Electric routing and concurrent flow cutting

We investigate an oblivious routing scheme, amenable to distributed computation and resilient to graph changes, based on electrical flow. Our main technical contribution is a new rounding method which we use to obtain a bound on the L1->L1 operator norm of the inverse graph Laplacian. We show how this norm reflects both latency and congestion of electric routing.Comment: 21 pages, 0 figures. To be published in Springer LNCS Book No. 5878, Proceedings of The 20th International Symposium on Algorithms and Computation (ISAAC'09

Метод ранжування об'єктів для пошуку за ключовими словами у децентралізованій мережi

Разработан метод ранжирования объектов для поиска по ключевым словам в децентрализованной сети. Представленный метод поддерживает 2 вида поиска: пин-поиск и поиск суперсета, которые могут быть использованы для построения поисковой инфраструктуры разного вида приложений, среди которых: система для поиска документов, обмен файлами и широкий спектр мультимедийных приложений.Search is a core part of any complete file and resource distribution system. Determining the location of unknown files by description – for example, based on keywords or meta-data resources – requires searching. In modern peer-to-peer systems end users cannot retrieve content unless it knows its unique name. In contrast, Web search services, such as Google, allow users to search for content (documents, images, videos, locations etc.). However, these services must actively and repeatedly index the Internet content by hyper-linking from one resource to another. Today, the exponential growth of Internet content makes it difficult to build and maintain a complete index of documents to support effective search. The breadth and rapid development of the Internet means that even the best search services will always be incomplete and inaccurate. The purpose of this work is to develop an efficient peer-to-peer search architecture and algorithm, that support object ranking and superset search. Proposed system involves hypercubes and spanning binomial trees to provide scalable and efficient multiple keyword mechanism, that supports features, that are highly desired for convenient and effective search infrastructure, while being fully decentralized and fault-tolerant. Dynamic Hash-Tables are used as an overlay network considering their popularity and a list of valuable characteristics. Proposed index scheme uses a clustered approach to group objects based on their keyword sets. This allows you to more evenly distribute workload across network nodes when performing general keyword searches. The described method scales well on a large scale network. In addition, low-level optimizations, such as route caching and search record information on adjacent nodes, can be used to further improve method scaling and reduce overall node load.Розроблено метод ранжування об’єктів для пошуку за ключовими словами у децентралізованій мережі. Представлений метод підтримує 2 види пошуку: пін-пошук та пошук суперсету, які можуть бути використанні для побудови пошукової інфраструктури різного виду застосунків, серед яких: система для пошуку документів, обміну файлами та широкий спектр мультимедійних застосунків

A simple, combinatorial algorithm for solving SDD systems in nearly-linear time

Original manuscript January 28, 2013In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.National Science Foundation (U.S.) (Award 0843915)National Science Foundation (U.S.) (Award 1111109)Alfred P. Sloan Foundation (Research Fellowship)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374

Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

We show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an $n \times n$ Eulerian directed Laplacian with $m$ nonzero entries, we show how to compute an $\epsilon$-approximate solution in time $O(m \log^{O(1)} (n) \log (1/\epsilon))$. Through reductions from [Cohen et al. FOCS'16] , this gives the first nearly-linear time algorithms for computing $\epsilon$-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing $\epsilon$-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time $O((m+n2^{O(\sqrt{\log n \log \log n})})\log^{O(1)}(n \epsilon^{-1}))$. To achieve our results, we provide a structural result that we believe is of independent interest. We show that Laplacians of all strongly connected directed graphs have sparse approximate LU-factorizations. That is, for every such directed Laplacian ${\mathbf{L}}$, there is a lower triangular matrix $\boldsymbol{\mathit{{\mathfrak{L}}}}$ and an upper triangular matrix $\boldsymbol{\mathit{{\mathfrak{U}}}}$, each with at most $\tilde{O}(n)$ nonzero entries, such that their product $\boldsymbol{\mathit{{\mathfrak{L}}}} \boldsymbol{\mathit{{\mathfrak{U}}}}$ spectrally approximates ${\mathbf{L}}$ in an appropriate norm. This claim can be viewed as an analogue of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that, once constructed, they yield nearly-linear time algorithms for solving directed Laplacian systems.Comment: Appeared in FOCS 201

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition -- a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus few extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in $O(mn)$ time into cycles of length at most $2\log n$, and at most $2n$ extra edges. We give an $m^{1+o(1)}$ time algorithm for constructing a short cycle decomposition, with cycles of length $n^{o(1)}$, and $n^{1+o(1)}$ extra edges. These decompositions enable us to make progress on several open questions: * We give an algorithm to find $(1\pm\epsilon)$-approximations to effective resistances of all edges in time $m^{1+o(1)}\epsilon^{-1.5}$, improving over the previous best of $\tilde{O}(\min\{m\epsilon^{-2},n^2 \epsilon^{-1}\})$. This gives an algorithm to approximate the determinant of a Laplacian up to $(1\pm\epsilon)$ in $m^{1 + o(1)} + n^{15/8+o(1)}\epsilon^{-7/4}$ time. * We show existence and efficient algorithms for constructing graphical spectral sketches -- a distribution over sparse graphs H such that for a fixed vector $x$, we have w.h.p. $x'L_Hx=(1\pm\epsilon)x'L_Gx$ and $x'L_H^+x=(1\pm\epsilon)x'L_G^+x$. This implies the existence of resistance-sparsifiers with about $n\epsilon^{-1}$ edges that preserve the effective resistances between every pair of vertices up to $(1\pm\epsilon).$ * By combining short cycle decompositions with known tools in graph sparsification, we show the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. Improved algorithms for constructing short cycle decompositions will lead to improvements for each of the above results.Comment: 80 page

Smaller steps for faster algorithms : a new approach to solving linear systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 81-85).In this thesis we study iterative algorithms with simple sublinear time update steps, and we show how a mix of of data structures, randomization, and results from numerical analysis allow us to achieve faster algorithms for solving linear systems in a variety of different regimes. First we present a simple combinatorial algorithm for solving symmetric diagonally dominant (SDD) systems of equations that improves upon the best previously known running time for solving such system in the standard unit-cost RAM model. Then we provide a general method for convex optimization that improves this simple algorithm's running time as special case. Our results include the following: -- We achieve the best known running time of ... for solving Symmetric Diagonally Dominant (SDD) system of equations in the standard unit-cost RAM model. -- We obtain a faster asymptotic running time than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. -- We achieve faster asymptotic convergence rates than the best known for Kaczmarz methods for solving overdetermined systems of equations, by accelerating an algorithm of Strohmer and Vershynin [55]. Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this thesis and we hope that they will open the door for further algorithmic improvements in the future. This work was done in collaboration with Jonathan Kelner, Yin Tat Lee, Lorenzo Orecchia, and Zeyuan Zhu, and is based on the content of [30] and [35].by Aaron Daniel Sidford.S.M

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