arXiv subjects

Michael T. Goodrich

Publications and source records attributed to Michael T. Goodrich.

At least 109 records · Page 6Linked to original sources

arXiv2010

Priority Range Trees

We describe a data structure, called a priority range tree, which accommodates fast orthogonal range reporting queries on prioritized points. Let $S$ be a set of $n$ points in the plane, where each point $p$ in $S$ is assigned a weight $w(p)$ that is polynomial in $n$, and define the rank of $p$ to be $r(p)=\lfloor \log w(p) \rfloor$. Then the priority range tree can be used to report all points in a three- or four-sided query range $R$ with rank at least $\lfloor \log w \rfloor$ in time $O(\log W/w + k)$, and report $k$ highest-rank points in $R$ in time $O(\log\log n + \log W/w' + k)$, where $W=\sum_{p\in S}{w(p)}$, $w'$ is the smallest weight of any point reported, and $k$ is the output size. All times assume the standard RAM model of computation. If the query range of interest is three sided, then the priority range tree occupies $O(n)$ space, otherwise $O(n\log n)$ space is used to answer four-sided queries. These queries are motivated by the Weber--Fechner Law, which states that humans perceive and interpret data on a logarithmic scale.

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arXiv2010

Privacy-Preserving Data-Oblivious Geometric Algorithms for Geographic Data

We give efficient data-oblivious algorithms for several fundamental geometric problems that are relevant to geographic information systems, including planar convex hulls and all-nearest neighbors. Our methods are "data-oblivious" in that they don't perform any data-dependent operations, with the exception of operations performed inside low-level blackbox circuits having a constant number of inputs and outputs. Thus, an adversary who observes the control flow of one of our algorithms, but who cannot see the inputs and outputs to the blackbox circuits, cannot learn anything about the input or output. This behavior makes our methods applicable to secure multiparty computation (SMC) protocols for geographic data used in location-based services. In SMC protocols, multiple parties wish to perform a computation on their combined data without revealing individual data to the other parties. For instance, our methods can be used to solve a problem posed by Du and Atallah, where Alice has a set, A, of m private points in the plane, Bob has another set, B, of n private points in the plane, and Alice and Bob want to jointly compute the convex hull of A u B without disclosing any more information than what can be derived from the answer. In particular, neither Alice nor Bob want to reveal any of their respective points that are in the interior of the convex hull of A u B.

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arXiv2010

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.

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arXiv2010

Extended h-Index Parameterized Data Structures for Computing Dynamic Subgraph Statistics

We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining statistics for undirected subgraphs of size three to directed subgraphs and to subgraphs of size four. For the directed case, we provide a data structure to maintain counts for all 3-vertex induced subgraphs in O(h) amortized time per update. For the undirected case, we maintain the counts of size-four subgraphs in O(h^2) amortized time per update. These extensions enable a number of new applications in Bioinformatics and Social Networking research.

cs.DS↗

arXiv2010

Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

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arXiv2010

Randomized Shellsort: A Simple Oblivious Sorting Algorithm

In this paper, we describe randomized Shellsort--a simple, randomized, data-oblivious version of the Shellsort algorithm that always runs in O(n log n) time and, as we show, succeeds in sorting any given input permutation with very high probability. Thus, randomized Shellsort is simultaneously simple, time-optimal, and data-oblivious. Taken together, these properties imply applications in the design of new efficient privacy-preserving computations based on the secure multi-party computation (SMC) paradigm. In addition, by a trivial conversion of this Monte Carlo algorithm to its Las Vegas equivalent, one gets the first version of Shellsort with a running time that is provably O(n log n) with very high probability.

cs.DS↗

arXiv2010

Cloning Voronoi Diagrams via Retroactive Data Structures

We address the problem of replicating a Voronoi diagram $V(S)$ of a planar point set $S$ by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in $S$; 2. the distance to and label(s) of the nearest site(s) in $S$; 3. a unique label for every nearest site in $S$. We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of $V(S)$ with $O(n)$ queries and $O(n \log^2 n)$ processing time. We also prove that queries of Type 3 can never exactly clone $V(S)$, but we show that with $O(n \log\frac{1}ε)$ queries we can construct an $ε$-approximate cloning of $V(S)$. In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures.

