Partial Reversal Acyclicity

Partial Reversal (PR) is a link reversal algorithm which ensures that the underlying graph structure is destination-oriented and acyclic. These properties of PR make it useful in routing protocols and algorithms for solving leader election and mutual exclusion. While proofs exist to establish the acyclicity property of PR, they rely on assigning labels to either the nodes or the edges in the graph. In this work we present simpler direct proof of the acyclicity property of partial reversal without using any external or dynamic labeling mechanism. First, we provide a simple variant of the PR algorithm, and show that it maintains acyclicity. Next, we present a binary relation which maps the original PR algorithm to the new algorithm, and finally, we conclude that the acyclicity proof applies to the original PR algorithm as well

Properties of link reversal algorithms for routing and leader election

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 75-77).We present two link-reversal algorithms and some interesting properties that they satisfy. First, we describe the Partial Reversal (PR) algorithm [13], which ensures that the underlying graph structure is destination-oriented and acyclic. These properties of PR make it useful in routing protocols and algorithms for solving leader election and mutual exclusion. While proofs exist to establish the acyclicity property of PR, they rely on assigning labels to either the nodes or the edges in the graph. In this work we present simpler direct proof of the acyclicity property of partial reversal without using any external or dynamic labeling mechanisms. Second, we describe the leader election (LE) algorithm of [16], which guarantees that a unique leader is elected in an asynchronous network with a dynamically-changing communication topology. The algorithm ensures that, no matter what pattern of topology changes occurs, if topology changes cease, then eventually every connected component contains a unique leader and all nodes have directed paths to that leader. Our contribution includes a complexity analysis of the algorithm showing that after topology changes stop, no more than 0(n) elections occur in the system. We also provide a discussion on certain situations in which a new leader is elected (unnecessarily) when there is already another leader in the same connected component. Finally, we show how the LE algorithm can be augmented in such a way that nodes also have the shortest path to the leader.by Tsvetomira RadevaS.M

Distributed House-Hunting in Ant Colonies

We introduce the study of the ant colony house-hunting problem from a distributed computing perspective. When an ant colony's nest becomes unsuitable due to size constraints or damage, the colony must relocate to a new nest. The task of identifying and evaluating the quality of potential new nests is distributed among all ants. The ants must additionally reach consensus on a final nest choice and the full colony must be transported to this single new nest. Our goal is to use tools and techniques from distributed computing theory in order to gain insight into the house-hunting process. We develop a formal model for the house-hunting problem inspired by the behavior of the Temnothorax genus of ants. We then show a \Omega(log n) lower bound on the time for all n ants to agree on one of k candidate nests. We also present two algorithms that solve the house-hunting problem in our model. The first algorithm solves the problem in optimal O(log n) time but exhibits some features not characteristic of natural ant behavior. The second algorithm runs in O(k log n) time and uses an extremely simple and natural rule for each ant to decide on the new nest.Comment: To appear in PODC 201

Trade-offs between Selection Complexity and Performance when Searching the Plane without Communication

We consider the ANTS problem [Feinerman et al.] in which a group of agents collaboratively search for a target in a two-dimensional plane. Because this problem is inspired by the behavior of biological species, we argue that in addition to studying the {\em time complexity} of solutions it is also important to study the {\em selection complexity}, a measure of how likely a given algorithmic strategy is to arise in nature due to selective pressures. In more detail, we propose a new selection complexity metric $\chi$, defined for algorithm ${\cal A}$ such that $\chi({\cal A}) = b + \log \ell$, where $b$ is the number of memory bits used by each agent and $\ell$ bounds the fineness of available probabilities (agents use probabilities of at least $1/2^\ell$). In this paper, we study the trade-off between the standard performance metric of speed-up, which measures how the expected time to find the target improves with $n$, and our new selection metric. In particular, consider $n$ agents searching for a treasure located at (unknown) distance $D$ from the origin (where $n$ is sub-exponential in $D$). For this problem, we identify $\log \log D$ as a crucial threshold for our selection complexity metric. We first prove a new upper bound that achieves a near-optimal speed-up of $(D^2/n +D) \cdot 2^{O(\ell)}$ for $\chi({\cal A}) \leq 3 \log \log D + O(1)$. In particular, for $\ell \in O(1)$, the speed-up is asymptotically optimal. By comparison, the existing results for this problem [Feinerman et al.] that achieve similar speed-up require $\chi({\cal A}) = \Omega(\log D)$. We then show that this threshold is tight by describing a lower bound showing that if $\chi({\cal A}) < \log \log D - \omega(1)$, then with high probability the target is not found within $D^{2-o(1)}$ moves per agent. Hence, there is a sizable gap to the straightforward $\Omega(D^2/n + D)$ lower bound in this setting.Comment: appears in PODC 201

