We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time

Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2-d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3-d runs in near O(n + k ) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the ( k)-level, previously used in proving combinatorial k-level bounds.

We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.

We present a near-quadratic time algorithm that computes a point inside a simple polygon P having approximately the largest visibility polygon inside P , and near-linear time algorithm for nding the point that will have approximately the largest Voronoi region when added to an n-point set. We apply the same technique to nd the translation that approximately maximizes the area of intersection of two polygonal regions in near-quadratic time.

We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of the problems we consider can be solved in time O(n 4=3+ffi ) or better, and there are\Omega\Gamma n 4=3 ) lower bounds for a few of them in specialized models of computation. However, the best known lower bound in any general model of computation is only\Omega\Gamma n log n). The paper is naturally divided into two parts. In the first part, we consider a large number of problems that are harder than Hopcroft's problem. These problems include various ray shooting problems, sorting line segments in IR 3 , collision detection in IR 3 , and halfspace emptiness checking in IR 5 . In the second, we survey known reductions among problems involving lines in three-space, and among highe...

Decision Problem (D): Given a set I R m of n points, decide if there exists a point c 2 R ` such that for all p 2 I, P (c; p) 0. We say P (x; y) has an order k linearization if there exists 2k+1 polynomials, X i = X i (x) (i = 1; : : : ; k) and Y i = Y i (y) (for i = 0; : : : ; k), such that P (x; y) = Y0 + P k i=1 X i Y i : Theorem 1 (i) If P (x; y) has an order k linearization, the decision problem (D) can be solved in O(n bk=2c ) in the algebraic model. schoemer@cs.uni-sb.de, Universitat des Saarlandes. y sellen@cs.uni-sb.de, Universitat des Saarlandes. z teichman@cs.nyu.edu, Courant Institute, NYU. x yap@cs.nyu.edu, Courant Institute, NYU. (ii) In the bit model, if each input coordinate has L bits, the problem (D) can be solved O((L)n bk=2c ). In our application, our focus is the xed-parameter problem to decide whether there exists an anchored cylinder of given radius r that encloses all input points. We obtain an order 9 linearization for this deci...

We consider the music pattern matching problem—to find occurrences of a small fragment of music called the “pattern” in a larger body of music called the “score”—as a problem of translating a set of horizontal line segments in the plane to find the best match in a larger set of horizontal line segments. Our contribution is that we use fairly general weight functions to measure the quality of a match, thus enabling approximate pattern matching. We give an algorithm with running time O(nm log m), where n is the size of the score and m is the size of the pattern. We show that the problem, in this geometric formulation, is unlikely to have a significantly faster algorithm because it is at least as hard as a basic problem called 3-SUM that is conjectured to have no subquadratic algorithm. We present some examples to show the potential of this method for finding minor variations of a theme, and for finding polyphonic musical patterns in a polyphonic score. 1.

We obtain subquadratic algorithms for 3SUM on integers and rationals in several models. On a standard word RAM with w-bit words, we obtain a running time of O(n 2 / max { w lg 2 w, lg 2 n (lg lg n) 2}). In the circuit RAM with one nonstandard AC0 operation, we obtain O(n2 / w2 lg2). In external w memory, we achieve O(n2 /(MB)), even under the standard assumption of data indivisibility. Cache-obliviously, we obtain a running time of O(n2 / MB lg2). In all cases, our speedup is almost M quadratic in the parallelism the model can afford, which may be the best possible. Our algorithms are Las Vegas randomized; time bounds hold in expectation, and in most cases, with high probability. 1

For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edge-weighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) node-weight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edge-weighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edge-weighted problem is much harder than its node-weighted version. 1

We examine a computational geometric problem concerning the structure of polymers. We model a polymer as a polygonal chain in three dimensions. Each edge splits the polymer into two subchains, and a dihedral rotation rotates one of these subchains rigidly about the edge.