cs.CG↗

arXiv2010

Round-Trip Voronoi Diagrams and Doubling Density in Geographic Networks

The round-trip distance function on a geographic network (such as a road network, flight network, or utility distribution grid) defines the "distance" from a single vertex to a pair of vertices as the minimum length tour visiting all three vertices and ending at the starting vertex. Given a geographic network and a subset of its vertices called "sites" (for example a road network with a list of grocery stores), a two-site round-trip Voronoi diagram labels each vertex in the network with the pair of sites that minimizes the round-trip distance from that vertex. Alternatively, given a geographic network and two sets of sites of different types (for example grocery stores and coffee shops), a two-color round-trip Voronoi diagram labels each vertex with the pair of sites of different types minimizing the round-trip distance. In this paper, we prove several new properties of two-site and two-color round-trip Voronoi diagrams in a geographic network, including a relationship between the "doubling density" of sites and an upper bound on the number of non-empty Voronoi regions. We show how those lemmas can be used in new algorithms asymptotically more efficient than previous known algorithms when the networks have reasonable distribution properties related to doubling density, and we provide experimental data suggesting that road networks with standard point-of-interest sites have these properties.

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arXiv2010

Simulating Parallel Algorithms in the MapReduce Framework with Applications to Parallel Computational Geometry

In this paper, we describe efficient MapReduce simulations of parallel algorithms specified in the BSP and PRAM models. We also provide some applications of these simulation results to problems in parallel computational geometry for the MapReduce framework, which result in efficient MapReduce algorithms for sorting, 1-dimensional all nearest-neighbors, 2-dimensional convex hulls, 3-dimensional convex hulls, and fixed-dimensional linear programming. For the case when reducers can have a buffer size of $B=O(n^ε)$, for a small constant $ε>0$, all of our MapReduce algorithms for these applications run in a constant number of rounds and have a linear-sized message complexity, with high probability, while guaranteeing with high probability that all reducer lists are of size $O(B)$.

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arXiv2010

Learning Character Strings via Mastermind Queries, with a Case Study Involving mtDNA

We study the degree to which a character string, $Q$, leaks details about itself any time it engages in comparison protocols with a strings provided by a querier, Bob, even if those protocols are cryptographically guaranteed to produce no additional information other than the scores that assess the degree to which $Q$ matches strings offered by Bob. We show that such scenarios allow Bob to play variants of the game of Mastermind with $Q$ so as to learn the complete identity of $Q$. We show that there are a number of efficient implementations for Bob to employ in these Mastermind attacks, depending on knowledge he has about the structure of $Q$, which show how quickly he can determine $Q$. Indeed, we show that Bob can discover $Q$ using a number of rounds of test comparisons that is much smaller than the length of $Q$, under reasonable assumptions regarding the types of scores that are returned by the cryptographic protocols and whether he can use knowledge about the distribution that $Q$ comes from. We also provide the results of a case study we performed on a database of mitochondrial DNA, showing the vulnerability of existing real-world DNA data to the Mastermind attack.

cs.DS↗

arXiv2009

Succinct Greedy Geometric Routing in the Euclidean Plane

In greedy geometric routing, messages are passed in a network embedded in a metric space according to the greedy strategy of always forwarding messages to nodes that are closer to the destination. We show that greedy geometric routing schemes exist for the Euclidean metric in R^2, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(log n) bits, where n is the number of vertices in the graph. Moreover, our embedding strategy introduces a coordinate system for R^2 that supports distance comparisons using our succinct coordinates. Thus, our scheme can be used to significantly reduce bandwidth, space, and header size over other recently discovered greedy geometric routing implementations for R^2.

