Showing papers by "David Eppstein published in 2011"
05 May 2011
TL;DR: In this paper, a fast algorithm for maximal cliques in large sparse graphs is proposed, which is based on the algorithm of Eppstein, Loffler, and Strash.
Abstract: We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.
216 citations
15 Aug 2011
TL;DR: A synopsis structure that allows two nodes to compute the elements belonging to the set difference in a single round with communication overhead proportional to the size of the difference times the logarithm of the keyspace is described.
Abstract: We describe a synopsis structure, the Difference Digest, that allows two nodes to compute the elements belonging to the set difference in a single round with communication overhead proportional to the size of the difference times the logarithm of the keyspace. While set reconciliation can be done efficiently using logs, logs require overhead for every update and scale poorly when multiple users are to be reconciled. By contrast, our abstraction assumes no prior context and is useful in networking and distributed systems applications such as trading blocks in a peer-to-peer network, and synchronizing link-state databases after a partition.Our basic set-reconciliation method has a similarity with the peeling algorithm used in Tornado codes [6], which is not surprising, as there is an intimate connection between set difference and coding. Beyond set reconciliation, an essential component in our Difference Digest is a new estimator for the size of the set difference that outperforms min-wise sketches [3] for small set differences.Our experiments show that the Difference Digest is more efficient than prior approaches such as Approximate Reconciliation Trees [5] and Characteristic Polynomial Interpolation [17]. We use Difference Digests to implement a generic KeyDiff service in Linux that runs over TCP and returns the sets of keys that differ between machines.
135 citations
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TL;DR: This work implements a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyzes its performance on a large corpus of real-world graphs to show that this algorithm is the first to offer a practical solution to listing allmaximal clique in large sparse graphs.
Abstract: We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.
82 citations
TL;DR: In this article, a deterministic solution to the straggler identification problem using only O(d log n) bits was proposed, based on a novel application of Newton's identities for symmetric polynomials.
Abstract: In this paper, we study the straggler identification problem, in which an algorithm must determine the identities of the remaining members of a set after it has had a large number of insertion and deletion operations performed on it, and now has relatively few remaining members. The goal is to do this in o(n) space, where n is the total number of identities. Straggler identification has applications, for example, in determining the unacknowledged packets in a high-bandwidth multicast data stream. We provide a deterministic solution to the straggler identification problem that uses only O(d log n) bits, based on a novel application of Newton's identities for symmetric polynomials. This solution can identify any subset of d stragglers from a set of n O(log n)-bit identifiers, assuming that there are no false deletions of identities not already in the set. Indeed, we give a lower bound argument that shows that any small-space deterministic solution to the straggler identification problem cannot be guaranteed to handle false deletions. Nevertheless, we provide a simple randomized solution, using O(d log n log (1/∈)) bits that can maintain a multiset and solve the straggler identification problem, tolerating false deletions, where ∈ > 0 is a user-defined parameter bounding the probability of an incorrect response. This randomized solution is based on a new type of Bloom filter, which we call the invertible Bloom filter.
76 citations
TL;DR: This work describes a method for performing greedy geometric routing for any n-vertex simple connected graph G in the hyperbolic plane, so that a message M between any pair of vertices may be routed by having each vertex that receives M pass it to a neighbor that is closer to M's destination.
Abstract: We describe a method for performing greedy geometric routing for any n-vertex simple connected graph G in the hyperbolic plane, so that a message M between any pair of vertices may be routed by having each vertex that receives M pass it to a neighbor that is closer to M's destination. Our algorithm produces succinct embeddings, where vertex positions are represented using O(\log n) bits and distance comparisons may be performed efficiently using these representations. These properties are useful, for example, for routing in sensor networks, where storage and bandwidth are limited.
62 citations
21 Sep 2011
TL;DR: It is shown that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.
Abstract: In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.
22 citations
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TL;DR: A privacy-enhanced protocol for set differencing that achieves absolute privacy for Bob (in the information theoretic sense), and a quantifiable degree of privacy protection for Alice is described.
