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1.1   어디에나 있는 알고리즘
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1.2   알고리즘이란 무엇인가
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1.3   요약
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제 2 장   복잡도 분석
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2.1   알고리즘 효율 평가
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2.1   알고리즘 효율 평가
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2.1.1   실제 측정
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2.1.2   이론적 추정
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2.2   반복과 재귀
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2.3   시간 복잡도
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2.4   공간 복잡도
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2.5   요약
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2.6   연습 문제
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제 3 장   자료 구조
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제 3 장   자료 구조
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3.1   자료 구조의 분류
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3.2   기본 자료형
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3.3   수의 인코딩 *
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3.4   문자 인코딩 *
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3.5   요약
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3.6   연습 문제
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제 4 장   배열과 연결 리스트
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제 4 장   배열과 연결 리스트
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4.1   배열
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4.2   연결 리스트
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4.3   리스트
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4.4   메모리와 캐시 *
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4.5   요약
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4.6   연습 문제
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제 5 장   스택과 큐
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제 5 장   스택과 큐
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5.1   스택
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5.2   큐
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5.3   덱
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5.4   요약
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5.5   연습 문제
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제 6 장   해시 테이블
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제 6 장   해시 테이블
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6.1   해시 테이블
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6.2   해시 충돌
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6.3   해시 알고리즘
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6.4   요약
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6.5   연습 문제
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제 7 장   트리
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제 7 장   트리
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7.1   이진 트리
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_tree/binary_tree_traversal/">
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7.2   이진 트리 순회
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7.3   이진 트리의 배열 표현
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7.4   이진 탐색 트리
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<a class="md-nav__link" href="../../chapter_tree/avl_tree/">
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7.5   AVL 트리 *
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7.6   요약
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7.7   연습 문제
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제 8 장   힙
</span>
</a>
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</label>
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<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
제 8 장   힙
</label>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/heap/">
<span class="md-ellipsis">
8.1   힙
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/build_heap/">
<span class="md-ellipsis">
8.2   힙 구성 연산
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/top_k/">
<span class="md-ellipsis">
8.3   Top-k 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/summary/">
<span class="md-ellipsis">
8.4   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/exercises/">
<span class="md-ellipsis">
8.5   연습 문제
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</a>
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</nav>
