https://re-st.github.io/blog/%EC%9D%B4%EC%82%B0%EC%88%98%ED%95%99/ Recent content in 이산수학 on Park, Geon (re-st) Hugo ko-kr Copyright © 2026, Geon Park. Fri, 01 May 2026 18:45:35 +0900 https://re-st.github.io/seminar/%EC%B9%B4%EB%93%9C-%EC%85%94%ED%94%8C%EC%9D%B4-%EC%9E%98-%EB%90%98%EB%8A%94-%EC%8B%9C%EA%B0%84-%EC%9E%84%EC%84%B1%ED%98%81-kaist-%EC%88%98%ED%95%99%EA%B3%BC/ Wed, 27 Feb 2019 16:00:00 +0900 https://re-st.github.io/seminar/%EC%B9%B4%EB%93%9C-%EC%85%94%ED%94%8C%EC%9D%B4-%EC%9E%98-%EB%90%98%EB%8A%94-%EC%8B%9C%EA%B0%84-%EC%9E%84%EC%84%B1%ED%98%81-kaist-%EC%88%98%ED%95%99%EA%B3%BC/ <blockquote> <p>2019-02-27 16:00, KAIST 오픈동방. 발표자: 임성혁 (KAIST 수학분과장) 참고 논문: Shuffling Cards and Stopping Times (&lsquo;86) — David Aldous, Persi Diaconis</p> </blockquote> <h1 id="트럼프-카드를-몇-번-섞어야-잘-섞인-것인가">트럼프 카드를 몇 번 섞어야 잘 섞인 것인가</h1> <p>발표자는 수학과장. Persi Diaconis 교수는 강의하기 전 마술사였음.</p> <p>핵심 질문: 트럼프 카드를 몇 번 섞어야 잘 섞였다고 말할 수 있는가?</p> <h2 id="정의-markov-chain">정의: Markov Chain</h2> <p>\((X_n)_{n=0}^{\infty}\)를 Markov chain이라 할 때:</p> <ul> <li>초기: \(N!\) 크기의 확률 vector (모든 배열 동일 확률)</li> <li>섞기: 다른 상태로 보내주는 함수 \(f\)</li> <li>\(Q^{*k}\): \(k\)번 섞은 후 카드 상태의 분포</li> <li>\(U(\pi) = 1/n!\): 균일 분포</li> </ul> <h2 id="거리-측정">거리 측정</h2> $$\|Q^{*k} - U\| = \frac{1}{2} \sum_{x \in S_n} |Q^{*k}(x) - U(x)| = \max_{A \subseteq S_n} |Q^{*k}(A) - U(A)|$$<ul> <li>0이면 잘 섞임, 1이면 전혀 안 섞임</li> </ul> <h2 id="top-in-at-random-shuffle-toy-model">Top-in-at-random Shuffle (toy model)</h2> <p>맨 위 카드를 빼서 랜덤한 위치에 꽂기.</p> https://re-st.github.io/seminar/%EC%A0%84%EC%82%B0-%EC%84%B8%EB%AF%B8%EB%82%98-%EA%B7%BC%EC%82%AC-%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98%EA%B3%BC-%EC%B5%9C%EC%A0%81%ED%99%94-%EB%AC%B8%EC%A0%9C-2018-%EA%B0%80%EC%9D%84-kaist/ Sat, 01 Sep 2018 00:00:00 +0900 https://re-st.github.io/seminar/%EC%A0%84%EC%82%B0-%EC%84%B8%EB%AF%B8%EB%82%98-%EA%B7%BC%EC%82%AC-%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98%EA%B3%BC-%EC%B5%9C%EC%A0%81%ED%99%94-%EB%AC%B8%EC%A0%9C-2018-%EA%B0%80%EC%9D%84-kaist/ <blockquote> <p>KAIST 전산 세미나, 2018 가을학기. 16pm. 발표자 미상.</p> </blockquote> <h1 id="approximation-algorithm-for-optimization-problem">Approximation Algorithm for Optimization Problem</h1> <h2 id="mathematical-optimization">Mathematical Optimization</h2> <ul> <li><strong>Input</strong>: feasible solutions 집합, objective function \(f\)</li> <li><strong>Output</strong>: \(f(x)\)를 최대화/최소화하는 \(x\)</li> <li><strong>Running time</strong>: \(\text{poly}(|l|)\) — \(l\)을 표현하는 비트 수</li> </ul> <h2 id="least-square-regression">Least Square Regression</h2> <ul> <li>Input: \(A \in \mathbb{R}^{m \times n}\), \(b \in \mathbb{R}^m\)</li> <li>Feasible set: \(x \in \mathbb{R}^n\)</li> <li>\(f(x) = \|Ax - b\|^2\)</li> </ul> <h2 id="discrete-optimization-vertex-cover">Discrete Optimization (Vertex Cover)</h2> <ul> <li>Input: \(G = (V, E)\)</li> <li>Feasible set: \(C \subseteq V\), \(C\)가 모든 edge를 cover</li> <li>\(f(C) = |C|\), 최소화 → <strong>NP-hard</strong> [Karp 72]</li> </ul> <h2 id="핵심-관찰">핵심 관찰</h2> <blockquote> <p>&ldquo;Almost every combinatory algorithm is either NP-complete or Polynomial solvable.&rdquo; — Williams 2013</p>