### Table 1: Average time (in milliseconds) for one point-location query.

2006

"... In PAGE 7: ...2 Results. Table1 shows the average query time associated with point location in arrange- ments of varying types and sizes using the dif- ferent point-location algorithms. The number of edges mentioned in these tables is the number of undirected edges of the arrangement.... ..."

Cited by 3

### Table 1: Average time (in milliseconds) for one point-location query.

2006

"... In PAGE 7: ...2 Results. Table1 shows the average query time associated with point location in arrange- ments of varying types and sizes using the dif- ferent point-location algorithms. The number of edges mentioned in these tables is the number of undirected edges of the arrangement.... ..."

Cited by 3

### Table 1. We show that previous methods for proxim- ity queries exhibit a sharp tradeo between degree and query time. Namely, low degree is achieved by the slow brute-force search method, while fast algorithms based on point-location in a preprocessed Voronoi diagram or on the extremal-search method have high degree. Our new technique gives instead both low degree and fast query time, and is optimal with respect to both cost measures for queries in sets of 2D point sites. For proofs and details omitted in this extended ab- stract, the reader is referred to the full paper [20].

1997

"... In PAGE 3: ...Method Degree Time brute-force distance comparison 2 * O(n) nearest neighbor point location in explicit Voronoi diagram 6 O(log n) * extremal-search method 4 O(log n) * point location in implicit Voronoi diagram 2 * O(log n) * k-nearest neighbors brute-force distance comparison 2 * O(n) and point location in explicit order-k Voronoi diagram 6 O(log n + k) * circular range search point location in implicit order-k Voronoi diagram 2 * O(log n + k) * nearest neighbor among brute-force distance comparison 6 O(n) points and segments point location in explicit Voronoi diagram 64 O(log n) * point location in implicit Voronoi diagram 6 O(log n) * brute-force distance comparison 2 * O(n) 3D nearest neighbor point location in explicit 3D Voronoi diagram 8 O(log2 n) point location in implicit 3D Voronoi diagram 3 O(log2 n) Table1 : Comparison of the degree and running time of algorithms for some fundamental proximity query problems. A * denotes optimality.... ..."

Cited by 33

### Table 1: Comparison of the degree and running time of algorithms for some fundamental proximity query problems. A * denotes optimality. The technique introduced in this paper (point location in an implicit Voronoi diagram) outperforms previous methods and is optimal for 2D queries.

1997

"... In PAGE 2: ... However, we shall show that the latter method fails to achieve optimal degree because the search is based on predicates requiring 4b bits of precision; moreover, the high overhead of the search technique (which uses the hierarchical polytope representation [8]) casts some doubts on the practicality of the method. The main results of this work are summarized in Table1 . We show that previous methods for proxim- ity queries exhibit a sharp tradeo between degree and query time.... ..."

Cited by 33

### Table 1: Benchmark Results for Non-Convex Polyhedra. Each column, respectively from left to right, denotes a benchmarking model, triangle counts of the model, a number of decomposed con- vex pieces in the model, average query time in msec for a point outside the model, average query time in msec for a point inside the model.

"... In PAGE 8: ... In our experiment, an average query time for a triangle takes 10 sec. The benchmarking results for polyhedra are also presented in Table1 . Depending on the location of a query point with respect to the polyhedron, the query time takes from 0.... ..."

### Table 3 shows the memory usage of the point-location strategies of the random line-segment arrangements from Tables 1 and 2.

"... In PAGE 11: ...0 258.9 Table3 : Memory usage (in MBytes) by the point location data structure. Number of Preprocessing Query % Queries Landmarks Time [sec] Time [msec] with AD=0 100 61.... ..."

### Table 1 Previous results for dynamic point location. N denotes the number of possible y-coordinates for edge endpoints in the subdivision. Also, we use O( ) to denote an amortized bound.

1991

"... In PAGE 2: ....1. Previous work. Before we describe our main results, let us brie y review previous work on dynamic point location, which we summarize in Table1 . Early work on dynamic point location includes a method by Overmars [35], which is based on a segment-tree [4] approach to planar-point location, and achieves an O(log2 n) query and update time with O(n log n) space.... ..."

Cited by 43

### Table 1: Comparison of the degree and time of algorithms for some fundamental proximity query problems. A * denotes optimality. The new technique introduced in this paper (point location in an implicit Voronoi diagram) always outperforms previous methods and is optimal for 2D queries. (point location in explicit Voronoi diagram) or 4 (extremal-search method), and we present our new technique, based on implicit Voronoi diagrams, which achieves optimal degree 2. In Sections 5{6, we extend our approach to nearest neighbor search queries in a set of 3D point sites and in a set of point and segment sites in the plane, respectively. Practical improvements are presented in Section 7. Finally, further research directions are discussed in Section 8.

1997

Cited by 33

### Table 4: Simple location query time

2003

"... In PAGE 21: ...Table 4: Simple location query time Table4 shows the response time of our prototype Person Location Provider (the average shown was computed using 20 runs of 20 queries each). In this test, the location of the user to be located was known to the People Location Provider.... ..."

### Table 4. Average time per a query (in second) Point Query Region Query Spatial Join Query

1996

"... In PAGE 14: ... Queries that we performed are classified into point queries, window queries and spatial join queries. Table4 presents the average time required for the evaluation of one single query. The time values are given in seconds.... In PAGE 15: ...Table4 suggest that the reasoning of the existence of Gopt is valid. The query performance of the no decomposition (i.... ..."

Cited by 1