28 research outputs found
On Greedily Packing Anchored Rectangles
Consider a set P of points in the unit square U, one of them being the
origin. For each point p in P you may draw a rectangle in U with its lower-left
corner in p. What is the maximum area such rectangles can cover without
overlapping each other? Freedman [1969] posed this problem in 1969, asking
whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu
and T\'oth [2011] achieved the first constant coverage of 9.1%; since then, no
significant progress was made. While 9.1% might seem low, the authors could not
find any instance where their algorithm covers less than 50%, nourishing the
hope to eventually prove a 50% bound. While we indeed significantly raise the
algorithm's coverage to 39%, we extinguish the hope of reaching 50% by giving
points for which the coverage is below 43.3%. Our analysis studies the
algorithm's average and worst-case density of so-called tiles, which represent
the area where a given point can freely choose its maximum-area rectangle. Our
approachis comparatively general and may potentially help in analyzing related
algorithms
Simple and Efficient Leader Election
We provide a simple and efficient population protocol for leader election that uses O(log n) states and elects exactly one leader in O(n (log n)^2) interactions with high probability and in expectation. Our analysis is simple and based on fundamental stochastic arguments. Our protocol combines the tournament based leader elimination by Alistarh and Gelashvili, ICALP\u2715, with the synthetic coin introduced by Alistarh et al., SODA\u2717
Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem
We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of n identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires O(log n) states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within O(n log n) interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly(n) parallel time w.h.p.
We use our clock to design a loosely-stabilizing protocol that solves the adaptive variant of the majority problem. We assume that the agents have either opinion A or B or they are undecided and agents can change their opinion at a rate of 1/n. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least ?(log n) agents and a sufficiently large bias is present, then the protocol converges to a correct output within O(n log n) interactions and stays in a correct configuration for poly(n) interactions, w.h.p
Discrete Load Balancing in Heterogeneous Networks with a Focus on Second-Order Diffusion
In this paper we consider a wide class of discrete diffusion load balancing
algorithms. The problem is defined as follows. We are given an interconnection
network and a number of load items, which are arbitrarily distributed among the
nodes of the network. The goal is to redistribute the load in iterative
discrete steps such that at the end each node has (almost) the same number of
items. In diffusion load balancing nodes are only allowed to balance their load
with their direct neighbors.
We show three main results. Firstly, we present a general framework for
randomly rounding the flow generated by continuous diffusion schemes over the
edges of a graph in order to obtain corresponding discrete schemes. Compared to
the results of Rabani, Sinclair, and Wanka, FOCS'98, which are only valid
w.r.t. the class of homogeneous first order schemes, our framework can be used
to analyze a larger class of diffusion algorithms, such as algorithms for
heterogeneous networks and second order schemes. Secondly, we bound the
deviation between randomized second order schemes and their continuous
counterparts. Finally, we provide a bound for the minimum initial load in a
network that is sufficient to prevent the occurrence of negative load at a node
during the execution of second order diffusion schemes.
Our theoretical results are complemented with extensive simulations on
different graph classes. We show empirically that second order schemes, which
are usually much faster than first order schemes, will not balance the load
completely on a number of networks within reasonable time. However, the maximum
load difference at the end seems to be bounded by a constant value, which can
be further decreased if first order scheme is applied once this value is
achieved by second order scheme.Comment: Full version of paper submitted to ICDCS 201
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
A simple algorithm for computing positively weighted straight skeletons of monotone polygons
We study the characteristics of straight skeletons of monotone polygonal chains and use them to devise an algorithm for computing positively weighted straight skeletons of monotone polygons. Our algorithm runs in O(nlogn) time and O(n) space, where n denotes the number of vertices of the polygon