6 research outputs found
Improved time-space trade-offs for computing Voronoi diagrams
Let P
be a planar set of n sites in general position. For kā{1,ā¦,nā1}, the Voronoi diagram of order k for P is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest k neighbors in P. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of k=1 and k=nā1, respectively. For any given Kā{1,ā¦,nā1}, the family of all higher-order Voronoi diagrams of order k=1,ā¦,K for P can be computed in total time O(nK2+nlogn) using O(K2(nāK)) space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for P can be computed in O(nlogn) time using O(n)
space [Preparata, Shamos, Springer'85].
For sā{1,ā¦,n}
, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words, of Ī(logn)
bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards.
We describe a deterministic s
-workspace algorithm for computing NVD and FVD for P that runs in O((n2/s)logs) time. Moreover, we generalize our s-workspace algorithm so that for any given KāO(sā), we compute the family of all higher-order Voronoi diagrams of order k=1,ā¦,K for P in total expected time O(n2K5s(logs+K2O(logāK))) or in total deterministic time O(n2K5s(logs+KlogK)). Previously, for Voronoi diagrams, the only known s-workspace algorithm runs in expected time O((n2/s)logs+nlogslogās) [Korman et al., WADS'15] and only works for NVD (i.e., k=1). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques
Improved Time-Space Trade-Offs for Computing Voronoi Diagrams
Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space.
For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards.
We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques
A Time-Space Tradeoff for Triangulations of Points in the Plane
In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm
Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees
In the limited-workspace model, we assume that the input of size lies in
a random access read-only memory. The output has to be reported sequentially,
and it cannot be accessed or modified. In addition, there is a read-write
workspace of words, where is a given parameter.
In a time-space trade-off, we are interested in how the running time of an
algorithm improves as varies from to .
We present a time-space trade-off for computing the Euclidean minimum
spanning tree (EMST) of a set of sites in the plane. We present an
algorithm that computes EMST using time and
words of workspace. Our algorithm uses the fact that EMST is a subgraph of
the bounded-degree relative neighborhood graph of , and applies Kruskal's
MST algorithm on it. To achieve this with limited workspace, we introduce a
compact representation of planar graphs, called an -net which allows us to
manipulate its component structure during the execution of the algorithm
Space Efficient Algorithms for Breadth-Depth Search
Continuing the recent trend, in this article we design several
space-efficient algorithms for two well-known graph search methods. Both these
search methods share the same name {\it breadth-depth search} (henceforth {\sf
BDS}), although they work entirely in different fashion. The classical
implementation for these graph search methods takes time and bits of space in the standard word RAM model (with word size being
bits), where and denotes the number of edges and
vertices of the input graph respectively. Our goal here is to beat the space
bound of the classical implementations, and design space
algorithms for these search methods by paying little to no penalty in the
running time. Note that our space bounds (i.e., with bits of
space) do not even allow us to explicitly store the required information to
implement the classical algorithms, yet our algorithms visits and reports all
the vertices of the input graph in correct order.Comment: 12 pages, This work will appear in FCT 201