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<input class="md-nav__toggle md-toggle" id="__nav_11" type="checkbox"/>
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제 9 장   그래프
</span>
</a>
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<span class="md-nav__icon md-icon"></span>
제 9 장   그래프
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph/">
<span class="md-ellipsis">
9.1   그래프
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph_operations/">
<span class="md-ellipsis">
9.2   그래프의 기본 연산
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph_traversal/">
<span class="md-ellipsis">
9.3   그래프 순회
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/summary/">
<span class="md-ellipsis">
9.4   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/exercises/">
<span class="md-ellipsis">
9.5   연습 문제
</span>
</a>
</li>
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<span class="md-ellipsis">
제 10 장   탐색
</span>
</a>
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
제 10 장   탐색
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search/">
<span class="md-ellipsis">
10.1   이진 탐색
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search_insertion/">
<span class="md-ellipsis">
10.2   이진 탐색으로 삽입 위치 찾기
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search_edge/">
<span class="md-ellipsis">
10.3   이진 탐색으로 경계 찾기
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/replace_linear_by_hashing/">
<span class="md-ellipsis">
10.4   해시를 이용한 최적화 전략
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/searching_algorithm_revisited/">
<span class="md-ellipsis">
10.5   탐색 알고리즘 다시 보기
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/summary/">
<span class="md-ellipsis">
10.6   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/exercises/">
<span class="md-ellipsis">
10.7   연습 문제
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle" id="__nav_13" type="checkbox"/>
<div class="md-nav__link md-nav__container">
<a class="md-nav__link" href="../../chapter_sorting/">
<svg viewbox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2z"></path></svg>
<span class="md-ellipsis">
제 11 장   정렬
</span>
</a>
<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
제 11 장   정렬
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/sorting_algorithm/">
<span class="md-ellipsis">
11.1   정렬 알고리즘
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/selection_sort/">
<span class="md-ellipsis">
11.2   선택 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/bubble_sort/">
<span class="md-ellipsis">
11.3   버블 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/insertion_sort/">
<span class="md-ellipsis">
11.4   삽입 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/quick_sort/">
<span class="md-ellipsis">
11.5   퀵 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/merge_sort/">
<span class="md-ellipsis">
11.6   병합 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/heap_sort/">
<span class="md-ellipsis">
11.7   힙 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/bucket_sort/">
<span class="md-ellipsis">
11.8   버킷 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/counting_sort/">
<span class="md-ellipsis">
11.9   계수 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/radix_sort/">
<span class="md-ellipsis">
11.10   기수 정렬
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/summary/">
<span class="md-ellipsis">
11.11   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/exercises/">
<span class="md-ellipsis">
11.12   연습 문제
</span>
</a>
</li>
</ul>
</nav>
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<span class="md-ellipsis">
제 12 장   분할 정복
</span>
</a>
<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
제 12 장   분할 정복
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/divide_and_conquer/">
<span class="md-ellipsis">
12.1   분할 정복 알고리즘
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/binary_search_recur/">
<span class="md-ellipsis">
12.2   분할 정복 탐색 전략
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/build_binary_tree_problem/">
<span class="md-ellipsis">
12.3   이진 트리 구성 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/hanota_problem/">
<span class="md-ellipsis">
12.4   하노이의 탑 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/summary/">
<span class="md-ellipsis">
12.5   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/exercises/">
<span class="md-ellipsis">
12.6   연습 문제
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
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제 13 장   백트래킹
</span>
