 
Summary: Binary Space Partitions for AxisParallel Line Segments:
SizeHeight Tradeoffs
Sunil Arya
Department of Computer Science
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong.
arya@cs.ust.hk
Abstract
We present worstcase lower bounds on the minimum size of a binary space partition (BSP)
tree as a function of its height, for a set S of n axisparallel line segments in the plane. We
assume that the BSP uses only axisparallel cutting lines. These lower bounds imply that, in
the worst case, a BSP tree of height O(log n) must have size (n log n) and a BSP tree of size
O(n) must have height (n
), where is a suitable constant.
Key words: Computational geometry, binary space partitions, line segments.
1 Introduction
Given a set S of n interiordisjoint objects in IRd
, the binary space partition (BSP) for S is a
subdivision obtained by recursively cutting space into two parts by a hyperplane, terminating when
each cell is intersected by at most a constant number of the given objects. BSPs can be represented