A leader election algorithm for dynamic networks with causal clocks

An algorithm for electing a leader in an asynchronous network with dynamically changing communication topology is presented. The algorithm ensures that, no matter what pattern of topology changes occurs, if topology changes cease, then eventually every connected component contains a unique leader. The algorithm combines ideas from the Temporally Ordered Routing Algorithm for mobile ad hoc networks (Park and Corson in Proceedings of the 16th IEEE Conference on Computer Communications (INFOCOM), pp. 1405–1413 (1997) with a wave algorithm (Tel in Introduction to distributed algorithms, 2nd edn. Cambridge University Press, Cambridge, MA, 2000), all within the framework of a height-based mechanism for reversing the logical direction of communication topology links (Gafni and Bertsekas in IEEE Trans Commun C–29(1), 11–18 1981). Moreover, a generic representation of time is used, which can be implemented using totally-ordered values that preserve the causality of events, such as logical clocks and perfect clocks. A correctness proof for the algorithm is provided, and it is ensured that in certain well-behaved situations, a new leader is not elected unnecessarily, that is, the algorithm satisfies a stability condition.National Science Foundation (U.S.) (0500265)Texas Higher Education Coordinating Board (ARP-00512-0007-2006)Texas Higher Education Coordinating Board (ARP 000512-0130-2007)National Science Foundation (U.S.) (IIS-0712911)National Science Foundation (U.S.) (CNS-0540631)National Science Foundation (U.S.) (Research Experience for Undergraduates (Program) (0649233)

A symbiotic perspective on distributed algorithms and social insects

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.Cataloged from PDF version of thesis.Includes bibliographical references (pages 211-219).Biological distributed algorithms are decentralized computer algorithms that solve problems related to real biological systems and provide insight into the behavior of actual biological species. The biological systems we consider are social insect colonies, and the problems we study include foraging for food (exploring the colony's surroundings), house hunting (reaching consensus on a new home for the colony), and task allocation (allocating workers to tasks in the colony). The goal is to combine the approaches used in understanding complex distributed and biological systems in order to develop (1) more formal and mathematical insights about the behavior of social insect colonies, and (2) new techniques to design simpler and more robust distributed algorithms. Our results introduce theoretical computer scientists to new metrics, new ways to think about models and lower bounds, and new types of robustness properties of algorithms. Moreover, we provide biologists with new tools and techniques to gain insight and generate hypotheses about real ant colony behavior.by Tsvetomira Radeva.Ph. D

Searching without communicating: tradeoffs between performance and selection complexity

© 2016, Springer-Verlag Berlin Heidelberg. We consider the ANTS problem (Feinerman et al.) in which a group of agents collaboratively search for a target in a two-dimensional plane. Because this problem is inspired by the behavior of biological species, we argue that in addition to studying the time complexity of solutions it is also important to study the selection complexity, a measure of how likely a given algorithmic strategy is to arise in nature due to selective pressures. Intuitively, the larger the χ value, the more complicated the algorithm, and therefore the less likely it is to arise in nature. In more detail, we propose a new selection complexity metric χ, defined for algorithm A such that χ(A) = b+ log ℓ, where b is the number of memory bits used by each agent and ℓ bounds the fineness of available probabilities (agents use probabilities of at least 1 / 2 ℓ). In this paper, we study the trade-off between the standard performance metric of speed-up, which measures how the expected time to find the target improves with n, and our new selection metric. Our goal is to determine the thresholds of algorithmic complexity needed to enable efficient search. In particular, consider n agents searching for a treasure located within some distance D from the origin (where n is sub-exponential in D). For this problem, we identify the threshold log log D to be crucial for our selection complexity metric. We first prove a new upper bound that achieves a near-optimal speed-up for χ(A) ≈ log log D+ O(1). In particular, for ℓ∈ O(1) , the speed-up is asymptotically optimal. By comparison, the existing results for this problem (Feinerman et al.) that achieve similar speed-up require χ(A) = Ω(log D). We then show that this threshold is tight by describing a lower bound showing that if χ(A) < log log D- ω(1) , then with high probability the target is not found in D2-o(1) moves per agent. Hence, there is a sizable gap with respect to the straightforward Ω(D2/ n+ D) lower bound in this setting