Data Structures and Algorithms 3: Multi-dimensional by K. Mehlhorn

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By K. Mehlhorn

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Exercise 14). Thus weight balanced dd-trees are only useful for exact match queries. A second problem is that weight-balanced dd-trees are hard to rebalance. Rotations are of no use since splitting is done with respect to different coordinates on different levels. Thus it is impossible to change the depth of a node as rotations do. There is a way out. Suppose that we followed path p = v o 'v 1 , ... to insert point x. Let i be minimal such that vi goes out of balance by the insertion. Then rebalance the tree by replacing the subtree rooted at vi by an ideal tree for set S(V i ).

A d-fold range tree for multiset 8 Uo x U1 x ... x Ud - 1 , 181 = n is defined as follows. If d = 1 then T is any BB[ a j-tree for 8. If d > 1 then T consists of a BB[ a jC tree To for po(8). To is called the primary tree. Furthermore, for every node v of To with depth (v) E m 7L there is an auxiliary tree Ta (v) . Ta(V) is a (d-1)-fold tree for set p(8(v), {l, ... ,d-1}). Here 8(v) is the set of x = (x o , ... ,x d _ 1 ) E 8 such that leaf xo is descendant of v o The pre c i se definition of range trees differs in two respects from the informal discussion.

X Ud - 1 and let P = {i 1 , ... ,ik} E {O, ... ,d-1}. , ... ,x. ); x E 8} is the projection of 8 onto coordiII lk nates P. If P = {i} then we also write Pi(8) instead of p(8, {i}). Definition: Let m E ~ 0 and let a E (1/4,1-/2/2). m is a slack parameter and a is a weight-balancing parameter. A d-fold range tree for multiset 8 Uo x U1 x ... x Ud - 1 , 181 = n is defined as follows. If d = 1 then T is any BB[ a j-tree for 8. If d > 1 then T consists of a BB[ a jC tree To for po(8). To is called the primary tree.

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