We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for one-dimensional keys. In the one-dimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expected-case search time, and further there exist search trees achieving expected search times of at most H + 2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.

Point location is the problem of preprocessing a planar polygonal subdivision S into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. Given the probabilities pz that the query point lies within each cell z 2 S, a natural question is how to design such a structure so as to minimize the expected-case query time. The entropy H of the probability distribution is the dominant term in the lower bound on the expected-case search time. Clearly the number of edges n of the subdivision is a lower bound on the space required. There is no known approach that simultaneously achieves the goals of H + o(H) query time and O(n) space. In this paper we introduce entropy-preserving cuttings and show how to use them to achieve query time H+o(H), using only O(n log n) space. 1 Introduction Planar point location is an important problem in computational geometry. We are given a polygonal subdivision S consisting of n edges, and the goal is ...

We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming languages. Our solution has a worst-case running time of O(1) for each push, pop, inject, eject and catenation. The best previously known solution has an O(log k) time bound for the k deque operation. Our solution is not only faster but simpler. A key idea used in our result is an algorithmic technique related to the redundant digital representations used to avoid carry propagation in binary counting.

What is the smallest constant c so that the planar point location queries can be answered in c log 2 n + o(log n) steps (i.e. point-line comparisons) in the worst case? In SODA 97 Goodrich, Orletsky, and Ramaiyer [6] showed that c = 2 is possible using linear space and conjectured this to be optimal. We disprove this conjecture and show that c = 1 can be achieved. Moreoever by giving upper and lower bounds we show that without space restrictions the worst case query complexity of planar point location differs from log 2 n + 2 p log 2 n at most by an additive factor of (1=2)log 2 log 2 n +O(1). For the case of linear space we show the query complexity to be bounded by log 2 n + 2 p log 2 n +O(log 1=4 n).

In this paper we consider the following problems: given a set T of triangles in 3-space, with jT j = n, a) answer the query " given a line l, does l stab the set of triangles?" (query problem). b) find whether a stabbing line exists for the set of triangles (existence problem). c) Given a ray ae, which is the first triangle in T hit by ae? The following results are shown. 1. There is an \Omega\Gamma n 3 ) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles. 2. The existence problem for triangles on a set of planes with g different plane inclinations can be solved in O(g 2 n 2 log n) time (Theorem 2). 3. The query problem is solvable in quasi-quadratic O(n 2+ffl ) preprocessing and storage and logarithmic O(log n) query time (Theorem 4). 4. All stabbing results for triangles extend, with the same asymptotic bounds, to sets of convex polyhedra with total complexity n. 5. Using O(n 3+ffl ) preprocessing time and storage we can det...

We address a longstanding open problem of [10, 9], and present a general transformation that transforms any pointer based data structure to be confluently persistent. Such transformations for fully persistent data structures are given in [10], greatly improving the performance compared to the naive scheme of simply copying the inputs. Unlike fully persistent data structures, where both the naive scheme and the fully persistent scheme of [10] are feasible, we show that the naive scheme for confluently persistent data structures is itself infeasible (requires exponential space and time). Thus, prior to this paper there was no feasible method for implementing confluently persistent data structures at all. Our methods give an exponential reduction in space and time compared to the naive method, placing confluently persistent data structures in the realm of possibility.

We present a method for maintaining biased search trees so as to support fast finger updates (i.e., updates in which one is given a pointer to the part of the tree being changed). We illustrate the power of such biased finger trees by showing how they can be used to derive an optimal O(n log n) algorithm for the 3-dimensional layers-of-maxima problem and also obtain an improved method for dynamic point location.

We present a data structure that can store a set of disjoint fat objects in d-space such that point location and bounded-size range searching with arbitrarily-shaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or non-convex polytopes. The multi-purpose data structure supports point location and range searching queries in time O(log d\Gamma1 n) and requires O(n log d\Gamma1 n) storage, after O(n log d\Gamma1 n log log n) preprocessing. The data structure and query algorithm are rather simple. 1 Introduction Fatness turns out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [3, 15, 22, 26, 32] and efficiency gains for algorithms [1, 4, 19, 28, 33] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of ...

0 A jA j A A 0 arrangement 1 Introduction U n d E U E d k k ! d U U E deterministically Marco Pellegrini May 20, 1996 U n d d d U U K U U O n K O n K n ffl ? U O n U d U d O n O n O n Key words. AMS subject classifications. 1.1 Point Location: definition and previous results On Point Location and Motion Planning among Simplices Proceedings of the 26th ACM Symposium on Theory of Computing A preliminary version of this work appeared in the . Research supported by C.N.R. within the 1993-1994 Senior Visiting Scientist program at ICSI. Affiliation: Istituto di Matematica Computazionale del C.N.R., via S. Maria 46, Pisa, Italy. e-mail: pellegrini@iei.pi.cnr.it arrangements of simplices, point location, sparse nets, motion planning, triangulations 68P05, 68Q25, 68Q40, 68U05 Point location is a central problem in computational geometry [PS85, Ede87, Meh84] and a continuous stream of research papers have been published on this topic from the early days of computational geometry till to...

Pathfinding for commercial games is a challenging problem, and many existing methods use abstractions which lose details of the environment and compromise path quality. Conversely, humans can ignore irrelevant details of an environment that modern search techniques still consider, while maintaining its topography. This thesis describes a technique for extracting features such as dead ends, corridors, and decision points from an environment represented using a constrained Delaunay triangulation. The result is that the pathfinding task is simplified to the point where the search algorithm need only decide to which side of each obstacle to go, while all features of the environment are retained. We present algorithms which search the triangles of the environment (Triangulation A*) and the decision points identified (Triangulation Reduction A*). We also explore a number of techniques which deal with finding paths for circular objects of nonzero radius and enhancements to various aspects of the search.