"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", code: "code", figure: "figure", h2: "h2", img: "img", li: "li", math: "math", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", ol: "ol", p: "p", pre: "pre", semantics: "semantics", span: "span", table: "table", tbody: "tbody", td: "td", th: "th", thead: "thead", tr: "tr", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: _jsx(_components.a, { href: "https://www.acmicpc.net/problem/2616", children: "문제 링크" }) }), "\n", _jsx(_components.h2, { children: "문제" }), "\n", _jsx(_components.p, { children: "기차는 맨 앞에 있는 기관차 1대가 손님이 탄 객차 여러 칸을 끌고 간다. 기관차가 고장나면 기차를 운행할 수 없게 되므로 최근 철도청은 기관차 고장에 대비하여 몇몇 역에 소형 기관차 3대를 배치하기로 결정하였다. 소형 기관차는 평소에 이용하는 기관차보다 훨씬 적은 수의 객차만을 끌 수 있다." }), "\n", _jsx(_components.p, { children: "기관차가 고장났을 때 끌고 가던 객차 모두를 소형 기관차 3대가 나누어 끌 수 없기 때문에, 소형 기관차들이 어떤 객차들을 끌고 가는 것이 좋을까하는 문제를 고민하다가 다음과 같이 하기로 결정하였다." }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "소형 기관차가 최대로 끌 수 있는 객차의 수를 미리 정해 놓고, 그보다 많은 수의 객차를 절대로 끌게 하지 않는다. 3대의 소형 기관차가 최대로 끌 수 있는 객차의 수는 서로 같다." }), "\n", _jsx(_components.li, { children: "소형 기관차 3대를 이용하여 최대한 많은 손님을 목적지까지 운송하도록 한다. 각 객차 마다 타고 있는 손님의 수는 미리 알고 있고, 다른 객차로 손님들이 이동하는 것은 허용하지 않는다." }), "\n", _jsx(_components.li, { children: "각 소형 기관차는 번호가 연속적으로 이어진 객차를 끌게 한다. 객차는 기관차 바로 뒤에 있는 객차부터 시작하여 1번 부터 차례로 번호가 붙어있다." }), "\n"] }), "\n", _jsx(_components.p, { children: "예를 들어 기관차가 끌고 가던 객차가 7칸이고, 소형 기관차 1대가 최대로 끌 수 있는 객차 수는 2칸이라고 하자. 그리고 1번 부터 7번까지 각 객차에 타고 있는 손님의 수가 아래 표와 같다고 하자. 괄호속에 있는 숫자는 객차 번호를 나타낸다." }), "\n", _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsx(_components.th, { children: "(1)" }), _jsx(_components.th, { children: "(2)" }), _jsx(_components.th, { children: "(3)" }), _jsx(_components.th, { children: "(4)" }), _jsx(_components.th, { children: "(5)" }), _jsx(_components.th, { children: "(6)" }), _jsx(_components.th, { children: "(7)" })] }) }), _jsx(_components.tbody, { children: _jsxs(_components.tr, { children: [_jsx(_components.td, { children: "35" }), _jsx(_components.td, { children: "40" }), _jsx(_components.td, { children: "50" }), _jsx(_components.td, { children: "10" }), _jsx(_components.td, { children: "30" }), _jsx(_components.td, { children: "45" }), _jsx(_components.td, { children: "60" })] }) })] }), "\n", _jsx(_components.p, { children: "소형 기관차 3대는 각각 1-2번, 3-4번, 그리고 6-7번 객차를 끌고 가면 손님 240명을 운송할 수 있고, 이보다 많은 수의 손님을 운송할 수 없다." }), "\n", _jsx(_components.p, { children: "기관차가 끌고 가던 객차의 수와 각 객차에 타고 있던 손님의 수, 그리고 소형 기관차가 최대로 끌수 있는 객차의 수가 주어질 때, 소형 기관차 3대를 이용하여 최대로 운송할 수 있는 손님 수를 구하는 프로그램을 작성하시오." }), "\n", _jsx(_components.h2, { children: "입력" }), "\n", _jsx(_components.p, { children: "첫째 줄에 기관차가 끌고 가던 객차의 수가 입력된다. 그 수는 50,000 이하이다. 둘째 줄에는 기관차가 끌고 가던 객차에 타고 있는 손님의 수가 1번 객차부터 차례로 입력된다. 한 객차에 타고 있는 손님의 수는 100명 이하이고, 입력되는 숫자들 사이에 빈칸이 하나씩 있다. 셋째 줄에는 소형 기관차가 최대로 끌 수 있는 객차의 수가 입력된다. 그 수는 기관차가 끌고 가던 객차 수의 1/3보다 적다." }), "\n", _jsx(_components.h2, { children: "출력" }), "\n", _jsx(_components.p, { children: "한 줄에 소형 기관차 3대를 이용하여 최대로 운송할 수 있는 손님 수를 출력한다." }), "\n", _jsx(_components.h2, { children: "풀이" }), "\n", _jsxs(_components.p, { children: ["연속 구간 합이 반복해서 필요하므로 누적합을 먼저 만든다. 이후 ", _jsx(_components.code, { children: "solve(train, car)" }), "를 \"", _jsx(_components.code, { children: "car" }), "번 객차부터 볼 때, ", _jsx(_components.code, { children: "train" }), "번째 기관차부터 얻을 수 있는 최대 승객 수\"로 두는 DP를 사용한다."] }), "\n", _jsxs(_components.p, { children: ["각 위치에서는 두 가지 선택만 있다. 현재 객차를 건너뛰고 다음 칸으로 가거나, 지금부터 길이 ", _jsx(_components.code, { children: "k" }), "의 구간을 하나 선택하고 다음 기관차로 넘어가는 것이다. 두 경우 중 큰 값을 택하면 된다."] }), "\n", _jsx(_components.p, { children: "기관차는 총 3대뿐이라 상태 수가 크지 않고, 메모이제이션으로 충분히 해결된다." }), "\n", _jsxs(_components.p, { children: ["현재 칸에서 기관차 하나를 쓰기로 했다면, 길이 ", _jsx(_components.code, { children: "k" }), "의 객차를 반드시 연속으로 묶어야 한다. 이 구간의 승객 수는 누적합으로 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(1)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "에 계산할 수 있고, 다음 상태는 ", _jsx(_components.code, { children: "train + 1" }), ", ", _jsx(_components.code, { children: "car + k" }), "가 된다. 