"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", mo: "mo", mrow: "mrow", 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/17619", children: "문제 링크" }) }), "\n", _jsx(_components.h2, { children: "문제" }), "\n", _jsx(_components.p, { children: "통나무 N개가 가로 (수평) 방향으로 연못에 떠 있다. 개구리는 한 통나무 A에서 다른 통나무 B로 정확히 수직 방향으로 점프할 수 있다. 단, 점프할 때 다른 통나무 위를 (끝 점 포함) 지나면 안된다." }), "\n", _jsx(_components.p, { children: "예를 들어 <그림 1>에서 1번 통나무에서 2번 통나무로 점선을 따라 개구리가 점프하는 것이 가능하다. 1번 통나무에서 2번 통나무로 점프한 후 다시 3번 통나무로 점프하면 1번 통나무에서 3번 통나무로 이동하는 것이 가능하다. (통나무 위에서 걸어서 움직이는 것은 언제든 가능하다.)" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/generated/posts/algorithm/boj-17619/assets/img1.png", alt: "" }) }), "\n", _jsx(_components.p, { children: "<그림 1>" }), "\n", _jsx(_components.p, { children: "통나무들의 위치를 입력받아 질문으로 주어진 통나무들의 쌍에 대해서 개구리가 한 통나무에서 다른 통나무로 한번 이상의 점프로 이동이 가능한지 판단하는 프로그램을 작성하라." }), "\n", _jsx(_components.h2, { children: "입력" }), "\n", _jsx(_components.p, { children: "첫 번째 줄에 통나무 개수 N과 질문의 개수 Q가 주어진다. 다음 N개의 줄에 각 통나무의 x₁, x₂, y가 주어진다. 주어진 통나무는 두 점 (x₁, y)와 (x₂, y)를 잇는 형태이며, 항상 x₁ < x₂를 만족한다. 모든 좌표는 0 이상 10^9 이하이다. 통나무들은 주어진 순서대로 1번부터 번호가 붙어 있다. 서로 다른 두 통나무는 끝점에서도 만나지 않는다. 다음 Q개의 줄에 서로 다른 두 통나무의 번호가 주어진다. (1 <= N <= 100,000, 1 <= Q <= 100,000)" }), "\n", _jsx(_components.h2, { children: "출력" }), "\n", _jsx(_components.p, { children: "Q개의 줄을 출력한다. 각 줄에는 주어진 순서대로 질문에 대한 대답이 출력되어야 한다. 질문에 주어진 두 통나무에 대해서 개구리가 한 통나무에서 다른 통나무로 한번 이상의 점프로 이동이 가능한 경우 대답은 1, 그렇지 않은 경우 대답은 0이다." }), "\n", _jsx(_components.h2, { children: "풀이" }), "\n", _jsx(_components.p, { children: "점프 가능 여부를 직접 그래프로 만들면 복잡해 보이지만, 실제로는 x축 구간이 이어져 있는 통나무들은 모두 같은 연결 요소로 볼 수 있다." }), "\n", _jsxs(_components.p, { children: ["코드에서는 통나무를 ", _jsx(_components.code, { children: "x1" }), " 기준으로 정렬한다. 그리고 왼쪽부터 보면서 현재까지 이어진 구간의 가장 큰 ", _jsx(_components.code, { children: "x2" }), " 값을 ", _jsx(_components.code, { children: "maxX2" }), "로 관리한다. 다음 통나무의 ", _jsx(_components.code, { children: "x1" }), "이 ", _jsx(_components.code, { children: "maxX2" }), " 이하라면 x축 구간이 겹치거나 닿아 있으므로 같은 그룹으로 union 한다."] }), "\n", _jsxs(_components.p, { children: ["반대로 ", _jsx(_components.code, { children: "x1 > maxX2" }), "라면 이전 묶음과는 완전히 끊긴 새로운 그룹이 시작된 것이다. 이렇게 한 번 훑고 나면, 이후 질의는 두 통나무의 대표가 같은지만 보면 된다."] }), "\n", _jsx(_components.p, { children: "즉 이 문제는 기하 문제처럼 보여도, 정렬 + 스위핑 + union-find로 정리할 수 있다." }), "\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": "java", "data-theme": "nord", children: _jsxs(_components.code, { "data-language": "java", "data-theme": "nord", style: { display: "grid" }, children: [_jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "import" }), _jsx(_components.span, { style: { color: "#8FBCBB" }, children: " java" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "." }), _jsx(_components.span, { style: { color: "#8FBCBB" }, children: "io" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "." }), _jsx(_components.span, { style: { 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}), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " =" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " a" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "\t\t\telse" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " parent" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "a" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " =" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " b" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "\t\t}" }) }), "\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: "#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.mi, { children: "N" }), _jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "Q" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(N \\log N + Q \\alpha(N))" })] }) }) }), _jsxs(_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: "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", children: "Q" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _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", _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", _jsx(_components.p, { children: "통나무의 높이보다 중요한 건 x축 구간이 이어지는지 여부였다. 한 번 정렬해서 묶음을 만들고 나면, 이후 질의는 union-find 조회로 아주 가볍게 끝난다." }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/generated/posts/algorithm/boj-17619/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 17619 - 개구리 점프