The worst-case performance of heuristics with bucketing techniques and/or spacefilling curves for the planar matching problem and the planar traveling salesman problem is analyzed. Two types of heuristics are investigated, one is to sequence given points in a spacefilling-curve order and the other is to sequence the points in the order of buckets which are arranged according to the spacefilling curve. The former heuristics take O(n log n) time, while the latter ones run in O(n) time when the number of buckets is O(n). It is shown that the worst-case performance of the former and that of the latter are the same if a sufficient number of O(n) buckets are provided, which is investigated in detail especially for the heuristics based on the Sierpinski curve. The worst-case performance of the heuristic employing the Hilbert curve is also analyzed.