반대로 지금 칸을 사용하지 않으면 ", _jsx(_components.code, { children: "car + 1" }), "로만 넘어가면 된다."] }), "\n", _jsx(_components.p, { children: "즉 이 문제는 \"지금 여기서 한 대를 쓰는 것이 이득인가, 다음 칸으로 넘기는 것이 이득인가\"를 반복해서 판단하는 구간 DP다. 기관차 수가 3대로 고정되어 있다는 점 덕분에 완전탐색처럼 보이는 경우의 수도 충분히 줄어든다." }), "\n", _jsx(_components.h2, { children: "코드" }), "\n", _jsx(_components.figure, { "data-rehype-pretty-code-figure": "", children: _jsx(_components.pre, { style: { backgroundColor: "#2e3440ff", color: "#d8dee9ff" }, tabIndex: "0", "data-language": "cpp", "data-theme": "nord", children: _jsxs(_components.code, { "data-language": "cpp", "data-theme": "nord", style: { display: "grid" }, children: [_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: "iostream" }), _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: "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: "algorithm" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ">" })] }), "\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: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " k" }), _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: " sum" }), _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: "vector" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ">" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " dp" }), _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: "#88C0D0" }, children: " solve" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " train" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " int" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " car" }), _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: "\tif" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "train " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ">" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 3" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " return" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 0" }), _jsx(_components.span, { style: { color: "#81A1C1" }, 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: "#D8DEE9FF" }, children: "car " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "+" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " k " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "-" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 1" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " >" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " n" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " return" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 0" }), _jsx(_components.span, { style: { color: "#81A1C1" }, 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: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "train" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "car" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " !=" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " -" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "1" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " return" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " dp" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "train" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "car" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#616E88" }, children: "\t// 객차를 건너뛴 경우" }) }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\tint" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " skip " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "=" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " solve" }), 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color: "#D8DEE9FF" }, children: "\t" }) }), "\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: "1" }), _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: ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(3n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] })] }), "\n", _jsxs(_components.li, { children: ["공간 복잡도: ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(3n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), 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BOJ 2616 - 소형기관차