3,387 research outputs found
Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs
Pebble games are single-player games on DAGs involving placing and moving
pebbles on nodes of the graph according to a certain set of rules. The goal is
to pebble a set of target nodes using a minimum number of pebbles. In this
paper, we present a possibly simpler proof of the result in [CLNV15] and
strengthen the result to show that it is PSPACE-hard to determine the minimum
number of pebbles to an additive term for all , which improves upon the currently known additive constant hardness of
approximation [CLNV15] in the standard pebble game. We also introduce a family
of explicit, constant indegree graphs with nodes where there exists a graph
in the family such that using constant pebbles requires moves
to pebble in both the standard and black-white pebble games. This independently
answers an open question summarized in [Nor15] of whether a family of DAGs
exists that meets the upper bound of moves using constant pebbles
with a different construction than that presented in [AdRNV17].Comment: Preliminary version in WADS 201
General Connectivity Distribution Functions for Growing Networks with Preferential Attachment of Fractional Power
We study the general connectivity distribution functions for growing networks
with preferential attachment of fractional power, ,
using the Simon's method. We first show that the heart of the previously known
methods of the rate equations for the connectivity distribution functions is
nothing but the Simon's method for word problem. Secondly, we show that the
case of fractional the -transformation of the rate equation
provides a fractional differential equation of new type, which coincides with
that for PA with linear power, when . We show that to solve such a
fractional differential equation we need define a transidental function
that we call {\it upsilon function}. Most of all
previously known results are obtained consistently in the frame work of a
unified theory.Comment: 10 page
Border trees of complex networks
The comprehensive characterization of the structure of complex networks is
essential to understand the dynamical processes which guide their evolution.
The discovery of the scale-free distribution and the small world property of
real networks were fundamental to stimulate more realistic models and to
understand some dynamical processes such as network growth. However, properties
related to the network borders (nodes with degree equal to one), one of its
most fragile parts, remain little investigated and understood. The border nodes
may be involved in the evolution of structures such as geographical networks.
Here we analyze complex networks by looking for border trees, which are defined
as the subgraphs without cycles connected to the remainder of the network
(containing cycles) and terminating into border nodes. In addition to
describing an algorithm for identification of such tree subgraphs, we also
consider a series of their measurements, including their number of vertices,
number of leaves, and depth. We investigate the properties of border trees for
several theoretical models as well as real-world networks.Comment: 5 pages, 1 figure, 2 tables. A working manuscript, comments and
suggestions welcome
Projeto de um gerador de atraso digital de cinco canais ajustĂĄvel via microcontrolador.
Entrada padronizada: VILLAS-BOAS, P. R
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