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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>ch7-efficiency</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
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<h1 id="Chapter-7-Efficiency">Chapter 7 Efficiency<a class="anchor-link" href="#Chapter-7-Efficiency">¶</a></h1><p>미국의 컴퓨터 과학자인 Alan Perlis가 애기하기를...</p>
<blockquote><p>A functional programmer was someone who knew the value of everything and the cost of nothing.</p>
</blockquote>
<p>functional programming의 <strong>성능</strong>과 <strong>효율성</strong>은?</p>
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<h1 id="7.1-Lazy-Evaluation">7.1 Lazy Evaluation<a class="anchor-link" href="#7.1-Lazy-Evaluation">¶</a></h1><h2 id="Lazy">Lazy<a class="anchor-link" href="#Lazy">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">sqr</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="n">x</span>
<span class="n">sqr</span> <span class="p">(</span><span class="n">sqr</span> <span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="p">))</span>
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<pre>2401</pre>
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<p><s>위 expression에 대해서 아래와 같이 lazy하게 reduction 된다.</s></p>
<pre><code>sqr (sqr (3+4))
= sqr (3+4) * sqr (3+4)
= ((3+4)*(3+4)) * ((3+4)*(3+4))
= ...
= 2401
</code></pre>
<p>lazy evaluation은 아래와 같이 reduction된다.</p>
<pre><code>sqr (sqr (3+4))
= let x = sqr (3+4) in sqr x
= let y = 3+4 in
let x = sqr y in sqr x
= let y = 7 in
let x = sqr y in sqr x
= let x = sqr 7 in sqr x
= let x = 7 * 7 in sqr x
= let x = 49 in sqr x
= 49 * 49
= 2401
</code></pre>
<blockquote><p><s>Lazy evaluation arguments는 필요한 시점에 단 한 번만 evaluate 된다.</s></p>
</blockquote>
<p>보다 정확하게 얘기하면</p>
<blockquote><p>Lazy evaluation에서 arguments는 <strong>필요한 시점</strong>에 단 <strong>한 번만</strong> <strong>head normal form</strong> 으로 evaluate 된다.</p>
</blockquote>
<h2 id="Head-Normal-Form">Head Normal Form<a class="anchor-link" href="#Head-Normal-Form">¶</a></h2><p>(Weak) Head Normal Form은 top-level이 아래로 구성되어 있다.</p>
<ul>
<li>data constructor</li>
<li>fully reduced lambda abstraction (weak: any lambda abstraction)</li>
</ul>
<p>NF는 HNF이나 HNF가 NF는 아니다.</p>
<ul>
<li>NF: 42, (2,"hello"), \x -> (x+1), e1, e2</li>
<li>HNF: (1+2,2+3), \x -> 2+2, 'h':("e"++"llo"), x, (e1, e2)</li>
<li>???: 1+1, (\x -> x+1) 2</li>
</ul>
<p>'sqr (head xs)'의 evaluation</p>
<pre><code>sqr (head xs)
= let a = head xs in sqr a
= let b = xs in
let a = head xs in sqr a
= let b = y:ys in
let a = head xs in sqr a
= let a = head (y:ys) in sqr a
= let a = y in sqr a
= sqr y
= y * y
</code></pre>
<h2 id="Common-Subexpression-elimination">Common Subexpression elimination<a class="anchor-link" href="#Common-Subexpression-elimination">¶</a></h2>
<pre><code>subseqs (x:xs) = subseqs xs ++ map (x:) (subseqs xs)
subseqs' (x:xs) = xss ++ map (x:) xss
where xss = subseqs' xs
</code></pre>
<p>subseqs는 'subseqs xs'가 2번 evaluate된다. 이것을 where로 분리하는 경우 1번만 연산할 수 있으나 <strong>haskell은 common subexpression elimination 을 자동으로 하지 않는다.</strong>
추가 공간을 사용하여 시간을 최적화하는 것이므로 프로그래머가 결정해야하는 사항임.</p>
<h2 id="Binding">Binding<a class="anchor-link" href="#Binding">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">foo1</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">sum</span> <span class="p">(</span><span class="n">take</span> <span class="n">n</span> <span class="n">primes</span><span class="p">)</span>
<span class="n">where</span>
<span class="n">primes</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="p">],</span> <span class="n">divisors</span> <span class="n">x</span> <span class="o">==</span> <span class="p">[</span><span class="n">x</span><span class="p">]]</span>
<span class="n">divisors</span> <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span> <span class="o">|</span> <span class="n">d</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="n">x</span><span class="p">],</span> <span class="n">x</span> <span class="err">`</span><span class="n">mod</span><span class="err">`</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">foo1</span> <span class="mi">100</span>
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<pre>24133</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">foo2</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">sum</span> <span class="p">(</span><span class="n">take</span> <span class="n">n</span> <span class="n">primes</span><span class="p">)</span>
