Time–space trade-offs for triangulations and Voronoi diagrams

Abstract

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that all points in a cell have the same nearest neighbor in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1, . . . , n}, an s-workspace algorithm has readonly access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing an arbitrary triangulation of S in time O(n 2 /s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n 2 /s) log s + n log s log * s). * WS and PS were supported in part by DFG Grants MU 3501/1 and MU 3501/2. YS was supported by the DFG within the research training group "Methods for Discrete Structures" (GRK 1408).The exact setting may vary, but there is a common theme: the input resides in read-only memory, the output must be written to a write-only structure, and we can use O(s) additional variables to find the solution (for a parameter s). The goal is to design algorithms whose running time decreases as s grows, giving a time-space trade-off [23]. One of the first problems considered in this model is sorting [19,20]. Here, the time-space product is known to be Ω(n 2 ) [8], and matching upper bounds for the case b ∈ Ω(log n) ∩ O(n/ log n) were obtained by Pagter and Rauhe [21] (b denotes the available workspace in bits).Our current notion of memory constrained algorithms was introduced to computational geometry by Asano et al. [4], who showed how to compute many classic geometric structures with O(1) workspace (related models were studied before [9]). Later, time-space trade-offs were given for problems on simple polygons, e.g., shortest paths [1], visibility [6], or the convex hull of the vertices [5].We consider a model in which the set S of n points is in an array such that random access to each input point is possible, but we may not change or even reorder the input. Additionally, we have O(s) variables (for a parameter s ∈ {1, . . . , n}). We assume that each variable or pointer contains a data word of Θ(log n) bits. Other than this, the model allows the usual word RAM operations. In this setting we study two problems: computing an arbitrary triangulation for S and computing the Voronoi diagram VD(S) for S. Since the output cannot be stored explicitly, the goal is to report the edges of the triangulation or the vertices of VD(S) successively, in no particular order. Dually, the latter goal may be phrased in terms of Delaunay triangulations. We focus on Voronoi diagrams, as they lead to a more natural presentation.Both problems can be solved in O(n 2 ) time with O(1) workspace [4] or in O(n log n) ...

A Time-Space Trade-Off for Triangulations of Points in the Plane

“…Our multi-pass algorithm also improves the previously best known algorithm under the random-access model by Korman et al which takes O(min{n 2 /s + n log n log s, n 2 log n/s}) time [4,12] although the multi-pass model is more restrictive than the random-access model. It seems unclear whether the algorithm by Korman et al [12] can be extended to a multi-pass streaming algorithm.…”

“…Our algorithm uses O(n/s) passes over the input array. It is not only the first algorithm for this problem under the multi-pass model, but it also improves the previously best known random-access algorithm [12]. Moreover, its running time is optimal under the multi-pass model.…”

“…The edge adjacency is essential information in representing and reconstructing the triangulation. In contrast, the algorithms in [4,12] report the edges of a triangulation in an arbitrary order with no adjacency information of them. Furthermore, the algorithm by Korman et al [12] uses the algorithm by Asano and Kirkpatrick [3] as a subprocedure, but it is unclear how to modify the subprocedure to report a triangulation together with edge adjacency information [2].…”

“…For the case that a space is given as a positive integer parameter s at most n, several results are known for the random-access model while no result is known for the multi-pass streaming model. Korman et al presented an sworkspace algorithm for computing a triangulation of S in O(n 2 /s+n log n log s) time [12]. In the same paper, they presented an s-workspace algorithm for computing the Delaunay triangulation of S in O((n 2 /s) log s + n log s log * s) expected time.…”

“…Recently, it is improved to O(n 2 log n/s) deterministic time [4]. Combining [4] and [12], a triangulation of S can be computed in O(min{n 2 /s + n log n log s, n 2 log n/s}) time.…”