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제 10 장 &nbsp; 탐색
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10.1 &nbsp; 이진 탐색
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10.2 &nbsp; 이진 탐색으로 삽입 위치 찾기
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10.3 &nbsp; 이진 탐색으로 경계 찾기
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10.4 &nbsp; 해시를 이용한 최적화 전략
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10.5 &nbsp; 탐색 알고리즘 다시 보기
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제 11 장 &nbsp; 정렬
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11.2 &nbsp; 선택 정렬
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11.3 &nbsp; 버블 정렬
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11.4 &nbsp; 삽입 정렬
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11.5 &nbsp; 퀵 정렬
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11.6 &nbsp; 병합 정렬
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11.8 &nbsp; 버킷 정렬
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11.9 &nbsp; 계수 정렬
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11.10 &nbsp; 기수 정렬
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11.11 &nbsp; 요약
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제 12 장 &nbsp; 분할 정복
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12.1 &nbsp; 분할 정복 알고리즘
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12.2 &nbsp; 분할 정복 탐색 전략
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12.3 &nbsp; 이진 트리 구성 문제
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12.4 &nbsp; 하노이의 탑 문제
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12.5 &nbsp; 요약
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제 13 장 &nbsp; 백트래킹
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13.4 &nbsp; n-퀸 문제
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14.2 &nbsp; 동적 계획법 문제의 특성
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<h1 id="64">6.4 &nbsp; 요약<a class="headerlink" href="#64" title="고유 링크">&para;</a></h1>
<h3 id="1">1. &nbsp; 핵심 정리<a class="headerlink" href="#1" title="고유 링크">&para;</a></h3>
<ul>
<li>해시 테이블은 <code>key</code>를 입력받아 <span class="arithmatex">\(O(1)\)</span> 시간에 <code>value</code>를 조회하므로 매우 효율적이다.</li>
<li>대표적인 연산에는 조회, 키-값 쌍 추가와 삭제, 해시 테이블 순회 등이 있다.</li>
<li>해시 함수는 <code>key</code>를 배열 인덱스로 매핑하여 해당 버킷에 접근하고 <code>value</code>를 얻도록 한다.</li>
<li>서로 다른 두 <code>key</code>가 해시 함수를 거쳐 같은 배열 인덱스를 얻으면 잘못된 조회 결과가 발생할 수 있다. 이를 해시 충돌이라고 한다.</li>
<li>해시 테이블 용량이 클수록 충돌 확률이 낮으므로 용량 확장으로 충돌을 완화할 수 있다. 배열과 마찬가지로 해시 테이블의 용량 확장 비용도 크다.</li>
<li>적재율은 원소 수를 버킷 수로 나눈 값이며, 해시 충돌의 심각도를 반영한다. 용량 확장의 시작 조건으로 자주 사용된다.</li>
<li>체이닝은 단일 원소를 연결 리스트로 바꾸어 충돌하는 모든 원소를 같은 리스트에 저장한다. 리스트가 너무 길면 조회 효율이 낮아지므로, 레드-블랙 트리로 바꾸어 효율을 높일 수 있다.</li>
<li>개방 주소법은 여러 차례 탐사하여 해시 충돌을 처리한다. 선형 탐사는 고정 간격을 사용하며, 원소를 삭제할 수 없고 군집이 생기기 쉽다는 단점이 있다. 다중 해싱은 여러 해시 함수로 탐사하므로 군집은 덜 생기지만 계산량이 늘어난다.</li>
