Dynamic Graph Stream Algorithms in o(n) Space
Dynamic Graph Stream Algorithms in o(n) Space
复制标题
o(n) 空间中的动态图流算法
DOI：
10.1007/s00453-018-0520-8
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发表时间：
2016
期刊：
影响因子：
1.1
通讯作者：
Pan Peng
中科院分区：
文献类型：
--
作者：
Zengfeng Huang;Pan Peng
在本文中，我们研究了动态流模型中的图形问题，其中输入是由一系列边缘插入和删除来定义的，因为许多自然问题需要ω（n）\ documentclass [12pt] {minimal} \ usepackage {amsmath} \ usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega （n）$$ \ end {document {document}空间，其中n是顶点的数量，现有作品主要集中于设计O（n·Polylogn）\ DocumentClass [12pt] {Minimal} \ usepackage {amsmath} \ use-package {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage { mathrm {poly} \ log n）$$ \ end {document}空间算法。尽管在密集图的边缘数量中，对于许多应用程序来说，它仍然可能太大（例如，n是巨大的或图形稀疏）。问题。 \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} graph and (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \ setLength {\ oddsidemargin} { - 69pt} \ begin {document} $$（1+ \ varepsilon）$$ \ end {document {document} - 适用于具有边界边缘权重的连接图的最小跨度树的重量，对于任何小常数ε> 0 \ documentclass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amssymb} {-69PT} \ BEGIN {document} $$ \ varepsilon > 0 $$ \ end {document}。 \ use-package {amsbsy} \ usepackage {Mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} \ begin {Document} Ahn等人（SODA 2012）给出的空间算法同一类图。我们在动态流模型中启动了近似图属性测试的研究，我们想区分满足属性的图形与ε\ documentclass [12pt] {minimal} \ usepackage {amsmath} Wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage} \ usepackage {upgreek} $$ \ end {document} -far具有属性。 （N1-ε·Polylogn）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ useym} \ usepackage {amsfonts} \ usepackage { k} \ setLength {\ oddSideMargin} { - 69pt} \ begin {document {document} $} $}（n^{1- \ varepsilon} \ cdot \ cdot \ cdot \ cdot \ mathrm {poly} \ log n）对于任何常数ε\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amsmsfonts} \ usepackage {amsymb} \ amssbage {amssbage {amssbage}升级} \ setLength {\ oddSidemargin} { - 69pt} \ begin {document} $$ \ varepsilon $$ \ end {document}。 AMSSYMB} \ USEPACKAGE {AMSBSY} \ USEPACKAGE {MATHRSFS} \ usePackage { $ \ end {document}这些问题的空间下限，这表明了这样的问题对ε\ documentclass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ useym} \ usepackage {amsfonts} \ usepackage {amssymb}长度{\ ODDSIDEMARGIN} { - 69pt} \ begin {document} $$ \ varepsilon $$ $ end \ end {document {document}是必要的。
In this paper we study graph problems in the dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (n)$$\end{document} space, where n is the number of vertices, existing works mainly focused on designing O(n·polylogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}(n\cdot \mathrm {poly}\log n)$$\end{document} space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g., n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error εn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon n$$\end{document} of a general graph and (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\varepsilon )$$\end{document}-approximating the weight of the minimum spanning tree of a connected graph with bounded edge weights, for any small constant ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}. The latter improves upon the previous O(n·polylogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\cdot \mathrm {poly}\log n)$$\end{document} space algorithm given by Ahn et al. (SODA 2012) for the same class of graphs. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly O(n1-ε·polylogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}(n^{1-\varepsilon }\cdot \mathrm {poly}\log n)$$\end{document} space, which is o(n) for any constant ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}. To complement our algorithms, we present Ω(n1-O(ε))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (n^{1-O(\varepsilon )})$$\end{document} space lower bounds for these problems, which show that such a dependence on ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is necessary.
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发表时间：
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