Two points p and q in a simple polygon P are k-visible when the line segment pq crosses the boundary of P at most k times. Given a query point q, a positive integer k, and a polygon P, we design an algorithm that computes the region of P that is k-visible from q in O(nk) time, where n denotes the number of vertices of P. This region is called the k-visibility region of q. This is the first algorithm parameterized in terms of k, resulting in an asymptotically faster worst-case running time compared to previous algorithms when k is \(o(\log {n})\), and bridging the gap between the O(n)-time algorithm for computing the 0-visibility region of q in P and the \(O(n\log n)\)-time algorithm for computing the k-visibility region of q in P for any k.

We also design a data structure of size O(n⁵) that supports visibility queries, returning the k-visible region of P for any arbitrary query point q in \(O(\log {n}+m)\) time, where m denotes the number of vertices on the boundary of the output visibility region.

当线段pq最多跨越K次P的边界时，简单多边形P中的两个点p和q是k可见的。给定查询点q，一个正整数ķ，和多边形P，我们设计一个计算的区域的算法P是ķ从-可见光q在Ö（Ñ ķ）时间，其中Ñ表示的顶点的数量P。该区域称为kq的可见区域。这是第一个用k参数化的算法，当k为\（o（\ log {n}）\）时，与以前的算法相比，渐近地加快了最坏情况的运行时间，并弥合了O（n）-time算法用于计算的0-可见区域q在P和\（O（N \ log n）的\） -time算法用于计算ķ的-visibility区域q中P为任何ķ。

我们还设计了一个大小为O（n ⁵）的数据结构，该结构支持可见性查询，并在\（O（\ log {n} + m）\）时间内返回任意查询点q的P的k可见区域，其中m表示输出可见性区域边界上的顶点数。