以最少的游程遍历网格（图形）的每个边缘

Question

I have an (mxn) grid where each edge has the same unit length of 1. Each run starts from the start-point (0, 0) and moves to the endpoint (m, n). Each run can only move rightwards or upwards, ie no backward travel allowed, and hereby each run must consist of a length of exactly (m+n).

How many runs do I need so that I can make sure that I have traversed every edge in this grid? Ie what's the minimum number of runs from (0, 0) to (m, n) is required to make sure that each edge (corner) of the (mxn) grid has been gone through for at least once? Note that I'm not aiming to find number of paths from (0, 0) or (m, n). I only want to explore all edges in the entire grid.

Thanks!

Answer 1

您必须遍历每一行和每一列，才能给出n + m条路径的答案。

Answer 2

Turn the grid so that (0,0) is at the top, and (m, n) is at the bottom. You now have a nice top-to-bottom flow that lets "gravity" provide the edge directions for a directed graph.

Adjust the diagram so that the edges are on the nearest 45-degree angle. (0,0) is at the top; the second layer consists of (0, 1) and (1, 0); the third is (0, 2), (1, 1), (2, 0), and so on. After i moves, you'll be at a node whose coordinates sum to i .

Now, since each node has 1 or 2 edges in, and 1 or 2 edges out, it's trivial to show that the maximum layer size is min(m, n) nodes, and the maximum number of edges at any given step is 2*(min(m, n) - 1) . It's also trivial to construct a series of paths that will traverse each layer or step in that number of runs. Number the nodes/edges from one side to the other; use corresponding numbers to plan each run. When you hit the limit in one dimension, just stick to the edge for the rest of that run.

So, if you need only to hit every node, you can do it in min(m, n) runs. If you need every edge, it's 2*(min(m, n) - 1) runs.

Answer 3

To cover every edge, you need m+(n-2) paths. A simple example of this consists of horizontal and vertical paths, as shown below. Note that the red paths are duplicates, so the red paths in the second diagram are not needed. That's the reason for the -2 in the equation.