39 research outputs found
A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems
We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC \u2716) that can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM \u2711), the high-temperature expansion of the Ising model, and careful probabilistic arguments
The Canonical Amoebot Model: Algorithms and Concurrency Control
The amoebot model abstracts active programmable matter as a collection of
simple computational elements called amoebots that interact locally to
collectively achieve tasks of coordination and movement. Since its introduction
at SPAA 2014, a growing body of literature has adapted its assumptions for a
variety of problems; however, without a standardized hierarchy of assumptions,
precise systematic comparison of results under the amoebot model is difficult.
We propose the canonical amoebot model, an updated formalization that
distinguishes between core model features and families of assumption variants.
A key improvement addressed by the canonical amoebot model is concurrency. Much
of the existing literature implicitly assumes amoebot actions are isolated and
reliable, reducing analysis to the sequential setting where at most one amoebot
is active at a time. However, real programmable matter systems are concurrent.
The canonical amoebot model formalizes all amoebot communication as message
passing, leveraging adversarial activation models of concurrent executions.
Under this granular treatment of time, we take two complementary approaches to
concurrent algorithm design. We first establish a set of sufficient conditions
for algorithm correctness under any concurrent execution, embedding concurrency
control directly in algorithm design. We then present a concurrency control
framework that uses locks to convert amoebot algorithms that terminate in the
sequential setting and satisfy certain conventions into algorithms that exhibit
equivalent behavior in the concurrent setting. As a case study, we demonstrate
both approaches using a simple algorithm for hexagon formation. Together, the
canonical amoebot model and these complementary approaches to concurrent
algorithm design open new directions for distributed computing research on
programmable matter.Comment: 48 pages, 7 figures, 2 table
Convex Hull Formation for Programmable Matter
We envision programmable matter as a system of nano-scale agents (called
particles) with very limited computational capabilities that move and compute
collectively to achieve a desired goal. We use the geometric amoebot model as
our computational framework, which assumes particles move on the triangular
lattice. Motivated by the problem of sealing an object using minimal resources,
we show how a particle system can self-organize to form an object's convex
hull. We give a distributed, local algorithm for convex hull formation and
prove that it runs in asynchronous rounds, where is the
length of the object's boundary. Within the same asymptotic runtime, this
algorithm can be extended to also form the object's (weak) -hull,
which uses the same number of particles but minimizes the area enclosed by the
hull. Our algorithms are the first to compute convex hulls with distributed
entities that have strictly local sensing, constant-size memory, and no shared
sense of orientation or coordinates. Ours is also the first distributed
approach to computing restricted-orientation convex hulls. This approach
involves coordinating particles as distributed memory; thus, as a supporting
but independent result, we present and analyze an algorithm for organizing
particles with constant-size memory as distributed binary counters that
efficiently support increments, decrements, and zero-tests --- even as the
particles move
Energy-Constrained Programmable Matter Under Unfair Adversaries
Individual modules of programmable matter participate in their system's
collective behavior by expending energy to perform actions. However, not all
modules may have access to the external energy source powering the system,
necessitating a local and distributed strategy for supplying energy to modules.
In this work, we present a general energy distribution framework for the
canonical amoebot model of programmable matter that transforms energy-agnostic
algorithms into energy-constrained ones with equivalent behavior and an
-round runtime overhead -- even under an unfair adversary --
provided the original algorithms satisfy certain conventions. We then prove
that existing amoebot algorithms for leader election (ICDCN 2023) and shape
formation (Distributed Computing, 2023) are compatible with this framework and
show simulations of their energy-constrained counterparts, demonstrating how
other unfair algorithms can be generalized to the energy-constrained setting
with relatively little effort. Finally, we show that our energy distribution
framework can be composed with the concurrency control framework for amoebot
algorithms (Distributed Computing, 2023), allowing algorithm designers to focus
on the simpler energy-agnostic, sequential setting but gain the general
applicability of energy-constrained, asynchronous correctness.Comment: 31 pages, 4 figures, 1 table. Submitted to OPODIS 202
Improved Bi-criteria Approximation for the All-or-Nothing Multicommodity Flow Problem in Arbitrary Networks
This paper addresses the following fundamental maximum throughput routing
problem: Given an arbitrary edge-capacitated -node directed network and a
set of commodities, with source-destination pairs and demands
, admit and route the largest possible number of commodities -- i.e.,
the maximum {\em throughput} -- to satisfy their demands. The main
contributions of this paper are two-fold: First, we present a bi-criteria
approximation algorithm for this all-or-nothing multicommodity flow (ANF)
problem. Our algorithm is the first to achieve a {\em constant approximation of
the maximum throughput} with an {\em edge capacity violation ratio that is at
most logarithmic in }, with high probability. Our approach is based on a
version of randomized rounding that keeps splittable flows, rather than
approximating those via a non-splittable path for each commodity: This allows
our approach to work for {\em arbitrary directed edge-capacitated graphs},
unlike most of the prior work on the ANF problem. Our algorithm also works if
we consider the weighted throughput, where the benefit gained by fully
satisfying the demand for commodity is determined by a given weight
. Second, we present a derandomization of our algorithm that maintains
the same approximation bounds, using novel pessimistic estimators for
Bernstein's inequality. In addition, we show how our framework can be adapted
to achieve a polylogarithmic fraction of the maximum throughput while
maintaining a constant edge capacity violation, if the network capacity is
large enough. One important aspect of our randomized and derandomized
algorithms is their {\em simplicity}, which lends to efficient implementations
in practice
Approximation algorithms for the mobile piercing set problem with applications to clustering,
Abstract. The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining efficient constantfactor approximation algorithms for the mobile piercing set (MPS) problem on unit-disks for standard metrics in fixed dimension vector spaces. More specifically, we provide low constant approximations for L 1 and L ∞ norms on a d-dimensional space, for any fixed d > 0, and for the L 2 norm on two-and three-dimensional spaces. Our framework provides a family of fully-distributed and decentralized algorithms, which adapt (asymptotically) optimally to the mobility of disks, at the expense of a low degradation on the best known approximation factors of the respective centralized algorithms: Our algorithms take O(1) time to update the piercing set maintained, per movement of a disk. We also present a family of fully-distributed algorithms for the MPS problem which either match or improve the best known approximation bounds of centralized algorithms for the respective norms and space dimensions. Second, we show how the proposed algorithms can be directly applied to provide theoretical performance analyses for two popular 1-hop clustering algorithms in ad-hoc networks: the lowest-id algorithm and the Least Cluster Change (LCC) algorithm. More specifically, we formally prove that the LCC algorithm adapts in constant time to the mobility of the network nodes, and minimizes (up to low constant factors) the number of 1-hop clusters maintained. While there is a vast literature on simulation results for the LCC and the lowest-id algorithms, these had not been formally analyzed prior to this work. We also present an O(log n)-approximation algorithm for the mobile piercing set problem for nonuniform disks (i.e., disks that may have different radii), with constant update time