In the limited-workspace model, we assume that the input of size n 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 O(s) words, where s ∈ {1, . . . , n} is a given parameter. In a time-space trade-off, we are interested in how the running time of an algorithm improves as s varies from 1 to n.We present a time-space trade-off for computing the Euclidean minimum spanning tree (EMST) of a set V of n sites in the plane. We present an algorithm that computes EMST(V ) using O(n 3 log s/s 2 ) time and O(s) words of workspace. Our algorithm uses the fact that EMST(V ) is a subgraph of the boundeddegree relative neighborhood graph of V , and applies Kruskal's MST algorithm on it. To achieve this with limited workspace, we introduce a compact representation of planar graphs, called an s-net which allows us to manipulate its component structure during the execution of the algorithm. * B.B. and W.M. were supported in part by DFG project MU/3501/2. L.B. was supported by the ETH Postdoctoral Fellowship.
We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex of P . Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.For any fixed ε > 0, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + ε. The labels have O(log n) bits, and the routing tables are of size O((ε −1 + h) log n). The preprocessing time is O(n 2 log n). It can be improved to O(n 2 ) for simple polygons.
In the limited workspace model, we consider algorithms whose input resides in read-only memory and that use only a constant or sublinear amount of writable memory to accomplish their task. We survey recent results in computational geometry that fall into this model and that strive to achieve the lowest possible running time. In addition to discussing the state of the art, we give some illustrative examples and mention open problems for further research.
Let P be a simple polygon with n vertices, and let q ∈ P be a point in P . Let k ∈ {0, . . . , n − 1}. A point p ∈ P is k-visible from q if and only if the line segment pq crosses the boundary of P at most k times. The k-visibility region of q in P is the set of all points that are k-visible from q. We study the problem of computing the k-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of O(n) words, each with Ω(log n) bits. The algorithm can read and write O(s) additional words of workspace, where s ∈ N is a parameter of the model. The output is written to a write-only stream.Given a simple polygon P with n vertices and a point q ∈ P , we present an algorithm that reports the k-visibility region of q in P in O(cn/s + c log s + min{ k/s n, n log log s n}) expected time using O(s) words of workspace. Here, c ∈ {1, . . . , n} is the number of critical vertices of P for q where the k-visibility region of q may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.
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