Abstract. An improved and general approach to connected-component labeling of images is presented. The algorithm presented in this paper processes images in predeterttuned order, which means that the processing order depends only on the image representation scheme and not on specific properties of the image. The algorithm handles a wide variety of image representation schemes (rasters, run lengths, quadtrees, bintrees, etc.). How to adapt the standard UNION-FIND algorithm to permit reuse of tempora~labels is shown. This is done using a technique called age balancing, in which, when two labels are merged, the older label becomes the father of the younger label. This technique can be made to coexist with the more conventional rule of wetghf balancing, in which the label with more descendants becomes the father of the label with fewer descendants. Various image scanning orders are examined and classified. It is also shown that when the algorithm is specialized to a pixel array scanned in raster order, the total processing time is linear in the number of pixels. The linear-time processing time follows from a special property of the UNION-FINDalgorithm, which maybe of independent interest. This property states that under certain restrictions on the input, UNION-FINDruns in time linear in the number of FIND and UNIONoperations. Under these restrictions, linear-time performance can be achieved without resorting to the more complicated Gabow-Tarj an algorithm for disjoint set union.

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A discussion is presented of the relationship between two solid representation schemes: constructive solid geometry (CSG trees) and recursive spatial subdivision exemplified by the bintree, a generalization of the quadtree and octree. Detailed algorithms are developed and analyzed for evaluating CSG trees by bintree conversion, i.e., by determining explicitly which parts of space are solid and which empty. These techniques enable the addition of the time dimension and motion to the approximate analysis of CSG trees in a simple manner to solve problems such as dynamic interference detection. For "well-behaved" CSG trees, the execution time of the conversion algorithm is directly related to the spatial complexity of the object represented by the CSG tree (i.e., asymptotically it is proportional to the number of bintree nodes as the resolution increases). The set of well-behaved CSG trees includes all trees that define multidimensional polyhedra in a manner that does not give rise to tangential intersections at CSG tree nodes.

Abstract.The extendible cell method is an application of order preserving extendible hashing to multidimensional point files. We derive some of its performance characteristics and show its expected case optimality for closest point problems.Ko'wor~L~: multidimensional search, order preserving extendible hashing.