full version of the paper from STACS 201
New Explicit Constant-Degree Lossless Expanders
We present a new explicit construction of onesided bipartite lossless
expanders of constant degree, with arbitrary constant ratio between the sizes
of the two vertex sets. Our construction is simpler to state and analyze than
the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002),
and achieves improvements in some parameters.
We construct our lossless expanders by imposing the structure of a
constant-sized lossless expander "gadget" within the neighborhoods of a large
bipartite spectral expander; similar constructions were previously used to
obtain the weaker notion of unique-neighbor expansion. Our analysis simply
consists of elementary counting arguments and an application of the expander
mixing lemma.Comment: Edits to expositio
Slow Emergence of Cooperation for Win-Stay Lose-Shift on Trees
We consider a group of agents on a graph who repeatedly play the prisoner’s dilemma game against their neighbors. The players adapt their actions to the past behavior of their opponents by applying the win-stay lose-shift strategy. On a finite connected graph, it is easy to see that the system learns to cooperate by converging to the all-cooperate state in a finite time. We analyze the rate of convergence in terms of the size and structure of the graph. Dyer et al. (2002) showed that the system converges rapidly on the cycle, but that it takes a time exponential in the size of the graph to converge to cooperation on the complete graph. We show that the emergence of cooperation is exponentially slow in some expander graphs. More surprisingly, we show that it is also exponentially slow in bounded-degree trees, where many other dynamics are known to converge rapidly