PAT/source/html/t1019.html

Separate the Animals
<div class="ques-view"><p>There are some animals in a zoo which can be described as a grid with N rows and M columns. Your task is to place some obstacles so that no pairs of animals can reach each other.</p>
<p>Two animals can <strong>reach each other</strong> if and only if their cells are <strong>4-connected</strong>. For example, in Figure 1, the central blue cell can be reached by the four red cells, and cannot be reached by the other four white cells.</p>
<p><img alt="sep1.JPG" src="https://images.ptausercontent.com/0eba9ce1-1391-424e-ab2c-2e67cea8a90c.JPG"/></p>
<p>Figure 1</p>
<p>What is more, you must put obstacles in exactly K cells, which are 4-connected and form exactly H holes. Here a <strong>hole</strong> is defined as a 4-connected part with finitely many open cells while the zoo is placed in an infinite open grid. For example, there are 2 holes (the green and the yellow areas) in Figure 2.</p>
<p><img alt="sep2.JPG" src="https://images.ptausercontent.com/be0f6b32-a181-4ee7-8410-0899291ce154.JPG"/></p>
<p>Figure 2</p>
<p>For the following grid with two animals:</p>
<p><img alt="sep3.jpg" src="https://images.ptausercontent.com/5265c6d9-1a9a-4ab1-8eb8-f880f8add579.jpg"/></p>
<p>Figure 3</p>
<p>If K = 8 and H = 1, one way to separate them is the following:</p>
<p><img alt="sep4.jpg" src="https://images.ptausercontent.com/0c3939f3-ba1e-46cd-a677-a9e122fd7202.jpg"/></p>
<p>Figure 4</p>
<p>Figure 5 is illegal because it contains no hole.</p>
<p><img alt="sep5.jpg" src="https://images.ptausercontent.com/5dbbae27-6155-480a-a7b9-48007deb653c.jpg"/></p>
<p>Figure 5</p>
<p>Figure 6 is also illegal because the obstacles are not 4-connected.</p>
<p><img alt="sep6.jpg" src="https://images.ptausercontent.com/927e59e8-4c19-4d77-8bc4-df2486228b50.jpg"/></p>
<p>Figure 6</p>
<p>Given some animals, you are supposed to count the number of different solutions.</p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case. For each case, the first line gives four integers: N, M, K, H (2 <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> N, M <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> 6; 1 <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> K <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> 12; 0 <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> H <span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo></mrow>\le</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> 2). All the numbers are separated by spaces.</p>
<p>Then N lines follow, each contains M characters, which are either <code>.</code> or <code>O</code>, representing an open cell or an animal, respectively. There will be at least 2 animals.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each case, print a single line containing a single integer: the number of solutions.</p>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">3 5 8 1
...O.
.O...
.....
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">8
</code></pre>
</div>