Abstract: In this paper, we study methods for improving the efficiency and privacy of compressed DNA sequence comparison computations, under various querying scenarios. For instance, one scenario involves a querier, Bob, who wants to test if his DNA string, $Q$, is close to a DNA string, $Y$, owned by a data owner, Alice, but Bob does not want to reveal $Q$ to Alice and Alice is willing to reveal $Y$ to Bob \emph{only if} it is close to $Q$. We describe a privacy-enhanced method for comparing two compressed DNA sequences, which can be used to achieve the goals of such a scenario. Our method involves a reduction to set differencing, and we describe a privacy-enhanced protocol for set differencing that achieves absolute privacy for Bob (in the information theoretic sense), and a quantifiable degree of privacy protection for Alice. One of the important features of our protocols, which makes them ideally suited to privacy-enhanced DNA sequence comparison problems, is that the communication complexity of our solutions is proportional to a threshold that bounds the cardinality of the set differences that are of interest, rather than the cardinality of the sets involved (which correlates to the length of the DNA sequences). Moreover, in our protocols, the querier, Bob, can easily compute the set difference only if its cardinality is close to or below a specified threshold.
16 citations
15 Aug 2011
TL;DR: This work describes algorithms for transforming a rectangular layout that does not have this hierarchical structure, together with a clustering of the rectangles of the layout, into a spatial treemap that respects the clustering and also respects to the extent possible the adjacencies of the input layout.
Abstract: Rectangular layouts, subdivisions of an outer rectangle into smaller rectangles, have many applications in visualizing spatial information, for instance in rectangular cartograms in which the rectangles represent geographic or political regions. A spatial treemap is a rectangular layout with a hierarchical structure: the outer rectangle is subdivided into rectangles that are in turn subdivided into smaller rectangles. We describe algorithms for transforming a rectangular layout that does not have this hierarchical structure, together with a clustering of the rectangles of the layout, into a spatial treemap that respects the clustering and also respects to the extent possible the adjacencies of the input layout.
15 citations
TL;DR: In this article, the authors show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2).
Abstract: We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time solutions
14 citations
TL;DR: It is proved that it is NP-complete to decide whether fdim (G) equals the isometric dimension of $G, and it is shown that no algorithm to approximate fdim(G) has approximation ratio below $741/740$, unless P$=$NP.
Abstract: The Fibonacci dimension ${\rm fdim}(G)$ of a graph $G$ is introduced as the smallest integer $f$ such that $G$ admits an isometric embedding into $\Gamma_f$, the $f$-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view, we prove that it is NP-complete to decide whether ${\rm fdim}(G)$ equals the isometric dimension of $G$, and show that no algorithm to approximate ${\rm fdim}(G)$ has approximation ratio below $741/740$, unless P$=$NP. We also give a $(3/2)$-approximation algorithm for ${\rm fdim}(G)$ in the general case and a $(1+\varepsilon)$-approximation algorithm for simplex graphs.
12 citations
TL;DR: This work gives an example of a planar 3-tree that has no planar Lombardi drawing and shows that all outerpaths do have a planars Lombardi Drawing, and generalizes the notion of Lombardi drawings to that of (smooth) $k-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of circular arcs.
Abstract: In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.
TL;DR: In this article, a data structure, a rectangular complex, is proposed to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane.
Abstract: We describe a data structure, a rectangular complex, that can be used represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n × n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O ( n 2 ), and to add a single point to a planar tight span in time O ( n ). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O( n 2 ), and add a single point to the space and re-test whether it has such an embedding in time O ( n ).
TL;DR: The show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two, and how to construct a drawing with features in an O(n) grid in time from a series-parallel decomposition of the partial order.
Abstract: We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with $O(n^2)$ features, in an $O(n) \times O(n)$ grid in $O(n^2)$ time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with $O(n)$ features in an $O(n) \times O(n)$ grid in $O(n)$ time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use.
21 Sep 2011
TL;DR: This work shows that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two, and shows how to construct a drawing with O(n) features in an O(N) ×O( n) grid in O(m) time from a series-parallel decomposition of the partial order.
Abstract: We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n) ×O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) features in an O(n) ×O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use.
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TL;DR: This work studies the online problem of assigning a moving point to a base-station region that contains it in terms of a competitive analysis measured as a function of Δ, the ply of the system of regions, that is, the maximum number of regions that cover any single point.
