In video games and robotics, one often discretizes a continuous 2D environment into a regular grid with blocked and unblocked cells and then finds shortest paths for the agents on the resulting grid graph. Shortest grid paths, of course, are not necessarily true shortest paths in the continuous 2D environment. In this article, we therefore study how much longer a shortest grid path can be than a corresponding true shortest path on all regular grids with blocked and unblocked cells that tessellate continuous 2D environments. We study 5 different vertex connectivities that result from both different tessellations and different definitions of the neighbors of a vertex. Our path-length analysis yields either tight or asymptotically tight worst-case bounds in a unified framework. Our results show that the percentage by which a shortest grid path can be longer than a corresponding true shortest path decreases as the vertex connectivity increases. Our path-length analysis is topical because it determines the largest path-length reduction possible for any-angle path-planning algorithms (and thus their benefit), a class of path-planning algorithms in artificial intelligence and robotics that has become popular.

在视频游戏和机器人技术中，人们经常将连续的 2D 环境离散化为具有阻塞和未阻塞单元格的规则网格，然后在生成的网格图上找到代理的最短路径。当然，最短网格路径不一定是连续二维环境中真正的最短路径。因此，在本文中，我们研究了最短网格路径可以比所有规则网格上的相应真实最短路径长多少，这些网格具有细分连续 2D 环境的阻塞和未阻塞单元。我们研究了 5 种不同的顶点连接性，它们是由不同的细分和顶点邻居的不同定义产生的。我们的路径长度分析在统一框架中产生紧密或渐近紧密的最坏情况边界。我们的结果表明，随着顶点连接性的增加，最短网格路径比相应的真实最短路径更长的百分比会降低。我们的路径长度分析是热门话题，因为它确定了任意角度路径规划算法可能的最大路径长度减少（以及它们的好处），人工智能和机器人技术中的一类路径规划算法已经变得流行。