A Fast and Simple Algorithm for Computing Approximate Euclidean Minimum Spanning Trees

The Euclidean minimum spanning tree (EMST) is a fundamental and widely studied structure. In the approximate version we are given an n-element point set P in R and an error parameter ε > 0, and the objective is to compute a spanning tree over P whose weight is at most (1 + ε) times that of the true minimum spanning tree. Assuming that d is a fixed constant, existing algorithms have running times that (up to logarithmic factors) grow as O ( n/ε ) . We present an algorithm whose running time is O ( n log n+ ( ε−2 log 1ε ) n ) . Thus, this is the first algorithm for approximate EMSTs that eliminates the exponential ε dependence on dimension. (Note that the O-notation conceals a constant factor of the form O(1).) The algorithm is deterministic and very simple.