<li>프로그래밍 언어마다 해시 테이블 구현이 다르다. 예를 들어 Java의 <code>HashMap</code>은 체이닝을, Python의 <code>dict</code>는 개방 주소법을 사용한다.</li>
<li>해시 테이블의 해시 알고리즘에는 결정성, 높은 효율, 균등 분포가 필요하다. 암호학에서는 충돌 저항성과 눈사태 효과도 갖추어야 한다.</li>
<li>해시 알고리즘은 해시값의 균등 분포를 최대한 보장하고 충돌을 줄이기 위해 보통 큰 소수를 모듈러 연산의 법으로 사용한다.</li>
<li>대표적인 해시 알고리즘으로 MD5, SHA-1, SHA-2, SHA-3 등이 있다. MD5는 파일 무결성 검사에, SHA-2는 보안 응용과 프로토콜에 자주 쓰인다.</li>
<li>프로그래밍 언어는 보통 자료형별 내장 해시 알고리즘을 제공하여 버킷 인덱스를 계산한다. 일반적으로 불변 객체만 해시 가능하다.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="고유 링크">&para;</a></h3>
<p><strong>Q</strong>: 해시 테이블의 시간 복잡도가 <span class="arithmatex">\(O(n)\)</span>이 되는 경우는 언제인가?</p>
<p>해시 충돌이 심하면 시간 복잡도가 <span class="arithmatex">\(O(n)\)</span>으로 나빠진다. 해시 함수를 잘 설계하고 용량을 적절히 정하며 충돌이 비교적 고르게 분포하면 <span class="arithmatex">\(O(1)\)</span>이다. 프로그래밍 언어의 내장 해시 테이블을 사용할 때는 보통 시간 복잡도를 <span class="arithmatex">\(O(1)\)</span>로 본다.</p>
<p><strong>Q</strong>: 해시 함수 <span class="arithmatex">\(f(x) = x\)</span>를 사용하면 충돌이 없을 텐데, 왜 그렇게 하지 않는가?</p>
<p><span class="arithmatex">\(f(x) = x\)</span>에서는 각 원소에 고유한 버킷 인덱스가 대응하므로 배열과 같다. 그러나 입력 공간은 보통 출력 공간(배열 길이)보다 훨씬 크므로, 해시 함수의 마지막 단계에서 대개 배열 길이로 나눈 나머지를 구한다. 즉, 해시 테이블의 목적은 큰 상태 공간을 작은 공간에 매핑하면서 <span class="arithmatex">\(O(1)\)</span>의 조회 효율을 제공하는 것이다.</p>
<p><strong>Q</strong>: 해시 테이블도 배열, 연결 리스트, 이진 트리로 구현되는데 어떻게 이들보다 효율이 높을 수 있는가?</p>
<p>첫째, 시간 효율은 높아지지만 공간 효율은 낮아진다. 해시 테이블에는 사용하지 않는 메모리가 상당 부분 있다.</p>
<p>둘째, 특정 활용 상황에서만 시간 효율이 높아진다. 같은 기능을 배열이나 연결 리스트로 같은 시간 복잡도에 구현할 수 있다면 보통 그쪽이 더 빠르다. 해시 함수 계산에 비용이 들어 시간 복잡도의 상수항이 더 크기 때문이다.</p>
<p>마지막으로 해시 테이블의 시간 복잡도도 나빠질 수 있다. 예를 들어 체이닝에서는 연결 리스트나 레드-블랙 트리에서 탐색하므로, 여전히 <span class="arithmatex">\(O(n)\)</span> 시간으로 나빠질 위험이 있다.</p>
<p><strong>Q</strong>: 다중 해싱도 원소를 곧바로 삭제할 수 없는가? 삭제 표시를 한 공간은 다시 사용할 수 있는가?</p>
<p>다중 해싱은 개방 주소법의 한 종류다. 개방 주소법은 모두 원소를 곧바로 삭제할 수 없으므로 삭제 표시를 해야 한다. 표시된 공간은 다시 사용할 수 있다. 새 원소를 삽입할 때 해시 함수로 삭제 표시가 있는 위치를 찾으면 그곳에 새 원소를 넣을 수 있다. 이렇게 하면 탐사 순서를 유지하면서 공간 활용률도 보장한다.</p>
<p><strong>Q</strong>: 선형 탐사에서는 원소를 찾을 때도 왜 해시 충돌이 발생하는가?</p>
<p>해시 함수로 해당 버킷과 키-값 쌍을 찾았는데 <code>key</code>가 일치하지 않으면 해시 충돌이 있다는 뜻이다. 따라서 미리 정한 간격으로 뒤를 차례로 탐색하다가 올바른 키-값 쌍을 찾거나, 찾을 수 없다고 판단하면 종료한다.</p>
<p><strong>Q</strong>: 해시 테이블의 용량 확장은 왜 해시 충돌을 완화하는가?</p>
<p>해시 함수는 마지막에 대개 배열 길이 <span class="arithmatex">\(n\)</span>으로 나눈 나머지를 구하여 출력이 인덱스 범위에 들어가도록 한다. 용량을 늘리면 <span class="arithmatex">\(n\)</span>이 바뀌고 <code>key</code>에 대응하는 인덱스도 달라질 수 있다. 원래 같은 버킷에 있던 여러 <code>key</code>가 여러 버킷으로 분산될 수 있으므로 충돌이 완화된다.</p>
<p><strong>Q</strong>: 효율적으로 저장하고 접근하려는 목적이라면 배열을 바로 사용하면 되지 않는가?</p>
<p><code>key</code>가 좁은 범위의 연속된 정수라면 배열을 바로 쓰는 것이 간단하고 효율적이다. 하지만 문자열처럼 다른 자료형이면 해시 함수로 <code>key</code>를 배열 인덱스에 매핑하고 버킷 배열에 원소를 저장해야 한다. 이것이 바로 해시 테이블이다.</p>
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