<span class="n">primes</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="p">],</span> <span class="n">divisors</span> <span class="n">x</span> <span class="o">==</span> <span class="p">[</span><span class="n">x</span><span class="p">]]</span>
<span class="n">divisors</span> <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span> <span class="o">|</span> <span class="n">d</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="n">x</span><span class="p">],</span> <span class="n">x</span> <span class="err">`</span><span class="n">mod</span><span class="err">`</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">foo2</span> <span class="mi">100</span>
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<pre>24133</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">foo3</span> <span class="o">=</span> \<span class="n">n</span> <span class="o">-></span> <span class="nb">sum</span> <span class="p">(</span><span class="n">take</span> <span class="n">n</span> <span class="n">primes</span><span class="p">)</span>
<span class="n">where</span>
<span class="n">primes</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="p">],</span> <span class="n">divisors</span> <span class="n">x</span> <span class="o">==</span> <span class="p">[</span><span class="n">x</span><span class="p">]]</span>
<span class="n">divisors</span> <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span> <span class="o">|</span> <span class="n">d</span> <span class="o"><-</span> <span class="p">[</span><span class="mf">2.</span><span class="o">.</span><span class="n">x</span><span class="p">],</span> <span class="n">x</span> <span class="err">`</span><span class="n">mod</span><span class="err">`</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">foo3</span> <span class="mi">100</span>
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<pre>24133</pre>
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<p>primes, divisors의 binding</p>
<ul>
<li>foo1: 'foo1 n'에 bind 됨</li>
<li>foo2: global에 bind 됨</li>
<li>foo3: 'foo3'에 bind 됨</li>
</ul>
<p>foo2, foo3의 경우 evaluation 결과를 재사용하나 공간 사용량이 증가함</p>
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<h1 id="7.2.-Controlling-space">7.2. Controlling space<a class="anchor-link" href="#7.2.-Controlling-space">¶</a></h1><h2 id="lazy-and-space">lazy and space<a class="anchor-link" href="#lazy-and-space">¶</a></h2><p>lazy evaluation</p>
<pre><code>sum [1..1000]
= foldl (+) 0 [1..1000]
= foldl (+) (0+1) [2..1000]
= foldl (+) ((0+1)+2) [3..1000]
= ...
= 500500
</code></pre>
<p>eager evaluation</p>
<pre><code>sum [1..1000]
= foldl (+) 0 [1..1000]
= foldl (+) (0+1) [2..1000]
= foldl (+) 1 [2..1000]
= foldl (+) (1+2) [3..1000]
= ...
= 500500
</code></pre>
<p><strong>space를 제어하기 위해서는 eager evaluation을 lazy evaluation과 함께 사용하는 것이 좋다.</strong></p>
<h2 id="eager-evaluation">eager evaluation<a class="anchor-link" href="#eager-evaluation">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="o">--</span> <span class="n">seq</span> <span class="p">::</span> <span class="n">a</span> <span class="o">-></span> <span class="n">b</span> <span class="o">-></span> <span class="n">b</span>
<span class="o">--</span> <span class="n">Data</span><span class="o">.</span><span class="n">List</span><span class="o">.</span><span class="n">foldl</span><span class="s">'</span>
<span class="n">foldl</span><span class="s">' :: (b -> a -> b) -> b -> [a] -> b</span>
<span class="n">foldl</span><span class="s">' f e [] = e</span>
<span class="n">foldl</span><span class="s">' f e (x:xs) = y `seq` foldl'</span> <span class="n">f</span> <span class="n">y</span> <span class="n">xs</span>
<span class="n">where</span> <span class="n">y</span> <span class="o">=</span> <span class="n">f</span> <span class="n">e</span> <span class="n">x</span>
<span class="n">foldl</span><span class="s">' (+) 0 [1..10]</span>
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<pre>55</pre>
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<p>Data.List.sum과 foldl'은 위 방식으로 구현됨.</p>
<p>lazy와 eager 구현은 strict function에 대해서는 동일하다. (f ⊥ = ⊥ 라면, f는 strict하다.)</p>
<h2 id="mean-예제">mean 예제<a class="anchor-link" href="#mean-예제">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">mean</span> <span class="p">[]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">mean</span> <span class="n">xs</span> <span class="o">=</span> <span class="nb">sum</span> <span class="n">xs</span> <span class="o">/</span> <span class="n">fromIntegral</span> <span class="p">(</span><span class="n">length</span> <span class="n">xs</span><span class="p">)</span>
