Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees

Abstract

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.

Simultaneous Optimization of Electrical Interconnection Configuration in Onshore Wind Fields

“…Given V, we can compute EMST(V) in O(n log n) time using O(n) words of workspace [11]. Given n sites in the plane, their EMST is the minimum spanning tree with the sites as vertices, where the weight of the edge between two sites is their Euclidean distance [12,13,14].…”

“…To do this with O(s) words of workspace, a succinct representation of its component structure was used. In the developed algorithm, we represent each edge ei by two directed half-edges [12]. The two half-edges are oriented in opposite directions such that the face incident a half-edge lies to the left of it.…”

“…The endpoint w = u; v of ej identifies the halfedges pw and sw as the predecessor and the successor of ej. They are shown in green and blue, respectively [12].…”

Geometric Algorithms with Limited Workspace: A Survey

“…Thus, the time-space trade-off from Theorem 3.4 does not immediately extend for computing EMST(S). Recently, Banyassady et al [19] revisited the problem and provided a time-space trade-off, building on the ideas in Theorem 3.6. Their algorithm computes EMST(S) in O(n 3 log s/s 2 ) time, provided that s cells of workspace are available.…”

“…It uses the workspace in two different ways: akin to Lemma 3.3, we check s edges in parallel for membership in EMST(S). Further, Banyassady et al [19] introduce s-nets, a compact representation of planar graphs. Using s-nets, one can speed up Kruskal's MST algorithm on S by making better use of the limited workspace.…”