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arXiv2009

Going Off-road: Transversal Complexity in Road Networks

A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of edges of that graph that are crossed by random line or line segment. In this paper, we study the average transversal complexity of road networks. By viewing road networks as multiscale-dispersed graphs, we show that a random line will cross the edges of such a graph O(sqrt(n)) times on average. In addition, we provide by empirical evidence from experiments on the road networks of the fifty states of United States and the District of Columbia that this bound holds in practice and has a small constant factor. Combining this result with data structuring techniques from computational geometry, allows us to show that we can then do point location and ray-shooting navigational queries with respect to road networks in O(sqrt(n) log n) expected time. Finally, we provide empirical justification for this claim as well.

cs.CG↗

arXiv2009

Straggler Identification in Round-Trip Data Streams via Newton's Identities and Invertible Bloom Filters

We introduce 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. The straggler identification problem has applications, for example, in determining the set of 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 and is 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 show that there is a simple randomized solution using O(d log n log(1/epsilon)) bits that can maintain a multiset and solve the straggler identification problem, tolerating false deletions, where epsilon>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.

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arXiv2009

Efficient Authenticated Data Structures for Graph Connectivity and Geometric Search Problems

Authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being controlled by a remote untrusted host. We present efficient techniques for authenticating data structures that represent graphs and collections of geometric objects. We introduce the path hash accumulator, a new primitive based on cryptographic hashing for efficiently authenticating various properties of structured data represented as paths, including any decomposable query over sequences of elements. We show how to employ our primitive to authenticate queries about properties of paths in graphs and search queries on multi-catalogs. This allows the design of new, efficient authenticated data structures for fundamental problems on networks, such as path and connectivity queries over graphs, and complex queries on two-dimensional geometric objects, such as intersection and containment queries.

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arXiv2009

Planar Drawings of Higher-Genus Graphs

In this paper, we give polynomial-time algorithms that can take a graph G with a given combinatorial embedding on an orientable surface S of genus g and produce a planar drawing of G in R^2, with a bounding face defined by a polygonal schema P for S. Our drawings are planar, but they allow for multiple copies of vertices and edges on P's boundary, which is a common way of visualizing higher-genus graphs in the plane. Our drawings can be defined with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeoff, since canonical schemas have fewer sides, and have a nice topological structure, but they can have many more repeated vertices and edges than general polygonal cutsets. As a side note, we show that it is NP-complete to determine whether a given graph embedded in a genus-g surface has a set of 2g fundamental cycles with vertex-disjoint interiors, which would be desirable from a graph-drawing perspective.

cs.CG↗

arXiv2009

Pipelined Algorithms to Detect Cheating in Long-Term Grid Computations

This paper studies pipelined algorithms for protecting distributed grid computations from cheating participants, who wish to be rewarded for tasks they receive but don't perform. We present improved cheater detection algorithms that utilize natural delays that exist in long-term grid computations. In particular, we partition the sequence of grid tasks into two interleaved sequences of task rounds, and we show how to use those rounds to devise the first general-purpose scheme that can catch all cheaters, even when cheaters collude. The main idea of this algorithm might at first seem counter-intuitive--we have the participants check each other's work. A naive implementation of this approach would, of course, be susceptible to collusion attacks, but we show that by, adapting efficient solutions to the parallel processor diagnosis problem, we can tolerate collusions of lazy cheaters, even if the number of such cheaters is a fraction of the total number of participants. We also include a simple economic analysis of cheaters in grid computations and a parameterization of the main deterrent that can be used against them--the probability of being caught.

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arXiv2009

Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.

cs.CG↗

arXiv2009

Studying Geometric Graph Properties of Road Networks Through an Algorithmic Lens

This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data. Unlike previous approaches, our study is not based on the assumption that road networks are planar graphs. Indeed, based on the a number of experiments we have performed on the road networks of the 50 United States and District of Columbia, we provide strong empirical evidence that road networks are quite non-planar. Our approach therefore instead is directed at finding algorithmically-motivated properties of road networks as non-planar geometric graphs, focusing on alternative properties of road networks that can still lead to efficient algorithms for such problems as shortest paths and Voronoi diagrams. In particular, we study road networks as multiscale-dispersed graphs, which is a concept we formalize in terms of disk neighborhood systems. This approach allows us to develop fast algorithms for road networks without making any additional assumptions about the distribution of edge weights. In fact, our algorithms can allow for non-metric weights.

cs.CG↗