"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", code: "code", figure: "figure", h2: "h2", img: "img", li: "li", p: "p", pre: "pre", span: "span", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: _jsx(_components.a, { href: "https://www.acmicpc.net/problem/16991", children: "문제 링크" }) }), "\n", _jsx(_components.h2, { children: "문제" }), "\n", _jsx(_components.p, { children: "외판원 순회 문제는 영어로 Traveling Salesman problem (TSP) 라고 불리는 문제로 computer science 분야에서 가장 중요하게 취급되는 문제 중 하나이다. 여러 가지 변종 문제가 있으나, 여기서는 가장 일반적인 형태의 문제를 살펴보자." }), "\n", _jsx(_components.p, { children: "1번부터 N번까지 번호가 매겨져 있는 도시들이 있고, 모든 도시 사이에는 길이 있다. 이제 한 외판원이 어느 한 도시에서 출발해 N개의 도시를 모두 거쳐 다시 원래의 도시로 돌아오는 순회 여행 경로를 계획하려고 한다. 단, 한 번 갔던 도시로는 다시 갈 수 없다. (맨 마지막에 여행을 출발했던 도시로 돌아오는 것은 예외) 이런 여행 경로는 여러 가지가 있을 수 있는데, 가장 적은 비용을 들이는 여행 계획을 세우고자 한다." }), "\n", _jsx(_components.p, { children: "도시 A에서 도시 B로 가는 비용은 두 도시 사이의 거리와 같다. 한 도시 A의 좌표가 (xA, yA), B의 좌표가 (xB, yB)라고 한다면, 두 도시의 거리는 √((xB-xA)^2 + (yB-yA)^2)와 같다." }), "\n", _jsx(_components.p, { children: "도시의 수 N과 모든 도시의 위치가 주어졌을 때, 가장 적은 비용을 들이는 외판원의 순회 여행 경로를 구하는 프로그램을 작성하시오." }), "\n", _jsx(_components.h2, { children: "입력" }), "\n", _jsx(_components.p, { children: "첫째 줄에 도시의 수 N이 주어진다. (2 ≤ N ≤ 16) 다음 N개의 줄에는 도시의 좌표 x, y가 주어진다. 모든 좌표는 -1,000보다 크거나 같고, 1,000보다 작거나 같은 정수이다. 두 도시의 위치가 같은 경우는 없다." }), "\n", _jsx(_components.h2, { children: "출력" }), "\n", _jsx(_components.p, { children: "첫째 줄에 외판원의 순회에 필요한 최소 비용을 출력한다. 절대/상대 오차는 10^-6까지 허용한다." }), "\n", _jsx(_components.h2, { children: "풀이" }), "\n", _jsx(_components.p, { children: "모든 도시를 한 번씩 방문하고 다시 시작점으로 돌아와야 하므로, 현재까지 방문한 도시 집합과 현재 위치를 함께 상태로 들고 가는 비트마스크 DP가 잘 맞는다. 단순 순열 완전탐색으로는 경우의 수가 너무 크기 때문이다." }), "\n", _jsx(_components.p, { children: "코드에서는 solve(cur, bitmask)를 현재 cur 도시에 있고, bitmask에 표시된 도시들을 이미 방문했을 때 남은 최소 비용으로 둔다. 모든 도시를 방문한 상태라면 시작점으로 돌아가는 비용을 그대로 반환하고, 아니라면 아직 방문하지 않은 다음 도시를 모두 시도하면서 최소값을 갱신한다." }), "\n", _jsx(_components.p, { children: "도시 사이 거리는 좌표로부터 직접 계산하므로 별도의 그래프 입력이 없어도 된다. 핵심은 같은 방문 집합과 같은 현재 위치 조합은 언제 도달했든 남은 최소 비용이 같다는 점이고, 이 성질 덕분에 메모이제이션이 정확하게 작동한다." }), "\n", _jsx(_components.h2, { children: "코드" }), "\n", _jsx(_components.figure, { "data-rehype-pretty-code-figure": "", children: 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}, children: " <" }), _jsx(_components.span, { style: { color: "#A3BE8C" }, children: "vector" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ">" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#5E81AC", fontWeight: "bold" }, children: "#" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "include" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " <" }), _jsx(_components.span, { style: { color: "#A3BE8C" }, children: "cmath" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ">" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#5E81AC", fontWeight: "bold" }, children: "#" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "include" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " <" }), _jsx(_components.span, { style: { color: "#A3BE8C" }, children: 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"\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "using" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " namespace" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " std" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " n" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "vector" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "pair" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ">" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " cities" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "double" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " dp" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "MAX" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "1" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " <<" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " MAX" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "double" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " getDistance" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "double" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " x1" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " double" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " y1" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " double" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " x2" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " double" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " y2" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " {" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\treturn" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " sqrt" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "((" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "x2 " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "-" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " x1" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: 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color: "#D8DEE9" }, children: " cities" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "0" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]." }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: "second" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" }), _jsx(_components.span, { style: { color: "#616E88" }, children: "\t// 시작 지점으로 돌아가기" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\tif" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: "dp" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), 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color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "bitmask" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" }), _jsx(_components.span, { style: { color: "#616E88" }, children: "\t// 메모리제이션" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9" }, children: "\tdp" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "cur" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "bitmask" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " =" }), 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{ style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "}" }) }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " main" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "()" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " {" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "\tios_base" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "::" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "sync_with_stdio" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), 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children: "\t" }) }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\tfor" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " i " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "=" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 0" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " i " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " n" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " i" }), _jsx(_components.span, { style: 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[_jsx(_components.span, { style: { color: "#D8DEE9" }, children: "\t\tcities" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "." }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "push_back" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "make_pair" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "x" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " y" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "))" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "\t}" }) }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "\t" }) }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#616E88" }, children: "\t// 소수점 6자리까지 표현" }) }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "\tcout " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<<" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " fixed" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9" }, children: "\tcout" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "." }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "precision" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "6" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "\tcout " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<<" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " solve" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "0" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 1" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " <<" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " '" }), _jsx(_components.span, { style: { color: "#EBCB8B" }, children: "\\n" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "'" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\treturn" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 0" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "}" }) })] }) }) }), "\n", _jsx(_components.h2, { children: "복잡도" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["시간 복잡도: 상태 수 기준 ", _jsx(_components.code, { children: "O(N^2 * 2^N)" })] }), "\n", _jsxs(_components.li, { children: ["공간 복잡도: ", _jsx(_components.code, { children: "O(N * 2^N)" })] }), "\n"] }), "\n", _jsx(_components.h2, { children: "마무리" }), "\n", _jsx(_components.p, { children: "외판원 순회는 순서를 전부 세기 시작하면 금방 폭발한다. 방문 집합과 현재 위치만 상태로 남기면 같은 부분 문제를 재사용할 수 있어 비트마스크 DP 구조가 선명하게 드러난다." }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/generated/posts/algorithm/boj-16991/assets/1.png", alt: "" }) })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };

BOJ 16991 - 외판원 순회 3