2,546 research outputs found
A recursive construction of t-wise uniform permutations
We present a recursive construction of a (2t + 1)-wise uniform set of
permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial
design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise
uniform set of permutations on n objects. Using the complete design in this
procedure gives a t-wise uniform set of permutations on n objects whose size is
at most t^2n, the first non-trivial construction of an infinite family of
t-wise uniform sets for t \geq 4. If a non-trivial design with suitable
parameters is found, it will imply a corresponding improvement in the
construction
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Higher Order Correlations in Quantum Chaotic Spectra
The statistical properties of the quantum chaotic spectra have been studied,
so far, only up to the second order correlation effects. The numerical as well
as the analytical evidence that random matrix theory can successfully model the
spectral fluctuatations of these systems is available only up to this order.
For a complete understanding of spectral properties it is highly desirable to
study the higher order spectral correlations. This will also inform us about
the limitations of random matrix theory in modelling the properties of quantum
chaotic systems. Our main purpose in this paper is to carry out this study by a
semiclassical calculation for the quantum maps; however results are also valid
for time-independent systems.Comment: Revtex, Four figures (Postscript files), Phys. Rev E (in press
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page
Low-Memory Algorithms for Online and W-Streaming Edge Coloring
For edge coloring, the online and the W-streaming models seem somewhat
orthogonal: the former needs edges to be assigned colors immediately after
insertion, typically without any space restrictions, while the latter limits
memory to sublinear in the input size but allows an edge's color to be
announced any time after its insertion. We aim for the best of both worlds by
designing small-space online algorithms for edge-coloring. We study the problem
under both (adversarial) edge arrivals and vertex arrivals. Our results
significantly improve upon the memory used by prior online algorithms while
achieving an -competitive ratio. In particular, for -node graphs with
maximum vertex-degree under edge arrivals, we obtain an online
-coloring in space. This is also the
first W-streaming edge-coloring algorithm for -coloring in sublinear
memory. All prior works either used linear memory or colors.
We also achieve a smooth color-space tradeoff: for any , we get an
-coloring in space,
improving upon the state of the art that used space for
the same number of colors. The improvements stem from extensive use of random
permutations that enable us to avoid previously used colors. Most of our
algorithms can be derandomized and extended to multigraphs, where edge coloring
is known to be considerably harder than for simple graphs.Comment: 32 pages, 1 figur
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