Abstract: We study the online problem of assigning a moving point to a base-station region that contains it. For instance, the moving object could represent a cellular phone and the base station could represent the coverage zones of cell towers. Our goal is to minimize the number of handovers that occur when the point moves outside its assigned region and must be assigned to a new region. We study this problem in terms of competitive analysis and we measure the competitive ratio of our algorithms as a function of the ply of the system of regions, that is, the maximum number of regions that cover any single point. In the offline version of this problem, when object motions are known in advance, a simple greedy strategy suffices to determine an optimal assignment of objects to base stations, with as few handovers as possible. For the online version of this problem for moving points in one dimension, we present a deterministic algorithm that achieves a competitive ratio of O(log ply) with respect to the optimal algorithm, and we show that no better ratio is possible. For two or more dimensions, we present a randomized online algorithm that achieves a competitive ratio of O(log ply) with respect to the optimal algorithm, and a deterministic algorithm that achieves a competitive ratio of O(ply); again, we show that no better ratio is possible.
15 Aug 2011
TL;DR: In this article, the authors studied the online problem of assigning a moving point to a base station region that contains it, and the goal is to minimize the number of handovers that occur when the point moves outside its assigned region and must be assigned to a new one.
Abstract: We study the online problem of assigning a moving point to a base-station region that contains it. Our goal is to minimize the number of handovers that occur when the point moves outside its assigned region and must be assigned to a new one. We study this problem in terms of a competitive analysis measured as a function of Δ, the ply of the system of regions, that is, the maximum number of regions that cover any single point.
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23 Aug 2011TL;DR: In this article, it was shown that several problems of compacting orthogonal graph drawings to use the minimum number of rows or the minimum possible area cannot be approximated to within better than a polynomial factor.
Abstract: We show that several problems of compacting orthogonal graph drawings to use the minimum number of rows or the minimum possible area cannot be approximated to within better than a polynomial factor in polynomial time unless P=NP. However, there is a fixed-parameter-tractable algorithm for testing whether a drawing can be compacted to a given number of rows.
TL;DR: A new type of Voronoi diagram, in which distance is measured from a point to a pair of points, is revisited, and the structure and complexity of the nearest- and furthest-neighbor 2-site Voronoa diagrams of a point set in the plane with respect to these distance functions are analyzed.
Abstract: We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the nearest- and furthest-neighbor Voronoi diagrams of a point set with respect to these distance functions.
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TL;DR: In this article, the authors describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the nodes within each pass of the algorithm.
Abstract: We describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.
01 Dec 2011
TL;DR: In this paper, the authors introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network, and show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.
Abstract: A classic experiment by Milgram shows that individuals can route messages along short paths in social networks, given only simple categorical information about recipients (such as “he is a prominent lawyer in Boston” or “she is a Freshman sociology major at Harvard”). That is, these networks have very short paths between pairs of nodes (the so-called small-world phenomenon); moreover, participants are able to route messages along these paths even though each person is only aware of a small part of the network topology. Some sociologists conjecture that participants in such scenarios use a greedy routing strategy in which they forward messages to acquaintances that have more categories in common with the recipient than they do, and similar strategies have recently been proposed for routing messages in dynamic ad-hoc networks of mobile devices. In this paper, we introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network. We show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.
TL;DR: The set of all vertices is computed, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time is shown by solving a number of linear programs that isPolynomial in $$n$$n.
Abstract: We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs.
13 Jun 2011
TL;DR: The combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces is studied, with the property that the highest dimension of any bounded face is much smaller than D.
Abstract: We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(nd) and the total number of bounded faces of the polyhedron is O(nd 2). For inputs in general position the number of bounded faces is O(nd). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs.
28 Jun 2011
TL;DR: A new type of a Voronoi diagram, in which distance is measured from a point to a pair of points, is revisited, and the structure and complexity of the nearest- and furthest-neighbor Vor onoi diagrams of a point set with respect to these distance functions are analyzed.
Abstract: We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the nearest- and furthest-neighbor Voronoi diagrams of a point set with respect to these distance functions.
21 Sep 2011
TL;DR: It is shown that several problems of compacting orthogonal graph drawings to use the minimum number of rows or the minimum possible area cannot be approximated to within better than a polynomial factor in polynometric time unless P=NP.
Abstract: We show that several problems of compacting orthogonal graph drawings to use the minimum number of rows or the minimum possible area cannot be approximated to within better than a polynomial factor in polynomial time unless P=NP However, there is a fixed-parameter-tractable algorithm for testing whether a drawing can be compacted to a given number of rows
TL;DR: This work describes algorithms for transforming a rectangular layout that does not have this hierarchical structure, together with a clustering of the rectangles of the layout, into a spatial treemap that respects the clustering and also respects to the extent possible the adjacencies of the input layout.
