A compact routing scheme and approximate distance oracle for power-law graphs
TL;DR: This work provides alabeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP/IP), routes messages along stretch--3 paths and obtains the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme.
Abstract: Compact routing addresses the tradeoff between table sizes and stretch, which is the worst-case ratio between the length of the path a packet is routed through by the scheme and the length of an actual shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick [2001] to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello et al. [2000]. Our result is the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme.Let n denote the number of nodes in the network. We provide a labeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP/IP), routes messages along stretch--3 paths. We prove that, instead of routing tables with O(n1/2) bits (O suppresses factors logarithmic in n) as in the general scheme by Thorup and Zwick, expected sizes of O(nγ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1+γ log n), with γ = τ-22/τ-3 + e, where τ∈(2,3) is the power-law exponent and e 0 (which implies e
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TL;DR: This work proposes a new exact method for shortest-path distance queries on large-scale networks that can handle social networks and web graphs with hundreds of millions of edges, which are two orders of magnitude larger than the limits of previous exact methods.
Abstract: We propose a new exact method for shortest-path distance queries on large-scale networks. Our method precomputes distance labels for vertices by performing a breadth-first search from every vertex. Seemingly too obvious and too inefficient at first glance, the key ingredient introduced here is pruning during breadth-first searches. While we can still answer the correct distance for any pair of vertices from the labels, it surprisingly reduces the search space and sizes of labels. Moreover, we show that we can perform 32 or 64 breadth-first searches simultaneously exploiting bitwise operations. We experimentally demonstrate that the combination of these two techniques is efficient and robust on various kinds of large-scale real-world networks. In particular, our method can handle social networks and web graphs with hundreds of millions of edges, which are two orders of magnitude larger than the limits of previous exact methods, with comparable query time to those of previous methods.
278 citations
Cites background from "A compact routing scheme and approx..."
...Therefore, by selecting central vertices as landmarks, the accuracy of estimates becomes much better than selecting landmarks randomly [29, 11]....
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22 Jun 2013
TL;DR: In this article, a new exact method for shortest-path distance queries on large-scale networks is proposed, where the key ingredient introduced here is pruning during breadth-first searches.
Abstract: We propose a new exact method for shortest-path distance queries on large-scale networks. Our method precomputes distance labels for vertices by performing a breadth-first search from every vertex. Seemingly too obvious and too inefficient at first glance, the key ingredient introduced here is pruning during breadth-first searches. While we can still answer the correct distance for any pair of vertices from the labels, it surprisingly reduces the search space and sizes of labels. Moreover, we show that we can perform 32 or 64 breadth-first searches simultaneously exploiting bitwise operations. We experimentally demonstrate that the combination of these two techniques is efficient and robust on various kinds of large-scale real-world networks. In particular, our method can handle social networks and web graphs with hundreds of millions of edges, which are two orders of magnitude larger than the limits of previous exact methods, with comparable query time to those of previous methods.
270 citations
TL;DR: This survey reviews selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time.
Abstract: We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.
249 citations
Cites background from "A compact routing scheme and approx..."
...I learned almost everything included in this survey from great collaborators while working on [Sommer et al. 2009; Honiden et al. 2010; Djidjev and Sommer 2011; Gavoille and Sommer 2011; Kawarabayashi et al. 2011; Akiba et al. 2012; Lim et al. 2012; Chen et al. 2012; Mozes and Sommer 2012; Kawarabayashi et al. 2013]....
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TL;DR: This survey studies and classifies shortest-path algorithms according to the proposed taxonomy and presents the challenges and proposed solutions associated with each category in the taxonomy.
Abstract: A shortest-path algorithm finds a path containing the minimal cost between two vertices in a graph. A plethora of shortest-path algorithms is studied in the literature that span across multiple disciplines. This paper presents a survey of shortest-path algorithms based on a taxonomy that is introduced in the paper. One dimension of this taxonomy is the various flavors of the shortest-path problem. There is no one general algorithm that is capable of solving all variants of the shortest-path problem due to the space and time complexities associated with each algorithm. Other important dimensions of the taxonomy include whether the shortest-path algorithm operates over a static or a dynamic graph, whether the shortest-path algorithm produces exact or approximate answers, and whether the objective of the shortest-path algorithm is to achieve time-dependence or is to only be goal directed. This survey studies and classifies shortest-path algorithms according to the proposed taxonomy. The survey also presents the challenges and proposed solutions associated with each category in the taxonomy.
78 citations
Cites methods from "A compact routing scheme and approx..."
...s [130]. In contrast, only a subset of vertices is covered using Kawarabayashi et al. approach. The approach is O(ǫ−1) times the number of paths in space complexity. For complex networks, Chen et al. [30] proposes a distance oracle over random power-law graphs [6] with 3 estimate that has a space complexity of O(n4/3). Their approach adopts the distance oracle proposed by Thorup and Zwick [132], where...
