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    <h2 id="번---행렬-곱셈-순서" class="fw-bold">11049번 - 행렬 곱셈 순서</h2>
<h4 id="분류-다이나믹-프로그래밍">분류 : 다이나믹 프로그래밍</h4>
<h4 id="문제">문제</h4>
<p>크기가 N×M인 행렬 A와 M×K인 B를 곱할 때 필요한 곱셈 연산의 수는 총
N×M×K번이다. 행렬 N개를 곱하는데 필요한 곱셈 연산의 수는 행렬을 곱하는
순서에 따라 달라지게 된다.</p>
<p>예를 들어, A의 크기가 5×3이고, B의 크기가 3×2, C의 크기가 2×6인
경우에 행렬의 곱 ABC를 구하는 경우를 생각해보자.</p>
<ul>
<li>AB를 먼저 곱하고 C를 곱하는 경우 (AB)C에 필요한 곱셈 연산의 수는
5×3×2 + 5×2×6 = 30 + 60 = 90번이다.</li>
<li>BC를 먼저 곱하고 A를 곱하는 경우 A(BC)에 필요한 곱셈 연산의 수는
3×2×6 + 5×3×6 = 36 + 90 = 126번이다.</li>
</ul>
<p>같은 곱셈이지만, 곱셈을 하는 순서에 따라서 곱셈 연산의 수가
달라진다.</p>
<p>행렬 N개의 크기가 주어졌을 때, 모든 행렬을 곱하는데 필요한 곱셈 연산
횟수의 최솟값을 구하는 프로그램을 작성하시오. 입력으로 주어진 행렬의
순서를 바꾸면 안 된다.</p>
<h4 id="풀이">풀이</h4>
<div class="sourceCode" id="cb1"><pre
class="sourceCode java"><code class="sourceCode java"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="im">java</span><span class="op">.</span><span class="im">util</span><span class="op">.</span><span class="im">Scanner</span><span class="op">;</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span> <span class="kw">class</span> Main <span class="op">{</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>    <span class="dt">static</span> <span class="dt">int</span> n<span class="op">;</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>    <span class="dt">static</span> <span class="dt">int</span><span class="op">[]</span> numbers<span class="op">;</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>    <span class="dt">static</span> <span class="dt">int</span><span class="op">[][]</span> dp<span class="op">;</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>    <span class="kw">public</span> <span class="dt">static</span> <span class="dt">void</span> <span class="fu">main</span><span class="op">(</span><span class="bu">String</span><span class="op">[]</span> args<span class="op">)</span> <span class="op">{</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>        <span class="bu">Scanner</span> sc <span class="op">=</span> <span class="kw">new</span> <span class="bu">Scanner</span><span class="op">(</span><span class="bu">System</span><span class="op">.</span><span class="fu">in</span><span class="op">);</span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>        n <span class="op">=</span> sc<span class="op">.</span><span class="fu">nextInt</span><span class="op">();</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>        dp <span class="op">=</span> <span class="kw">new</span> <span class="dt">int</span><span class="op">[</span>n<span class="op">+</span><span class="dv">1</span><span class="op">][</span>n<span class="op">+</span><span class="dv">1</span><span class="op">];</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>        numbers <span class="op">=</span> <span class="kw">new</span> <span class="dt">int</span><span class="op">[</span>n<span class="op">+</span><span class="dv">1</span><span class="op">];</span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a>        <span class="cf">for</span> <span class="op">(</span><span class="dt">int</span> i <span class="op">=</span> <span class="dv">0</span><span class="op">;</span> i <span class="op">&lt;</span> n<span class="op">;</span> i<span class="op">++)</span> <span class="op">{</span></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a>            numbers<span class="op">[</span>i<span class="op">]</span> <span class="op">=</span> sc<span class="op">.</span><span class="fu">nextInt</span><span class="op">();</span></span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>            <span class="dt">int</span> tmp <span class="op">=</span> sc<span class="op">.</span><span class="fu">nextInt</span><span class="op">();</span></span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a>            <span class="cf">if</span> <span class="op">(</span>i <span class="op">==</span> n<span class="op">-</span><span class="dv">1</span><span class="op">)</span></span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a>                numbers<span class="op">[</span>n<span class="op">]</span> <span class="op">=</span> tmp<span class="op">;</span></span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a>        <span class="op">}</span></span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a>        <span class="cf">for</span> <span class="op">(</span><span class="dt">int</span> i <span class="op">=</span> <span class="dv">2</span><span class="op">;</span> i <span class="op">&lt;</span> n<span class="op">+</span><span class="dv">1</span><span class="op">;</span> i<span class="op">++)</span></span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a>            <span class="cf">for</span> <span class="op">(</span><span class="dt">int</span> j <span class="op">=</span> <span class="dv">0</span><span class="op">;</span> j <span class="op">&lt;</span> n<span class="op">-</span>i<span class="op">+</span><span class="dv">1</span><span class="op">;</span> j<span class="op">++)</span> <span class="op">{</span></span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a>                dp<span class="op">[</span>j<span class="op">][</span>j <span class="op">+</span> i<span class="op">]</span> <span class="op">=</span> <span class="op">(</span><span class="dt">int</span><span class="op">)</span> <span class="fl">1e9</span><span class="op">;</span></span>
<span id="cb1-25"><a href="#cb1-25" aria-hidden="true" tabindex="-1"></a>                <span class="cf">for</span> <span class="op">(</span><span class="dt">int</span> k <span class="op">=</span> <span class="dv">0</span><span class="op">;</span> k <span class="op">&lt;</span> i<span class="op">;</span> k<span class="op">++)</span> <span class="op">{</span></span>
<span id="cb1-26"><a href="#cb1-26" aria-hidden="true" tabindex="-1"></a>                    dp<span class="op">[</span>j<span class="op">][</span>j <span class="op">+</span> i<span class="op">]</span> <span class="op">=</span> <span class="bu">Math</span><span class="op">.</span><span class="fu">min</span><span class="op">(</span>dp<span class="op">[</span>j<span class="op">][</span>j <span class="op">+</span> i<span class="op">],</span></span>
<span id="cb1-27"><a href="#cb1-27" aria-hidden="true" tabindex="-1"></a>                            dp<span class="op">[</span>j<span class="op">][</span>j <span class="op">+</span> k<span class="op">]</span> <span class="op">+</span> dp<span class="op">[</span>j <span class="op">+</span> k<span class="op">][</span>j <span class="op">+</span> i<span class="op">]</span> <span class="op">+</span> numbers<span class="op">[</span>j<span class="op">]</span> <span class="op">*</span> numbers<span class="op">[</span>j <span class="op">+</span> k<span class="op">]</span> <span class="op">*</span> numbers<span class="op">[</span>j <span class="op">+</span> i<span class="op">]);</span></span>
<span id="cb1-28"><a href="#cb1-28" aria-hidden="true" tabindex="-1"></a>                <span class="op">}</span></span>
<span id="cb1-29"><a href="#cb1-29" aria-hidden="true" tabindex="-1"></a>            <span class="op">}</span></span>
<span id="cb1-30"><a href="#cb1-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-31"><a href="#cb1-31" aria-hidden="true" tabindex="-1"></a>        <span class="bu">System</span><span class="op">.</span><span class="fu">out</span><span class="op">.</span><span class="fu">println</span><span class="op">(</span>dp<span class="op">[</span><span class="dv">0</span><span class="op">][</span>n<span class="op">]);</span></span>
<span id="cb1-32"><a href="#cb1-32" aria-hidden="true" tabindex="-1"></a>    <span class="op">}</span></span>
<span id="cb1-33"><a href="#cb1-33" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
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