Abstract: Rectangular layouts, subdivisions of an outer rectangle into smaller rectangles, have many applications in visualizing spatial information, for instance in rectangular cartograms in which the rectangles represent geographic or political regions. A spatial treemap is a rectangular layout with a hierarchical structure: the outer rectangle is subdivided into rectangles that are in turn subdivided into smaller rectangles. We describe algorithms for transforming a rectangular layout that does not have this hierarchical structure, together with a clustering of the rectangles of the layout, into a spatial treemap that respects the clustering and also respects to the extent possible the adjacencies of the input layout.
TL;DR: In this article, it was shown that several problems of compacting orthogonal graph drawings to use the minimum number of rows, area, length of longest edge or total edge length cannot be approximated better than within a polynomial factor of optimal in polynomially time unless P = NP.
Abstract: We show that several problems of compacting orthogonal graph drawings to use the minimum number of rows, area, length of longest edge or total edge length cannot be approximated better than within a polynomial factor of optimal in polynomial time unless P = NP. We also provide a fixed-parameter-tractable algorithm for testing whether a drawing can be compacted to a small number of rows.
01 Jan 2011
TL;DR: It is shown that n-vertex connected geometric graphs in R2 can have O (n/ log( c) n)(for constant c) edge crossings and yet the authors can still compute single-shortest paths and graph Voronoi diagrams on them in expected linear time.
Abstract: One of the main goals of theoretical computer science is to develop algorithms that solve a given problem optimally. However, the definition of what it means to optimally solve a problem can change if we make additional assumptions about the input. In particular, if the input to a given problem meets certain real-world properties, then it is often possible to develop algorithms that are more efficient than in the general case. We show that this is the case for three problems on geometric graphs and social networks: (1) We show that n-vertex connected geometric graphs in R2 can have O (n/ log( c) n)(for constant c) edge crossings and yet we can still compute single-shortest paths and graph Voronoi diagrams on them in expected linear time. (2) We show that any 3-connected planar graph can be embedded in R2 so that (a) routing can be accomplished by the simple rule of always forwarding a message to a neighbor closer to the destination and (b) these coordinates can be represented with O(log n) bits. (3) For graphs with low degeneracy (a measure of uniform sparsity), we describe an algorithm to list all maximal cliques whose theoretical running time matches the worst-case output size of the problem, and we show that our algorithm works well in practice.
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TL;DR: In this paper, the authors introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network, and show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.
Abstract: A classic experiment by Milgram shows that individuals can route messages along short paths in social networks, given only simple categorical information about recipients (such as "he is a prominent lawyer in Boston" or "she is a Freshman sociology major at Harvard"). That is, these networks have very short paths between pairs of nodes (the so-called small-world phenomenon); moreover, participants are able to route messages along these paths even though each person is only aware of a small part of the network topology. Some sociologists conjecture that participants in such scenarios use a greedy routing strategy in which they forward messages to acquaintances that have more categories in common with the recipient than they do, and similar strategies have recently been proposed for routing messages in dynamic ad-hoc networks of mobile devices. In this paper, we introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network. We show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.
TL;DR: Eppstein and Eades as discussed by the authors showed how to find a drawing of this type that maximizes the angular resolution of the drawing, the minimum angle between any two incident edges, in polynomial time, by reducing the problem to one of finding parametric shortest paths in an auxiliary graph.
Abstract: Let G be a graph that may be drawn in the plane in such a way that all internal faces are centrally symmetric convex polygons. We show how to find a drawing of this type that maximizes the angular resolution of the drawing, the minimum angle between any two incident edges, in polynomial time, by reducing the problem to one of finding parametric shortest paths in an auxiliary graph. The running time is at most O(t), where t is a parameter of the input graph that is at most O(n). Submitted: March 2009 Reviewed: June 2009 Revised: July 2010 Accepted: August 2011 Final: September 2011 Published: September 2011 Article type: Regular paper Communicated by: P. Eades E-mail addresses: eppstein@ics.uci.edu (David Eppstein) kwortman@fullerton.edu (Kevin A.