Simultaneous containment of several polygons

Simultaneous Containment of Several Polygons: Analysis of the Contact Configurations

“…The time complexity of this algorithm is O(n 2 m 1 m 2 log m 1 m 2 ) which improves over the result of Avnaim and Boissonnat [2] (O(n 2 m 1 m 2 log nm 1 log nm 2 )).…”

“…In the case of two convex polygons, we can achieve the computation in time O(n + m) and the size is also O(n + m). [1] Avnaim [2] has established the following results:…”

“…The free space of two convex polygons moving in a convex environment can be found in linear time. [1] In the case of two or three general polygons in general polygonal environment Avnaim and Boissonnat [2,3] gave algorithms to compute the free space or to only find one solution. The complexity depends on the different cases of convexity of objects and environment, in particular, they found one solution for the containment of two (resp.…”

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

“…1 Problem (P2) was studied recently for the special case K = 2 [11]; we consider the general case of arbitrary K (Section 3). On the contrary, for problem (P6), algorithms for general K are known [19,47]; we improve their running times for the special case K = 2 (Section 4.2). See Section 1.6 for in-depth survey of earlier work.…”

“…In particular, for packing K convex items into a minimum-area fixed-orientation rectangle, he gives an O(n K−1 log n) algorithm based on linear programming techniques. The most general problem for disjoint packing under translations was solved by Avnaim and Boissonnat [19], who presented an O((m 2 + mn) 2K log n) algorithm for packing K arbitrary simple n-gons into a simple m-gon. For our special case of K = 2, m = 4 (problem (P6)), the algorithms of [19] and [47] run in O(n 4 log n) and O(n log n) time, respectively; we present a linear-time algorithm (Section 4.2).…”