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

Jordan Jorgensen

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 ($\mathrm{EMST}$) of a set $V$ of $n$ sites in the plane. We present an algorithm that computes $\mathrm{EMST}(V)$ using $O\bigl(n^3\tfrac{\log s}{s^2}\bigr)$ time and $O(s)$ words of workspace. Our algorithm uses the fact that $\mathrm{EMST}(V)$ 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.