"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", msub: "msub", ol: "ol", p: "p", pre: "pre", semantics: "semantics", span: "span", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: _jsx(_components.a, { href: "https://www.acmicpc.net/problem/22988", children: "문제 링크" }) }), "\n", _jsx(_components.h2, { children: "문제" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/generated/posts/algorithm/boj-22988/assets/img1.png", alt: "" }) }), "\n", _jsx(_components.p, { children: "고등학생 때였다. 친구의 손에 이끌려 처음으로 가게 된 화장품 가게는 문을 열자마자 은은한 향기가 코끝을 스치는 곳이었다. 친구를 따라가기에 급급했던 한별이의 발걸음은 이제 누가 말하지 않아도 무언가에 빨려 들어간 듯이 앞을 향했다. 파운데이션을 바르고, 블러셔를 두드리고. 거울을 본 한별이는 처음 보는 자신의 모습에 푹 빠져버렸다. 얼굴에 띈 홍조는 블러셔가 무색해질 정도였다. 그런 기분에 감화된 탓인지 화장품 가게를 나서서 집에 돌아갈 때까지도 거울의 그 모습을 잊을 수가 없었다. 마치 자기가 주목받는 느낌이 들어 볼이 한번 더 붉어져 왔다. 한별이는 이 기분을 다른 사람에게도 전해주고 싶어서 나중에 꼭 화장품 가게를 열겠다고 다짐했다." }), "\n", _jsx(_components.p, { children: "그동안 이룬 결실도, 못다 한 각오도 있었다. 하지만 화장품 가게를 열겠다는 노력의 결과는 눈앞으로 다가와서, 내일은 한별이의 화장품 가게가 개점한 지 꼬박 1년 되는 날이 된다. 한별이의 화장품 가게는 만들어진 지 얼마 안 되었지만 많은 인기를 얻어 멀리서까지도 화장품을 사러 찾아온다. 이 화장품 가게의 인기 상품은 총 용량이 X㎖인 헤어에센스이고, 특유의 매력적인 딸기향은 모발의 상처만큼이나 마음의 상처를 치유하는 데에도 유용하다." }), "\n", _jsx(_components.p, { children: "한별이의 화장품 가게에서는 불필요한 쓰레기를 줄이자는 재활용 캠페인을 하고 있다. 한별이의 가게의 모든 상품은 재활용이 가능한 용기에 담긴다. 가게로 사용하다 남은 헤어에센스 용기 두 개를 반납하면 새로운 용기에다가 남은 헤어에센스를 모아 주고, 추가로 총 용량의 절반만큼의 헤어에센스를 추가로 채워준다. 단, 총 용량을 넘쳐서 채워주지는 않는다. 다시 말해, 용량이 각각 A㎖와 B㎖ 남은 헤어에센스를 가져가면 min(A + B + X / 2, X)㎖의 헤어에센스가 담긴 새로운 용기로 바꿔준다." }), "\n", _jsx(_components.p, { children: "한별이의 화장품 가게 단골인 히나는 이제까지 사모은 헤어에센스 용기가 N개 있다. i번째 용기에는 헤어에센스가 Cᵢ㎖ 담겨 있다. 히나는 한별이의 화장품 가게에 적당한 순서로 헤어에센스를 교환해서 용량이 꽉 찬, 즉 X㎖가 담겨 있는 헤어에센스 용기를 최대한 많이 만들고 싶다. 히나가 용량이 꽉 찬 헤어에센스를 최대 몇 개 만들 수 있는지 알려주자." }), "\n", _jsx(_components.h2, { children: "입력" }), "\n", _jsx(_components.p, { children: "다음과 같이 입력이 주어진다." }), "\n", _jsx(_components.p, { children: "N X" }), "\n", _jsx(_components.p, { children: "C₁ C₂ ... Cₙ" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "N은 히나가 가진 헤어에센스 용기의 수이다. (1 <= N <= 100000)" }), "\n", _jsx(_components.li, { children: "X는 헤어에센스 용기의 총 용량이다. (1 <= X <= 10^18)" }), "\n", _jsx(_components.li, { children: "Cᵢ는 i번째 용기에 담겨 있는 헤어에센스의 용량이 Cᵢ㎖라는 의미이다. (0 <= Cᵢ <= X)" }), "\n", _jsx(_components.li, { children: "입력으로 주어지는 모든 수는 정수다." }), "\n"] }), "\n", _jsx(_components.h2, { children: "출력" }), "\n", _jsx(_components.p, { children: "한별이의 화장품 가게에 가서 적당한 순서로 헤어에센스를 교환해서 용량이 꽉 찬 헤어에센스 용기를 몇 개 만들 수 있는지 출력하여라." }), "\n", _jsx(_components.h2, { children: "풀이" }), "\n", _jsx(_components.p, { children: "핵심은 꽉 찬 용기를 만드는 경우를 세 가지로 나눠 생각하는 것이다." }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["이미 ", _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: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "X" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_i = X" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0785em" }, children: "X" })] })] })] }), "인 용기는 그대로 정답에 포함된다."] }), "\n", _jsxs(_components.li, { children: ["두 용기 ", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), ", ", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "B" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "B" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" })] }) })] }), "에 