<span class="n">mean</span> <span class="p">[</span><span class="mf">1.</span><span class="o">.</span><span class="mi">10</span><span class="p">]</span>
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<pre>5.5</pre>
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<p>xs의 순회가 2번 발생한다. tupling을 사용하여 1번 순회로 합과 길이를 구함.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">sumlen</span> <span class="p">::</span> <span class="p">[</span><span class="n">Float</span><span class="p">]</span> <span class="o">-></span> <span class="p">(</span><span class="n">Float</span><span class="p">,</span><span class="n">Int</span><span class="p">)</span>
<span class="n">sumlen</span> <span class="o">=</span> <span class="n">foldr</span> <span class="n">f</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="n">where</span> <span class="n">f</span> <span class="n">x</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">sumlen</span> <span class="p">[</span><span class="mf">1.</span><span class="o">.</span><span class="mi">10</span><span class="p">]</span>
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<pre>(55.0,10)</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">mean</span><span class="s">' [] = 0</span>
<span class="n">mean</span><span class="s">' xs = s / fromIntegral n</span>
<span class="n">where</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">sumlen</span> <span class="n">xs</span>
<span class="n">mean</span><span class="s">' [1..10]</span>
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<pre>5.5</pre>
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<p>space leak을 제거하기 위해서 foldl'을 사용</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">sumlen</span> <span class="o">=</span> <span class="n">foldl</span><span class="s">' g (0,0)</span>
<span class="n">where</span> <span class="n">g</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="n">x</span> <span class="o">=</span> <span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">sumlen</span> <span class="p">[</span><span class="mf">1.</span><span class="o">.</span><span class="mi">10</span><span class="p">]</span>
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<pre>(55,10)</pre>
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<p>HNF인 (s+x,n+1)에 eager evaluation 적용</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">sumlen</span> <span class="o">=</span> <span class="n">foldl</span><span class="s">' f (0,0)</span>
<span class="n">where</span> <span class="n">f</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="n">x</span> <span class="o">=</span> <span class="n">s</span> <span class="err">`</span><span class="n">seq</span><span class="err">`</span> <span class="n">n</span> <span class="err">`</span><span class="n">seq</span><span class="err">`</span> <span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">sumlen</span> <span class="p">[</span><span class="mf">1.</span><span class="o">.</span><span class="mi">10</span><span class="p">]</span>
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<pre>(55,10)</pre>
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<p>two more application operators</p>
<ul>
<li>$: lazy evaluation</li>
<li>\$!: eager evaluation</li>
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<h1 id="7.3-Controlling-time">7.3 Controlling time<a class="anchor-link" href="#7.3-Controlling-time">¶</a></h1><h2 id="tips-for-time">tips for time<a class="anchor-link" href="#tips-for-time">¶</a></h2><p>eager evalution을 사용하여 쉽게 제어할 수 있으나 속도는 그렇지 못 하다.</p>
<p>GHC 문서가 제시하는 속도향상의 키 포인트 3가지</p>
<ol>
<li>GHC Profiling 도구를 사용해라</li>
<li>알고리즘을 개선해라</li>
<li>가능하면 제공되는 라이브러리를 사용하라. (매우 최적화되어 있고, 컴파일되어 있음.)</li>
</ol>
<p>추가적인 팁 2가지</p>
<ul>
<li>'Strict funcions are your dear friends': 보통 eager evaluation이 lazy evaluation보다 overhead가 적다.</li>
<li>요구사항을 만족하는 최적의 type을 명시적으로 사용할 것: Integer보다는 Int가 좋다.</li>
</ul>
<p>이러한 방법들은 asymptotic time complexity을 변환시키지 못 하므로 효과가 작다. 그러나 잘못된 코드는 asymptotic complexity에 영향을 줄 수도 있다.</p>
<h2 id="cartesian-product">cartesian product<a class="anchor-link" href="#cartesian-product">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">cp</span> <span class="p">[]</span> <span class="o">=</span> <span class="p">[[]]</span>
