Approximate motion planning and the complexity of the boundary of the union of simple geometric figures

Abstract

Abstract. We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of f~(n2), where n is the number of obstacle corners. We introduce the tightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time of O((a/b. 1/eorlt + 1)n(log n)2), where a > b are the lengths of the sides of a rectangle and ecrit is the tightness of the problem. We show further that the complexity ( = number of vertices) of the boundary of n bow ties (see Figure 1) is O(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.

On the Union of κ-Round Objects in Three and Four Dimensions

“…The fat-object model addresses the observation that in some applications long and skinny objects are rarely encountered and objects with bounded aspect ratio are predominant. Under the assumption that the input objects are fat (see below for the several possible precise definitions), improved results were obtained for various algorithmic problems [3,4,7,13,22,27,28,30,33,34,40]. As this list of applications indicates, there is a strong practical motivation to study the complexity of the union of fat objects, in addition to the intrinsic interest in the problem itself.…”

“…The union of α-fat wedges, i.e., wedges whose opening angle is at least some constant α > 0, has complexity O(n) [7,24]; here and hereafter the implied constants depend on the fatness parameters. The union of α-fat triangles, i.e., triangles all of whose angles are at least some positive constant α, has complexity O(n log log n) [32,37].…”

Improved Bounds for the Union of Locally Fat Objects in the Plane

“…Any tower T ∈ T rest (σ i ) must lie in the vertical strip delimited by the lines supporting the left and right sides of σ i . We partition the set T rest (σ i ) into two subsets, defined momentarily and denoted T (1) rest (σ i ) and T (2) rest (σ i ). We first bound their union complexities (inside σ i ) separately, and then analyze the complexity of the merged union.…”

“…Efrat and Sharir proved that the union complexity of any set F of n such objects, each of constant complexity, is O(n 1+ε ), for any fixed ε > 0. 2 Efrat and Katz [16] obtained a better bound, namely, O(λ s+2 (n) log n), for so-called κ-curved objects (an object o is κ-curved if for every point p of its boundary there exists a disk D of diameter κ · diam(o) passing through p and contained in o); here s is the maximum number of times any two object boundaries intersect, and λ t (q) is the maximum length of Davenport-Schinzel sequences of order t on q symbols; it is near-linear in q for any fixed t [33]. Unfortunately, the class of κ-curved objects is rather restricted, since it does not allow convex vertices; in particular, fat triangles are not κ-curved.…”

Improved Bounds on the Union Complexity of Fat Objects

“…For this case it has been shown [2,10] that the union complexity is O(n). Matoušek et al [15] considered the case of δ-fat triangles, that is, triangles all of whose angles are at least δ for some fixed constant δ.…”