65 research outputs found

    A Cut-Matching Game for Constant-Hop Expanders

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    This paper provides a cut-strategy that produces constant-hop expanders in the well-known cut-matching game framework. Constant-hop expanders strengthen expanders with constant conductance by guaranteeing that any demand can be (obliviously) routed along constant-hop paths - in contrast to the Ω(logn)\Omega(\log n)-hop routes in expanders. Cut-matching games for expanders are key tools for obtaining close-to-linear-time approximation algorithms for many hard problems, including finding (balanced or approximately-largest) sparse cuts, certifying the expansion of a graph by embedding an (explicit) expander, as well as computing expander decompositions, hierarchical cut decompositions, oblivious routings, multi-cuts, and multicommodity flows. The cut-matching game provided in this paper is crucial in extending this versatile and powerful machinery to constant-hop expanders. It is also a key ingredient towards close-to-linear time algorithms for computing a constant approximation of multicommodity-flows and multi-cuts - the approximation factor being a constant relies on the expanders being constant-hop

    Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player

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    A (ϕ,ϵ)(\phi,\epsilon)-expander-decomposition of a graph GG is a partition of VV into clusters V1,,VkV_1,\ldots,V_k with conductance Φ(G[Vi])ϕ\Phi(G[V_i]) \ge \phi, such that there are at most ϵm\epsilon m inter-cluster edges. We consider the problem of computing such a decomposition for a given GG and ϕ\phi, in near linear time, while minimizing ϵ\epsilon as a function of ϕ\phi. Saranurak and Wang [SW19] gave a randomized O(mlog4mϕ)O(\frac{m\log^4m}{\phi}) algorithm for computing a (ϕ,ϕlog3n)(\phi,\phi\log^3n)-expander decomposition. There are graphs that do not admit an expander decomposition with less than Ω(mϕlogn)\Omega(m\cdot\phi\log n) inter-cluster edges, so the number of inter-cluster edges in [SW19] is within two logarithmic factors of optimal. As a main building block, [SW19] use an adaptation of the algorithm of R\"{a}cke et al. [RST14] for computing an approximate balanced sparse cut. Both algorithms rely on the cut-matching game of Khandekar et al. [KRV09]. Orecchia et al. [OSVV08], using spectral analysis, improved upon [KRV09] by giving a fast algorithm that computes a sparse cut with better approximation guarantee. Using the technique of [OSVV08] for computing expander decompositions or balanced cuts [RST14, SW19], encounters many hurdles. In particular, in [RST14, SW19] the relevant part of the graph constantly changes, making it difficult to perform spectral analysis. In this paper, we manage to exploit the technique of [OSVV08] to compute an expander decomposition, improving the result by Saranurak and Wang [SW19]. Specifically, we give a randomized algorithm for computing a (ϕ,ϕlog2n)(\phi,\phi\log^2n)-expander decomposition of a graph, in O(mlog7m+mlog4mϕ)O(m\log^7m+\frac{m\log^4m}{\phi}) time. Our new result is achieved by using a novel combination of a symmetric version of the potential functions of [OSVV08, RST14, SW19] with a new variation of Cheeger's inequality for the notion of near-expansion.Comment: 43 pages; Revised the introduction, fixed typo

    Locality-Aware Qubit Routing for the Grid Architecture

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    Due to the short decohorence time of qubits available in the NISQ-era, it is essential to pack (minimize the size and or the depth of) a logical quantum circuit as efficiently as possible given a sparsely coupled physical architecture. In this work we introduce a locality-aware qubit routing algorithm based on a graph theoretic framework. Our algorithm is designed for the grid and certain \u27grid-like\u27 architectures. We experimentally show the competitiveness of algorithm by comparing it against the approximate token swapping algorithm, which is used as a primitive in many state-of-the-art quantum trans pilers. Our algorithm produces circuits of comparable depth (better on random permutations) while being an order of magnitude faster than a typical implementation of the approximate token swapping algorithm

    Dynamic Maxflow via Dynamic Interior Point Methods

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    In this paper we provide an algorithm for maintaining a (1ϵ)(1-\epsilon)-approximate maximum flow in a dynamic, capacitated graph undergoing edge additions. Over a sequence of mm-additions to an nn-node graph where every edge has capacity O(poly(m))O(\mathrm{poly}(m)) our algorithm runs in time O^(mnϵ1)\widehat{O}(m \sqrt{n} \cdot \epsilon^{-1}). To obtain this result we design dynamic data structures for the more general problem of detecting when the value of the minimum cost circulation in a dynamic graph undergoing edge additions obtains value at most FF (exactly) for a given threshold FF. Over a sequence mm-additions to an nn-node graph where every edge has capacity O(poly(m))O(\mathrm{poly}(m)) and cost O(poly(m))O(\mathrm{poly}(m)) we solve this thresholded minimum cost flow problem in O^(mn)\widehat{O}(m \sqrt{n}). Both of our algorithms succeed with high probability against an adaptive adversary. We obtain these results by dynamizing the recent interior point method used to obtain an almost linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for maintaining minimum ratio cycles in an undirected graph that succeeds with high probability against adaptive adversaries.Comment: 30 page

    Undergraduate Catalog, 2022-2023

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    https://scholar.valpo.edu/undergradcatalogs/1097/thumbnail.jp

    Undergraduate Catalog, 2021-2022

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    https://scholar.valpo.edu/undergradcatalogs/1096/thumbnail.jp

    Faster Parallel Algorithm for Approximate Shortest Path

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    We present the first mpolylog(n)m\,\text{polylog}(n) work, polylog(n)\text{polylog}(n) time algorithm in the PRAM model that computes (1+ϵ)(1+\epsilon)-approximate single-source shortest paths on weighted, undirected graphs. This improves upon the breakthrough result of Cohen~[JACM'00] that achieves O(m1+ϵ0)O(m^{1+\epsilon_0}) work and polylog(n)\text{polylog}(n) time. While most previous approaches, including Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent developments in \emph{continuous optimization}, studying the shortest path problem from the lens of the closely-related \emph{minimum transshipment} problem. To obtain our algorithm, we demonstrate a series of near-linear work, polylogarithmic-time reductions between the problems of approximate shortest path, approximate transshipment, and 1\ell_1-embeddings, and establish a recursive algorithm that cycles through the three problems and reduces the graph size on each cycle. As a consequence, we also obtain faster parallel algorithms for approximate transshipment and 1\ell_1-embeddings with polylogarithmic distortion. The minimum transshipment algorithm in particular improves upon the previous best m1+o(1)m^{1+o(1)} work sequential algorithm of Sherman~[SODA'17]. To improve readability, the paper is almost entirely self-contained, save for several staple theorems in algorithms and combinatorics.Comment: 53 pages, STOC 202

    Undergraduate Catalog, 2020-2021

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    https://scholar.valpo.edu/undergradcatalogs/1095/thumbnail.jp
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