Dynamics of spectral algorithms for distributed routing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-117).In the past few decades distributed systems have evolved from man-made machines to organically changing social, economic and protein networks. This transition has been overwhelming in many ways at once. Dynamic, heterogeneous, irregular topologies have taken the place of static, homogeneous, regular ones. Asynchronous, ad hoc peer-to-peer networks have replaced carefully engineered super-computers, governed by globally synchronized clocks. Modern network scales have demanded distributed data structures in place of traditionally centralized ones. While the core problems of routing remain mostly unchanged, the sweeping changes of the computing environment invoke an altogether new science of algorithmic and analytic techniques. It is these techniques that are the focus of the present work. We address the re-design of routing algorithms in three classical domains: multi-commodity routing, broadcast routing and all-pairs route representation. Beyond their practical value, our results make pleasing contributions to Mathematics and Theoretical Computer Science. We exploit surprising connections to NP-hard approximation, and we introduce new techniques in metric embeddings and spectral graph theory. The distributed computability of "oblivious routes", a core combinatorial property of every graph and a key ingredient in route engineering, opens interesting questions in the natural and experimental sciences as well. Oblivious routes are "universal" communication pathways in networks which are essentially unique. They are magically robust as their quality degrades smoothly and gracefully with changes in topology or blemishes in the computational processes. While we have only recently learned how to find them algorithmically, their power begs the question whether naturally occurring networks from Biology to Sociology to Economics have their own mechanisms of finding and utilizing these pathways. Our discoveries constitute a significant progress towards the design of a self-organizing Internet, whose infrastructure is fueled entirely by its participants on an equal citizen basis. This grand engineering challenge is believed to be a potential technological solution to a long line of pressing social and human rights issues in the digital age. Some prominent examples include non-censorship, fair bandwidth allocation, privacy and ownership of social data, the right to copy information, non-discrimination based on identity, and many others.by Petar Maymounkov.Ph.D

Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance

In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the widely studied GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In this paper, we give an algorithm that solves the information dissemination problem in at most $O(D+\text{polylog}{(n)})$ rounds in a network of diameter $D$, withno dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of $D$, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires $T$ rounds in the LOCAL model can be simulated in $O(T +\mathrm{polylog}(n))$ rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent

Rumor Spreading with No Dependence on Conductance

National Science Foundation (U.S.) (CCF-0843915

Greedy embeddings, trees, and euclidean vs. lobachevsky geometry

A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f: V → X such that for every source-sink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper we take a refined look at greedy embeddings, previously addressed in [1, 2], by examining their description complexity as a key parameter in conjunction with their dimensionality. We give arguments showing that the dimensionality lower-bounds for monotone maps do not extend to greedy embeddings. We prove a unified O(log n) lower-bound on the dimension of no-stretch greedy embeddings when the host metric is Euclidean or Lobachevsky geometry. The essence of the lower bound entails showing that low-dimensional spaces lack the topological capacity to realize the embeddings of certain graphs with “hard crossroads. ” This technique might be of independent interest. We develop new methods for building concise embeddings of trees (and some other graphs) in 3-dimensional Lobachevsky spaces using recursive applications of hyperbolic isometries guided by caterpillar-like decompositions. Our embeddings improve over prior work [1] by achieving O(κ(T) · log n) description complexity, where κ(T) is the caterpillar dimension. We further demonstrate concise O(log n)-dimensional greedy embeddings of trees into Euclidean space using techniques inspired by [3], thereby strengthening our belief and intuition that all O(log n) graphs can be embedded with no stretch in ℓ. ∗ PhD candidate.

On Radhakrishnan’s Proof of PCP Gap Amplification

This exposition gives a simplified version of Radhakrishnan’s construction of a PCP gap amplification transform [4], which is itself a simplified version of Dinur’s construction [3]. The preprocessing and alphabet reduction transforms are left unchanged, and are thus not addressed here. Briefly, we simplify the analysis by pretending that all vertices have opinions about the color of all other vertices. This helps avoid conditioning in the analysis of the non-truncated random walk, and makes our arguments conceptually crisper. The “truncation argument ” is reduced to a couple of lines as opposed to over two pages in [4]. We also include a proof, due to [6], to a somewhat counter-intuitive random walk lemma used in [4]. Our exposition contains various fixes of small “bugs ” as well as restated, clarified and detailed versions of the more important arguments in [4]