Compact routing schemes
03 Jul 2001-pp 1-10
TL;DR: Several compact routing schemes for general weighted undirected networks are described, which achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch.
Abstract: We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extremely short. The routing decision at each node takes constant time. Yet, the stretch of these routing schemes, i.e., the worst ratio between the cost of the path on which a packet is routed and the cost of the cheapest path from source to destination, is a small constant. Our schemes achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch. More specifically, we obtain: A routing scheme that uses only O (n 1/2) bits of memory at each node of an n-node network that has stretch 3. The space is optimal, up to logarithmic factors, in the sense that every routing scheme with stretch n2), and every routing scheme with stretch n3/2). The headers used are only (1 + O(1)) log2> n-bits long and each routing decision takes constant time. A variant of this scheme with [log2 n] -bit headers makes routing decisions in O(log log n) time. Also, for every integer k > 2, a general handshaking based routing scheme that uses O (n1/k) bits of memory at each node that has stretch 2k - 1. A conjecture of Erdos from 1963, settled for k = 3, 5, implies that the routing tables are of near-optimal size relative to the stretch. The handshaking is similar in spirit to a DNS lookup in TCP/IP. Headers are O(log2 n) bits long and each routing decision takes constant time. Without handshaking, the stretch of the scheme increases to 4k - 5. One ingredient used to obtain the routing schemes mentioned above, may be of independent practical and theoretical interest: A shortest path routing scheme for trees of arbitrary degree and diameter that assigns each vertex of an n-node tree a (1 + O(1)) log2 n-bit label. Given the label of a source node and the label of a destination it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination. The general scheme for k > 2 also uses a clustering technique introduced recently by the authors. The clusters obtained using this technique induce a sparse and low stretch tree cover of the network. This essentially reduces routing in general networks into routing problems in trees that could be solved using the above technique.
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AT&T1
TL;DR: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.), and is a "must-have" reference for every serious programmer's digital library.
Abstract: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.). One of the best-selling programming books published in the last fifty years, "K&R" has been called everything from the "bible" to "a landmark in computer science" and it has influenced generations of programmers. Available now for all leading ebook platforms, this concise and beautifully written text is a "must-have" reference for every serious programmers digital library.
As modestly described by the authors in the Preface to the First Edition, this "is not an introductory programming manual; it assumes some familiarity with basic programming concepts like variables, assignment statements, loops, and functions. Nonetheless, a novice programmer should be able to read along and pick up the language, although access to a more knowledgeable colleague will help."
2,120 citations
Book•
23 Aug 2010TL;DR: This chapter discusses random network models, which are based on the Erdos-Renyi models, and their application in the context of complex networks, where distances in scale-free networks are small and distances in complex networks are large.
Abstract: 1. Introduction Part I. Random Network Models: 2. The Erdos-Renyi models 3. Observations in real-world networks 4. Models for complex networks 5. Growing network models Part II. Structure and Robustness of Complex Networks: 6. Distances in scale-free networks - the ultra small world 7. Self-similarity in complex networks 8. Distances in geographically embedded networks 9. The network's structure - the generating function method 10. Percolation on complex networks 11. Structure of random directed networks - the bow tie 12. Introducing weights - bandwidth allocation and multimedia broadcasting Part III. Network Function - Dynamics and Applications: 13. Optimization of the network structure 14. Epidemiological models 15. Immunization 16. Thermodynamic models on networks 17. Spectral properties, transport, diffusion and dynamics 18. Searching in networks 19. Biological networks and network motifs Part IV. Appendices References Index.
891 citations
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name "oracle", and it provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1/k) expected time, constructing a data structure of size O(kn1p1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1/k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1/k) space had a query time of Ω(n1/k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
618 citations
06 Jul 2001
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name ``oracle', which provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G=(V,E) be an undirected weighted graph with |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/k}), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k-1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that ω(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name oracle. Previously, data structures that used only O(n^{1+1/k}) space had a query time of ω(n^{1/k}) and a slightly larger, non-optimal, stretch. Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.}
563 citations
TL;DR: It is shown that a planar digraph can be preprocessed in near-linear time, producing a near- linear space oracle that can answer reachability queries in constant time.
Abstract: It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only.The approach generalizes to give a near-linear space approximate distances oracle for a weighted planar digraph. With weights drawn from l0, …, Nr, it approximates distances within a factor (1 + e) in O(log log (nN) + 1/e) time. Our scheme can be extended to find and route along correspondingly short dipaths.
334 citations
References
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01 Jan 1991TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.
45,034 citations
"Compact routing schemes" refers methods in this paper
...[7] T. Cover and J. Thomas....
[...]
...It follows easily from the Kraft inequality (see Cover and Thomas [7]), that the encoding is almost optimal....
[...]
...It follows easily from the Kraft inequality (see Cover and Thomas [7]), that the encoding is almost optimal....
[...]
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
"Compact routing schemes" refers background in this paper
...[13]), meaning that some nodes have comparatively high degrees....
[...]
AT&T1
TL;DR: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.), and is a "must-have" reference for every serious programmer's digital library.
Abstract: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.). One of the best-selling programming books published in the last fifty years, "K&R" has been called everything from the "bible" to "a landmark in computer science" and it has influenced generations of programmers. Available now for all leading ebook platforms, this concise and beautifully written text is a "must-have" reference for every serious programmers digital library.
As modestly described by the authors in the Preface to the First Edition, this "is not an introductory programming manual; it assumes some familiarity with basic programming concepts like variables, assignment statements, loops, and functions. Nonetheless, a novice programmer should be able to read along and pick up the language, although access to a more knowledgeable colleague will help."
2,120 citations
"Compact routing schemes" refers methods in this paper
...When a message with header (v; Lv) arrives at w, the router can find the port number on which the packet should be forwarded using the following simple C expression (see [21]):...
[...]
01 Dec 1995
TL;DR: In this paper, the authors specify version 6 of the Internet Protocol (IPv6), also referred to as IP Next Generation or IPng, and propose a new protocol called IPng.
Abstract: This document specifies version 6 of the Internet Protocol (IPv6), also sometimes referred to as IP Next Generation or IPng.
2,112 citations
Book•
AT&T1
TL;DR: The C Programming Language (2nd Ed.) as discussed by the authors is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Languages (1st Ed.).
Abstract: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.). One of the best-selling programming books published in the last fifty years, "K&R" has been called everything from the "bible" to "a landmark in computer science" and it has influenced generations of programmers. Available now for all leading ebook platforms, this concise and beautifully written text is a "must-have" reference for every serious programmers digital library.
As modestly described by the authors in the Preface to the First Edition, this "is not an introductory programming manual; it assumes some familiarity with basic programming concepts like variables, assignment statements, loops, and functions. Nonetheless, a novice programmer should be able to read along and pick up the language, although access to a more knowledgeable colleague will help."
1,861 citations