A quadtree may be represented without pointers by encoding each black node with a quaternary integer whose digits reflect successive quadrant subdivisions. We refer to the sorted array of black nodes as the “linear quadtree” and show that it introduces a saving of at least 66 percent of the computer storage required by regular quadtrees. Some algorithms using linear quadtrees are presented, namely, ( i ) encoding a pixel from a 2 n × 2 >n array (or screen) into its quaternary code; ( ii ) finding adjacent nodes; ( iii ) determining the color of a node; ( iv ) superposing two images. It is shown that algorithms ( i )-( iii ) can be executed in logarithmic time, while superposition can be carried out in linear time with respect to the total number of black nodes. The paper also shows that the dynamic capability of a quadtree can be effectively simulated.

Summary. Suppose all zeros of a polynomial p but one are known to lie in specified circular regions, and the value of the logarithmic derivative p,p-1 is known at a point. What can be said about the location of the remaining zero ? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newton's method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.

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An algorithm is presented which finds the first intersection of a directed semi‐infinite straight‐line (called ray) with an octree, without resorting to the evaluation of neighbouring nodes. Given a pointer‐based region octree, intersections of the ray with a node's bisecting planes are first evaluated to determine in which sub‐octants the ray‐node intersections may lie; a local ordering then determines the sequence in which these sub‐octants should be examined so that the intersection closest to the ray's origin can be selected. This idea is applied recursively starting from the root of the octree; the novelty of the approach consists of the fact that the choice of each node's child in the octree is directly derived from the intersection of the ray‐segment within the tested sub‐octant, thus avoiding searching for any neighbouring nodes. The problem of selecting the correct octant while using the commonly available floating‐point arithmetic is also addressed in a ray tracing environment. Comparisons of this approach with the method given by Samet17 are carried out in a variety of cases involving millions of voxels. The improvement ranged from 32% to 62% in terms of execution time.