Given a two dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied "forces." We apply compaction to improve the material utilization of an already tightly packed layout. Compaction can be modeled as a motion of the polygons that reduces the value of some functional on their positions. Optimal compaction, planning a motion that reaches a layout that has the global minimum functional value among all reachable layouts, is shown to be NP-complete under certain assumptions. We first present a compaction algorithm based on existing physical simulation approaches. This algorithm uses a new velocity-based optimization model. Our experimental results reveal the limitation of physical simulation: even though our new model improves the running time of our algorithm over previous simulation algorithms, the algorithm still can not compact typical layouts of one hundred or more polygons in ...

We investigate 3-d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering ray-shooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.

Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial two-dimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of non-convex polygons are not previously known. This dissertation offers the first systematic study of compaction of non-convex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACE-hard. The major contribution of this dissertation is a position-based optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first ...

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of n low-degree algebraic surface patches in 3-space. We show that this complexity is O(n

. We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewise-linear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions, and (b) to obtain improved bounds on the complexity of the vertical decomposition of a single cell in an arrangement of triangles in 3-space, and of several other substructures in such an arrangement (the entire arrangement, all nonconvex cells, and any collection of cells). The latter results also lead to improved algorithms for computing substructures in arrangements of triangles and for translational motion planning in three dimensions. 1. Introduction The study of arrangements of curves or surfaces is an important area of research in computational and combinatorial geometry, because many...

. Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 0-1 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function. 1. Introduction A configuration, C = (c ij ) (1 i u; 1 j v), is a partial matrix with 1's and blanks at the entries. All the matrices we are going to work with will be 0 \Gamma 1 matrices. We say that a matrix M = (m ij ) does have the configuration C if one can find u rows i 1 ; i 2 ; . . . ; i u ; i 1 ! \Delta \Delta \Delta ! i u and v columns j 1 ; j 2 ; . . . ; j v ; j 1 ! \Delta \Delta \Delta ! j v in M such that the corresponding submatrix contains C, i.e. m i ff ;j fi = 1 whenever c ff;fi = 1. Let f(n; m;C) denote the maximum number of 1's in an n \Theta m matrix M not containing C. In the case of n = m we writ...

These are partitions of [l] = f1; 2; : : : ; lg into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by Davenport-Schinzel sequences. We summarize and extend known enumerative results for the pattern p = abab and give an explicit formula for the number p(abab; n; l; k) of such partitions. Our main tool are generating functions. We determine the corresponding generating function for p = abba and k = 1; 2; 3. For k = 2 there is a connection with the number of directed animals. We solve exactly two related extremal problems.

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting --- preprocess a set K of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure --- preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching --- preprocess a set C of n convex fat polygons, so that the k objects intersecting a `not-too-large' query polygon can be reported efficiently, and (iv) bounded-size segment shooting --- preprocess a set C as in (iii), so that the first object (if exists) hit by a `not-too-long' oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(s (n) log 3 n)...

We call a line ` a separator for a set S of objects in the plane if ` avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above ` and the other of objects lying below `. In this paper we present an O(n log n)- time algorithm for finding a separator line for a set of n segments, provided the ratio between the diameter of the set of segments and the length of the smallest segment is bounded. No subquadratic algorithms are known for the general case. Our algorithm is based on the recent results of [13], concerning the union of `fat ' triangles, but we also include an analysis which improves the bounds obtained in [13].

Let K be a set of n non-intersecting objects in 3-space. A depth order of K, if exists, is a linear order ! of the objects in K such that if K;L 2 K and K lies vertically below L then K ! L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xy-projections are all `fat', then a depth order for K can be computed in time O(n log 5 n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all `fat' and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(n 1=2 s (n) log 4 n), where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K, and s (n) is the maximum length of (n; s) Davenport-Schinzel sequences.