대해 ", _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: "A" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "B" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mi, { children: "X" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A + B + X / 2 \\ge X" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _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: "0.0785em" }, children: "X" }), _jsx(_components.span, { className: "mord", children: "/2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0785em" }, children: "X" })] })] })] }), "이면 한 번의 교환으로 꽉 찬 용기를 하나 만들 수 있다. 이는 ", _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.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "A" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "B" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mi, { children: "X" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2 \\times (A + B) \\ge X" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _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: "0.0502em" }, children: "B" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0785em" }, children: "X" })] })] })] }), "와 같으므로, 정렬 뒤 작은 값과 큰 값을 투 포인터로 묶어 최대한 많이 처리할 수 있다."] }), "\n", _jsxs(_components.li, { children: ["위 두 경우를 처리하고도 남은 용기들은 어느 두 개를 골라도 ", _jsx(_components.code, { children: "X / 2" }), "를 넘기지 못한다. 이때는 세 개를 모으면 최소한 하나의 꽉 찬 용기를 만들 수 있으므로, 남은 개수를 3으로 나눈 몫만큼 정답에 더할 수 있다."] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["코드에서는 먼저 ", _jsx(_components.code, { children: "lower_bound" }), "로 ", _jsx(_components.code, { children: "X" }), " 이상인 첫 위치를 찾아 이미 꽉 찬 용기의 개수를 바로 센다. 그보다 작은 값들만 대상으로 투 포인터를 돌리면서 ", _jsx(_components.code, { children: "2 * (C[left] + C[right]) >= X" }), "를 만족하면 한 쌍을 사용해 정답을 증가시킨다. 이렇게 짝지어 사용한 용기를 제외하고 남은 용기 수를 세어 ", _jsx(_components.code, { children: "remain / 3" }), "을 더하면 된다."] }), "\n", _jsxs(_components.p, { children: ["또한 ", _jsx(_components.code, { children: "X" }), "의 범위가 ", _jsx(_components.code, { children: "10^18" }), "까지 가능하므로 ", _jsx(_components.code, { children: "long long" }), " 사용이 필요하다. 비교도 ", _jsx(_components.code, { children: "X / 2" }), "를 직접 쓰기보다 ", _jsx(_components.code, { children: "(A + B) * 2 >= X" }), " 형태로 처리해 정수 연산으로 깔끔하게 비교했다."] }), "\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, { 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stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(N \\log N)" })] }) }) }), _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 mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, 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.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(N)" })] }) }) }), _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 mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] })] }), "\n"] }), "\n", _jsx(_components.h2, { children: "마무리" }), "\n", _jsxs(_components.p, { children: ["정렬 이후에는 이미 꽉 찬 용기, 두 개를 묶어 꽉 찰 수 있는 경우, 끝까지 남는 용기 세 그룹이라는 세 단계로 정리가 된다. 조건을 ", _jsx(_components.code, { children: "2 * (A + B) >= X" }), "로 바꿔서 보면 투 포인터 판단도 단순해져 구현이 깔끔하다."] }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/generated/posts/algorithm/boj-22988/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 22988 - 재활용 캠페인