Bayesian Generalized Network Design

We study network coordination problems, as captured by the setting of generalized network design (Emek et al., STOC 2018), in the face of uncertainty resulting from partial information that the network users hold regarding the actions of their peers. This uncertainty is formalized using Alon et al.\u27s Bayesian ignorance framework (TCS 2012). While the approach of Alon et al. is purely combinatorial, the current paper takes into account computational considerations: Our main technical contribution is the development of (strongly) polynomial time algorithms for local decision making in the face of Bayesian uncertainty

Estimating the matrix p→qp \rightarrow q norm

The matrix $p \rightarrow q$ norm is a fundamental quantity appearing in a variety of areas of mathematics. This quantity is known to be efficiently computable in only a few special cases. The best known algorithms for approximately computing this quantity with theoretical guarantees essentially consist of computing the $p\to q$ norm for $p,q$ where this quantity can be computed exactly or up to a constant, and applying interpolation. We analyze the matrix $2 \to q$ norm problem and provide an improved approximation algorithm via a simple argument involving the rows of a given matrix. For example, we improve the best-known $2\to 4$ norm approximation from $m^{1/8}$ to $m^{1/12}$. This insight for the $2\to q$ norm improves the best known $p \to q$ approximation algorithm for the region $p \le 2 \le q$, and leads to an overall improvement in the best-known approximation for $p \to q$ norms from $m^{25/128}$ to $m^{3 - 2 \sqrt{2}}$

Distributed algorithms for low stretch spanning trees

Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log3 n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D) rounds, where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e., Õ(min{√nD, √nD1/4 + n3/5 + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time

Electrical Flows for Polylogarithmic Competitive Oblivious Routing

Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well. In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with $n$ vertices and $m$ edges admit a routing scheme that has competitive ratio $O(\log^2 n)$ and consists of a convex combination of only $O(\sqrt{m})$ electrical routings. This immediately leads to an improved construction algorithm with time $\tilde{O}(m^{3/2})$ that can also be implemented in parallel with $\tilde{O}(\sqrt{m})$ depth.Comment: ITCS 202

Hop-Constrained Oblivious Routing

We prove the existence of an oblivious routing scheme that is $\mathrm{poly}(\log n)$-competitive in terms of $(congestion + dilation)$, thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize $(congestion + dilation)$, defined as follows: The dilation is the maximum path hop-length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have $(congestion + dilation)$ within a $\mathrm{poly}(\log n)$ factor of the best possible value. More precisely, for any integer hop-bound $h$, this oblivious routing scheme selects paths of length at most $h \cdot \mathrm{poly}(\log n)$ and is $\mathrm{poly}(\log n)$-competitive in terms of $congestion$ in comparison to the best possible $congestion$ achievable via paths of length at most $h$ hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of R\"{a}cke [FOCS 2002, STOC 2008], which are $O(\log n)$-competitive in terms of $congestion$, but are not competitive in terms of $dilation$

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