"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/11444", children: "문제 링크" }) }), "\n", _jsx(_components.h2, { children: "문제" }), "\n", _jsx(_components.p, { children: "피보나치 수는 0과 1로 시작한다. 0번째 피보나치 수는 0이고, 1번째 피보나치 수는 1이다. 그 다음 2번째 부터는 바로 앞 두 피보나치 수의 합이 된다." }), "\n", _jsx(_components.p, { children: "이를 식으로 써보면 Fn = Fn-1 + Fn-2 (n ≥ 2)가 된다." }), "\n", _jsx(_components.p, { children: "n=17일때 까지 피보나치 수를 써보면 다음과 같다." }), "\n", _jsx(_components.p, { children: "0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597" }), "\n", _jsx(_components.p, { children: "n이 주어졌을 때, n번째 피보나치 수를 구하는 프로그램을 작성하시오." }), "\n", _jsx(_components.h2, { children: "입력" }), "\n", _jsx(_components.p, { children: "첫째 줄에 n이 주어진다. n은 1,000,000,000,000,000,000보다 작거나 같은 자연수이다." }), "\n", _jsx(_components.h2, { children: "출력" }), "\n", _jsx(_components.p, { children: "첫째 줄에 n번째 피보나치 수를 1,000,000,007으로 나눈 나머지를 출력한다." }), "\n", _jsx(_components.h2, { children: "풀이" }), "\n", _jsx(_components.p, { children: "피보나치 수는 선형 점화식이라서 행렬 거듭제곱으로 빠르게 구할 수 있다. [[1,1],[1,0]] 행렬의 거듭제곱을 이용하면 n번째 피보나치 수를 O(log n)에 계산할 수 있다." }), "\n", _jsx(_components.p, { children: "코드에서는 빠른 거듭제곱처럼 지수를 절반으로 줄여 가며 행렬을 제곱한다. 짝수 지수는 절반 결과를 두 번 곱하고, 홀수 지수는 한 번 더 기본 행렬을 곱하면 된다." }), "\n", _jsx(_components.p, { children: "점화식을 행렬 형태로 바꿔 빠르게 압축하는 문제다." }), "\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: "#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: "typedef" }), _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: " matrix" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "const" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " long" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " long" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " MAXN " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "=" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 1000000007" }), _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: "//행렬 곱" }) }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "matrix " }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "matrixMultiply" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "matrix" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "&" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " a" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " matrix" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "&" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " b" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, 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: "#D8DEE9FF" }, children: " matrix " }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: "result" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "2" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " vector" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "<" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "long" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " long" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ">(" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: "2" }), _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: "#81A1C1" }, children: ";" })] }), "\n", _jsx(_components.span, { "data-line": "", children: " " }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: " for" }), _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: "#B48EAD" }, children: " 2" }), _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: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " {" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: " for" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " j " }), _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: " j " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 2" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " j" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "++" }), _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: " for" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "int" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " k " }), _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: " k " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "<" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 2" }), _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: "#ECEFF4" }, children: ")" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " {" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9" }, children: " result" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "i" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "j" }), _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: "#D8DEE9" }, children: "a" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "i" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "k" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " *" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " b" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "k" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "j" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "])" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " %" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " MAXN" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ";" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9" }, children: " result" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "[" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "i" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "][" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "j" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "]" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " %=" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " MAXN" }), _jsx(_components.span, { 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: _jsx(_components.span, { style: { color: "#ECEFF4" }, 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: " return" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " result" }), _jsx(_components.span, { 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", _jsx(_components.span, { "data-line": "", children: _jsx(_components.span, { style: { color: "#616E88" }, children: "//행렬 A의 거듭제곱" }) }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#81A1C1" }, children: "void" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " matrixPower" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "matrix" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "&" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " A" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " matrix" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "&" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " result" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "," }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " long" }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: " long" }), _jsx(_components.span, { style: { color: "#D8DEE9" }, children: " n" }), _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: " while" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "n " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: ">" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 0" }), _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: " if" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: " (" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "n " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "%" }), _jsx(_components.span, { style: { color: "#B48EAD" }, children: " 2" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: ")" })] }), "\n", _jsxs(_components.span, { "data-line": "", children: [_jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: " result " }), _jsx(_components.span, { style: { color: "#81A1C1" }, children: "=" }), _jsx(_components.span, { style: { color: "#88C0D0" }, children: " matrixMultiply" }), _jsx(_components.span, { style: { color: "#ECEFF4" }, children: "(" }), _jsx(_components.span, { style: { color: "#D8DEE9FF" }, children: "result" }), _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: ";" 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BOJ 11444 - 피보나치 수 6