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

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 $s \in \{1, \dots, n\}$ 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 bounded-degree 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

Time-space trade-off for finding the K-visibility region of a point in a polygon

We study the problem of computing the k-visibility region in the memory-constrained model. In this model, the input resides in a randomly accessible read-only memory of O(n) words, with O(log n) bits each. An algorithm can read and write O(s) additional words of workspace during its execution, and it writes its output to write-only memory. In a given polygon P and for a given point q ∈ P, we say that a point p is inside the k-visibility region of q, if and only if the line segment pq intersects the boundary of P at most k times. Given a simple n-vertex polygon P stored in a read-only input array and a point q ∈ P, we give a time-space trade-off algorithm which reports the kvisibility region of q in P in O(cn/s+n log s+min{⌈k/s⌉n, n log logs n}) expected time using O(s) words of workspace. Here c ≤ n is the number of critical vertices for q, i.e., the vertices of P where the visibility region may change. We also show how to generalize this result for polygons with holes and for sets of non-crossing line segments