Notes on Binary Dumbbell Trees

Dumbbell trees were introduced in [1]. A detailed description of non-binary dumbbell trees appears in Chapter 11 of [3]. These notes show how binary dumbbell trees can be obtained, and how they can be used to construct, in O(n log n) time, a spanner of bounded degree and weight proportional to O(log n) times the weight of a minimum spanning tree. The reader is assumed to be familiar with the split tree and the well-separated pair decomposition (WSPD), as described, e.g., in Chapter 9 of [3]. 1 Dumbbells Let S be a set of n points in R. For any subset A of S, we denote its bounding box by R(A). Let T be the split tree for S. For any node u in T , let Su be the set of all points in S that are stored in the subtree of u. Consider the WSPD that is obtained from T . Each of the O(n) pairs {A,B} in this WSPD is represented by two nodes, say u and v, of T : The set A is equal to the set Su, whereas the set B is equal to the set Sv. For each pair {A,B} in the WSPD, we define its dumbbell to be the geometric region consisting of the two bounding boxes R(A) and R(B), together with the line segment joining the centers of these boxes. The two bounding boxes are called the heads of the dumbbell. The length of the dumbbell is defined to be the length of the line segment joining the centers of the two heads. The size of a head is defined to be the length of its longest side.