</a>
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제 13 장   백트래킹
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/backtracking_algorithm/">
<span class="md-ellipsis">
13.1   백트래킹 알고리즘
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/permutations_problem/">
<span class="md-ellipsis">
13.2   모든 순열 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/subset_sum_problem/">
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13.3   부분 집합 합 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/n_queens_problem/">
<span class="md-ellipsis">
13.4   n-퀸 문제
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/summary/">
<span class="md-ellipsis">
13.5   요약
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/exercises/">
<span class="md-ellipsis">
13.6   연습 문제
</span>
</a>
</li>
</ul>
</nav>
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제 14 장   동적 계획법
</span>
</a>
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제 14 장   동적 계획법
</label>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_dynamic_programming/intro_to_dynamic_programming/">
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14.1   동적 계획법 첫걸음
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_dynamic_programming/dp_problem_features/">
<span class="md-ellipsis">
14.2   동적 계획법 문제의 특성
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_dynamic_programming/dp_solution_pipeline/">
<span class="md-ellipsis">
14.3   동적 계획법 문제 풀이 방법
</span>
</a>
</li>
<li class="md-nav__item">
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<h1 id="21">2.1   알고리즘 효율 평가<a class="headerlink" href="#21" title="고유 링크"></a></h1>
<p>알고리즘을 설계할 때는 다음 두 단계의 목표를 차례로 추구한다.</p>
<ol>
<li><strong>문제의 해법 찾기</strong>: 알고리즘은 정해진 입력 범위 안에서 문제의 올바른 해를 안정적으로 구해야 한다.</li>
<li><strong>최적의 해법 찾기</strong>: 같은 문제에도 여러 해법이 있을 수 있으므로, 가능한 한 효율적인 알고리즘을 찾고자 한다.</li>
</ol>
<p>즉, 문제를 해결할 수 있다는 전제하에 알고리즘 효율은 알고리즘의 우수성을 평가하는 주요 지표가 된다. 여기에는 다음 두 가지 측면이 있다.</p>
<ul>
<li><strong>시간 효율</strong>: 알고리즘의 실행 시간이 얼마나 긴가.</li>
<li><strong>공간 효율</strong>: 알고리즘이 차지하는 메모리 공간이 얼마나 큰가.</li>
</ul>
<p>간단히 말해, <strong>우리의 목표는 ‘빠르면서도 메모리를 적게 사용하는’ 자료 구조와 알고리즘을 설계하는 것이다</strong>. 알고리즘 효율을 효과적으로 평가하는 일도 매우 중요하다. 그래야 여러 알고리즘을 비교하고, 이를 바탕으로 알고리즘을 설계하고 최적화할 수 있기 때문이다.</p>
<p>효율 평가 방법은 크게 실제 측정과 이론적 추정으로 나뉜다.</p>
<h2 id="211">2.1.1   실제 측정<a class="headerlink" href="#211" title="고유 링크"></a></h2>
<p>같은 문제를 해결할 수 있는 알고리즘 <code>A</code>와 알고리즘 <code>B</code>가 있으며, 두 알고리즘의 효율을 비교해야 한다고 하자. 가장 직접적인 방법은 컴퓨터에서 두 알고리즘을 실행하고 실행 시간과 메모리 사용량을 관찰해 기록하는 것이다. 이러한 평가 방식은 실제 상황을 반영할 수 있지만 한계도 크다.</p>
<p>한편으로는 <strong>측정 환경에 따른 영향을 배제하기 어렵다</strong>. 하드웨어 구성은 알고리즘 성능에 영향을 준다. 예를 들어 병렬성이 높은 알고리즘은 멀티코어 CPU에서 실행하기에 더 적합하고, 메모리 연산이 집중된 알고리즘은 고성능 메모리에서 더 좋은 성능을 보인다. 즉, 같은 알고리즘도 컴퓨터에 따라 측정 결과가 다를 수 있다. 따라서 다양한 컴퓨터에서 측정해 평균 효율을 구해야 하지만, 이는 현실적이지 않다.</p>
<p>다른 한편으로는 <strong>충분한 측정을 수행하는 데 매우 많은 자원이 든다</strong>. 입력 데이터의 양이 달라지면 알고리즘의 효율도 달라진다. 예를 들어 입력 데이터가 적을 때는 알고리즘 <code>A</code>의 실행 시간이 알고리즘 <code>B</code>보다 짧지만, 입력 데이터가 많을 때는 결과가 정반대일 수 있다. 따라서 설득력 있는 결론을 얻으려면 다양한 크기의 입력 데이터를 측정해야 하며, 여기에는 많은 컴퓨팅 자원이 필요하다.</p>
<h2 id="212">2.1.2   이론적 추정<a class="headerlink" href="#212" title="고유 링크"></a></h2>
<p>실제 측정에는 큰 한계가 있으므로, 계산만으로 알고리즘 효율을 평가하는 방법을 생각해 볼 수 있다. 이러한 추정 방법을 <u>점근적 복잡도 분석(asymptotic complexity analysis)</u>, 줄여서 <u>복잡도 분석</u>이라고 한다.</p>
<p>복잡도 분석은 알고리즘 실행에 필요한 시간 및 공간 자원과 입력 데이터 크기의 관계를 나타낸다. <strong>입력 데이터 크기가 증가할 때 알고리즘 실행에 필요한 시간과 공간이 증가하는 추세를 설명한다</strong>. 이 정의는 다소 이해하기 어려우므로 세 가지 핵심으로 나누어 살펴보자.</p>
<ul>
<li>‘시간 및 공간 자원’은 각각 <u>시간 복잡도(time complexity)</u>와 <u>공간 복잡도(space complexity)</u>에 해당한다.</li>
<li>‘입력 데이터 크기가 증가할 때’는 복잡도가 알고리즘의 실행 효율과 입력 데이터 크기 사이의 관계를 반영한다는 뜻이다.</li>
<li>‘시간과 공간이 증가하는 추세’는 복잡도 분석이 실행 시간이나 사용 공간의 구체적인 값보다 시간 또는 공간이 얼마나 빠르게 증가하는지에 주목한다는 뜻이다.</li>
</ul>
<p><strong>복잡도 분석은 실제 측정 방법의 단점을 극복한다</strong>. 구체적으로 다음과 같은 장점이 있다.</p>
<ul>
<li>실제로 코드를 실행하지 않아도 되므로 에너지를 절약하고 환경 부담을 줄인다.</li>
<li>측정 환경에 의존하지 않아 분석 결과를 모든 실행 플랫폼에 적용할 수 있다.</li>
<li>데이터 양에 따른 알고리즘 효율, 특히 데이터가 많을 때의 알고리즘 성능을 보여 준다.</li>
</ul>
<div class="admonition tip">
<p class="admonition-title">참고</p>
<p>복잡도 개념이 아직 혼란스러워도 걱정할 필요 없다. 이후 절에서 자세히 소개한다.</p>
</div>
<p>복잡도 분석은 알고리즘 효율을 평가하는 ‘자’와 같다. 이를 통해 알고리즘 실행에 필요한 시간과 공간 자원을 가늠하고 여러 알고리즘의 효율을 비교할 수 있다.</p>
<p>복잡도는 수학적 개념이므로 초보자에게는 추상적으로 느껴지고 배우기도 다소 어려울 수 있다. 이런 관점에서는 복잡도 분석이 맨 처음 소개하기에 적절하지 않을 수도 있다. 그러나 어떤 자료 구조나 알고리즘의 특징을 논할 때는 실행 속도와 공간 사용량에 대한 분석을 피하기 어렵다.</p>
<p>따라서 자료 구조와 알고리즘을 깊이 학습하기 전에, <strong>복잡도 분석을 기초적으로 이해하여 간단한 알고리즘의 복잡도를 분석할 수 있도록 하는 것을 권한다</strong>.</p>
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