Simultaneous containment of several polygons

IINTRODUCTION We investigate the problem of whether one or several Polygons I, . ..lk can be translated to fit inside another polygon E without overlapping each other. The containment problem for one polygon, say I, has been studied Previously by Chazelle (C), Baker, Fortune and Mahaney (BFM) and Fortune (F). Chazelle derived an algorithm that runs in time O(n+m) to solve lhis problem, in the case that both polygons E and I are convex. Here n is the number of edges of E and m the number of edges of I. The case that I is non-convex but E is convex, can be easily reduced to the previous one. The case that E is non-convex but I is convex is considered in @FM) and (F). The best algorithm is obtained by Fortune who presents an O(nm lognm) algorithm. This algorithm is surely close to optimal (in the worst-case sense) since the feasible region may have O(mn) vertices. This paper presents an algorithm that solve the general case where I and E may be non-convex and may have (non-convex) holes. The algorithm provides the set of all the possible placements. Its complexity is O(CiCeNlOgN) where N=cin+cem, Ci=l +number of concave vertices of I, c,=number of convex vertices of E which are not vertices of the convex hull of E plus number of edges of the convex hull of E which are not edges of E This algorithm is close to optimal. in the worst-case where c,=O(n) and ci=O(m), since the feasible region may have O(n2m2) vertices. In the case of rectilinear polygons the algorithm runs in O(n2m2) time and, thus, is optimal. The problem of coordinating the movements of several disks has been studied by Schwartz and Sharir (SS) and by Yap(Y). Yap has obtained an O(nk) algorithm for the coordination of k disks