<span class="n">cp</span> <span class="p">(</span><span class="n">xs</span><span class="p">:</span><span class="n">xss</span><span class="p">)</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">:</span><span class="n">ys</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="n">xs</span><span class="p">,</span> <span class="n">ys</span> <span class="o"><-</span> <span class="n">cp</span> <span class="n">xss</span><span class="p">]</span>
<span class="n">cp</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]]</span>
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<pre>[[1,3,5],[1,3,6],[1,4,5],[1,4,6],[2,3,5],[2,3,6],[2,4,5],[2,4,6]]</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">cp</span><span class="s">' = foldr op [[]]</span>
<span class="n">where</span> <span class="n">op</span> <span class="n">xs</span> <span class="n">yss</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">:</span><span class="n">ys</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="n">xs</span><span class="p">,</span> <span class="n">ys</span> <span class="o"><-</span> <span class="n">yss</span><span class="p">]</span>
<span class="n">cp</span><span class="s">' [[1,2],[3,4],[5,6]]</span>
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<pre>[[1,3,5],[1,3,6],[1,4,5],[1,4,6],[2,3,5],[2,3,6],[2,4,5],[2,4,6]]</pre>
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<p>cp보다 cp'의 evaluation time이 훨씬 적다. cp는 cp xss를 xs의 길이만큼 반복 계산한다. cp의 list comprehension을 제거하면 아래 형태로 이해가 쉬워진다.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">cp</span> <span class="p">[]</span> <span class="o">=</span> <span class="p">[[]]</span>
<span class="n">cp</span> <span class="p">(</span><span class="n">xs</span><span class="p">:</span><span class="n">xss</span><span class="p">)</span> <span class="o">=</span> <span class="n">concat</span> <span class="p">(</span><span class="nb">map</span> <span class="n">f</span> <span class="n">xs</span><span class="p">)</span>
<span class="n">where</span> <span class="n">f</span> <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">:</span><span class="n">ys</span> <span class="o">|</span> <span class="n">ys</span> <span class="o"><-</span> <span class="n">cp</span> <span class="n">xss</span><span class="p">]</span>
<span class="n">cp</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]]</span>
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<pre>[[1,3,5],[1,3,6],[1,4,5],[1,4,6],[2,3,5],[2,3,6],[2,4,5],[2,4,6]]</pre>
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<p>아래 cp''는 cp'와 동일한 성능을 가진다.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">cp</span><span class="s">''</span> <span class="p">[]</span> <span class="o">=</span> <span class="p">[[]]</span>
<span class="n">cp</span><span class="s">''</span> <span class="p">(</span><span class="n">xs</span><span class="p">:</span><span class="n">xss</span><span class="p">)</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">:</span><span class="n">ys</span> <span class="o">|</span> <span class="n">x</span> <span class="o"><-</span> <span class="n">xs</span><span class="p">,</span> <span class="n">ys</span> <span class="o"><-</span> <span class="n">yss</span><span class="p">]</span>
<span class="n">where</span> <span class="n">yss</span> <span class="o">=</span> <span class="n">cp</span> <span class="n">xss</span>
<span class="n">cp</span><span class="s">''</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]]</span>
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<pre>[[1,3,5],[1,3,6],[1,4,5],[1,4,6],[2,3,5],[2,3,6],[2,4,5],[2,4,6]]</pre>
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<h1 id="7.4-Analysing-time">7.4 Analysing time<a class="anchor-link" href="#7.4-Analysing-time">¶</a></h1><ul>
<li>시간 복잡도는 expression의 속성이지 value의 속성은 아니다.</li>
<li>GHCi에서는 reduction step이 아닌 소요시간만을 측정한다. reduction 단계는 소요 시간과 반드시 일치하지는 않는다.</li>
<li>상황에 따라서 다른 측정 방법이 필요하다. 예를 들면 concat xss의 경우 n보다는 (m,n)이 적합하다.</li>
<li>시간 복잡도 측정은 eager evaluation으로 행한다. lazy evaluation은 측정이 어려우며 일반적으로 eager evaluation의 time이 lazy evaluation의 time 보다 tight boundary를 가진다.</li>
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<h2 id="Concat">Concat<a class="anchor-link" href="#Concat">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">concat</span> <span class="n">xss</span> <span class="o">=</span> <span class="n">foldr</span> <span class="p">(</span><span class="o">++</span><span class="p">)</span> <span class="p">[]</span> <span class="n">xss</span>
<span class="n">concat</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]]</span>
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