PKU3020：Antenna Placement

Antenna Placement

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 2712		Accepted: 1287

Description

The Global Aerial Research Centre has been allotted the task of building the fifth generation of mobile phone nets in Sweden. The most striking reason why they got the job, is their discovery of a new, highly noise resistant, antenna. It is called 4DAir, and comes in four types. Each type can only transmit and receive signals in a direction aligned with a (slightly skewed) latitudinal and longitudinal grid, because of the interacting electromagnetic field of the earth. The four types correspond to antennas operating in the directions north, west, south, and east, respectively. Below is an example picture of places of interest, depicted by twelve small rings, and nine 4DAir antennas depicted by ellipses covering them.

Obviously, it is desirable to use as few antennas as possible, but still provide coverage for each place of interest. We model the problem as follows: Let A be a rectangular matrix describing the surface of Sweden, where an entry of A either is a point of interest, which must be covered by at least one antenna, or empty space. Antennas can only be positioned at an entry in A. When an antenna is placed at row r and column c, this entry is considered covered, but also one of the neighbouring entries (c+1,r),(c,r+1),(c-1,r), or (c,r-1), is covered depending on the type chosen for this particular antenna. What is the least number of antennas for which there exists a placement in A such that all points of interest are covered?

Input

On the first row of input is a single positive integer n, specifying the number of scenarios that follow. Each scenario begins with a row containing two positive integers h and w, with 1 <= h <= 40 and 0 < w <= 10. Thereafter is a matrix presented, describing the points of interest in Sweden in the form of h lines, each containing w characters from the set ['*','o']. A '*'-character symbolises a point of interest, whereas a 'o'-character represents open space.

Output

For each scenario, output the minimum number of antennas necessary to cover all '*'-entries in the scenario's matrix, on a row of its own.

 

Sample Input

2
7 9
ooo**oooo
**oo*ooo*
o*oo**o**
ooooooooo
*******oo
o*o*oo*oo
*******oo
10 1
*
*
*
o
*
*
*
*
*
*

Sample Output

175

原题地址：http://acm.pku.edu.cn/JudgeOnline/problem?id=3020

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题解：

在一个n*m的棋盘上，有一些标志，问最少用多少个1*2的矩形可以把它们全部套住。

网上搜到的解法：

可以将每一个标志与其相邻的四个标志建一条边，这样就形成了一个无向图，题目的意思即要求最小的边数。使得所有的点都在这些边数。这样就转换成了最小路径覆盖问题。

最小路径覆盖=顶点数-最大二分匹配

本题是无向图，即二分图中的边是双向边，若1和2匹配的话，那么2和1也匹配，所以本题=顶点数-最大二分匹配/2

【对于这个解法，我有很大的疑问，最小路径覆盖最后求的是路径数而不是边数，题目要求的是边数，另外最小路径覆盖是有向图，虽然解法中说双向边要除以2，但是理解上还是很有问题，

到底为什么是路径覆盖？】

另附一大牛解法：

将横纵坐标和为奇数的放到x几何，偶数放到Y集合，然后连边。答案即为最大匹配加上为匹配的点。