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01 Aug 2014
TL;DR: This work proposes to build an index for answering point-to-point distance querying for massive scale-free graphs based on a novel hop-doubling labeling technique, and derives bounds on the index size, the computation costs and I/O costs based on the properties of unweighted scale- free graphs.
Abstract: We study the problem of point-to-point distance querying for massive scale-free graphs, which is important for numerous applications. Given a directed or undirected graph, we propose to build an index for answering such queries based on a novel hop-doubling labeling technique. We derive bounds on the index size, the computation costs and I/O costs based on the properties of unweighted scale-free graphs. We show that our method is much more efficient and effective compared to the state-of-the-art techniques, in terms of both querying time and indexing costs. Our empirical study shows that our method can handle graphs that are orders of magnitude larger than existing methods.
67 citations
Cites background or methods from "A compact routing scheme and approx..."
...In [13], it is shown that high-degree vertices in power-law g raphs are useful for finding approximate shortest paths by a compac t routing scheme....
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...However, the query evaluation in [13] does not return exact answers....
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References
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TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
Abstract: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
33,771 citations
"A compact routing scheme and approx..." refers background or methods in this paper
...We provide the detailed parameters used to generate the graphs using BRITE [30], based on the Barabási [7] and Waxman [38] models....
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...Besides the random power-law graph model of Aiello, Chung, and Lu [6, 10, 11, 28], other mathematical models for power-law graphs include the configuration model [33], the Poissonian process [34], and the preferential attachment model [7, 26]....
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...…et al. [2000], Chung and Lu [2002, 2006], and Lu [2002b], other mathematical models for power-law graphs include the con.guration model [Newman et al. 2001], the Poissonian process [Norros and Reittu 2006], and the preferential attachment model [Barab´ asi and Albert 1999; Kumar et al. 2000]....
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...For each of the remaining graphs (Barab´ asi, Waxman, CAIDA), both schemes were constructed at least 10 times....
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...Multiplicative Stretch: Mean and Standard Deviation Graph CAIDA ASBarabasi RTBarabasi ASWaxman RTWaxman random 1.28 ±0.16 1.38 ±0.28 1.38 ±0.25 1.37 ±0.25 1.38 ±0.16 highdeg, ln. l 1.12 ±0.14 1.15 ±0.21 1.20 ±0.22 1.36 ±0.26 1.35 ±0.24 BC Graphs t = 2.1 t = 2.2 t = 2.3 t = 2.4 t = 2.5 random, p = n-1/2 1.340±0.240 1.350±0.243 1.347±0.251 1.342±0.259 1.335±0.261 highdeg, ln. l 1.300±0.239 1.264±0.230 1.226±0.227 1.211±0.226 1.183±0.221 BC Graphs t = 2.6 t = 2.7 t = 2.8 t = 2.9 random, p = n-1/2 1.330±0.275 1.306±0.281 1.290±0.286 1.247±0.284 highdeg, ln. l 1.160±0.218 1.151±0.222 1.147±0.237 1.111±0.216 Table III....
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to
22,704 citations
"A compact routing scheme and approx..." refers background in this paper
...Dijkstra’s algorithm [15] finds a shortest path in any graph with non-negative edge weights in time O(n log n + m), where n and m denote the number of nodes and edges respectively....
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TL;DR: This work proposes a principled statistical framework for discerning and quantifying power-law behavior in empirical data by combining maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios.
Abstract: Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.
8,753 citations
"A compact routing scheme and approx..." refers background in this paper
...Power-law graphs [31] constitute an important family of networks appearing in various realworld scenarios such as the Internet, the World Wide Web, collaboration networks, and social networks [12, 18]....
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...…2003] constitute an important family of networks appearing in various real-world scenarios such as social networks and many more are claimed to be power-law graphs [Clauset et al. 2009; Faloutsos et al. 1999], sometimes rather controversially [Achlioptas et al. 2009; Roughan et al. 2011]....
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5,331 citations
30 Aug 1999
TL;DR: These power-laws hold for three snapshots of the Internet, between November 1997 and December 1998, despite a 45% growth of its size during that period, and can be used to generate and select realistic topologies for simulation purposes.
Abstract: Despite the apparent randomness of the Internet, we discover some surprisingly simple power-laws of the Internet topology. These power-laws hold for three snapshots of the Internet, between November 1997 and December 1998, despite a 45% growth of its size during that period. We show that our power-laws fit the real data very well resulting in correlation coefficients of 96% or higher.Our observations provide a novel perspective of the structure of the Internet. The power-laws describe concisely skewed distributions of graph properties such as the node outdegree. In addition, these power-laws can be used to estimate important parameters such as the average neighborhood size, and facilitate the design and the performance analysis of protocols. Furthermore, we can use them to generate and select realistic topologies for simulation purposes.
5,023 citations