大斐波那契数

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题目描述

定义：
f(1)=1, f(2)=1, f(n>2)=f(n-1)+f(n-2)
我们把符合以上定义的序列称为斐波那契序列，现在给你一个数字n，请你求出f(n)。
输入格式

输入包含多组测试数据。每组数据为一个正整数n。
输出

输出对应的f(n)。题目保证结果不会超过1000位数字。
样例输入

100
样例输出

354224848179261915075

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对于斐波拉契数列，一看就会想到递归，我们直接看递归式：

然后编写递归程序，当数据比较大时，我们采用Java的大数运算表示：

package org.jian.graph;import java.math.BigInteger;import java.util.Scanner;public class F2 {private static BigInteger f1 = new BigInteger("1") ;private static BigInteger f2 = new BigInteger("2") ;public static void main(String[] args) {Scanner s = new Scanner(System.in);while (s.hasNext()) {BigInteger n = s.nextBigInteger();long start = System.currentTimeMillis() ;System.out.println(f(n));long time = System.currentTimeMillis()-start ;System.out.println(time+"ms");}}public static BigInteger f(BigInteger n) {if(n.equals(f1)||n.equals(f2)){return new BigInteger("1") ;}else {return f(n.subtract(f1)).add(f(n.subtract(f2))) ;}}}

运行结果：

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35
9227465
2900ms

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显然仅仅只是很小的一个数，所耗时间就很大了。

当然，这里的大数运算也许会影响一下性能，我们换成普通的运算：

package org.jian.graph;import java.util.Scanner;public class F3 {public static void main(String[] args) {Scanner s = new Scanner(System.in);while (s.hasNext()) {int n = s.nextInt();long start = System.currentTimeMillis() ;System.out.println(f(n));long time = System.currentTimeMillis()-start ;System.out.println(time+"ms");}}public static int f(int n) {if(n==1||n==2){return 1 ;}else {return f(n-1)+f(n-2) ;}}}

运算结果：

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35
9227465
85ms

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显然比大数运算的快很多，但数字大一点就无能为力了。

而我们换一种非递归的方式：

package org.jian.graph;import java.math.BigInteger;import java.util.Scanner;public class F {public static void main(String[] args) {Scanner s = new Scanner(System.in);while (s.hasNext()) {int n = s.nextInt();long start = System.currentTimeMillis() ;System.out.println(f(n));long time = System.currentTimeMillis()-start ;System.out.println(time+"ms");}}public static BigInteger f(int n) {BigInteger f1 = new BigInteger("1");BigInteger f2 = new BigInteger("1");BigInteger temp = new BigInteger("0");for (int i = 2; i < n; i++) {temp = f2;f2 = f1.add(f2);f1 = temp;}return f2;}}

运行结果：

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35
9227465
3ms

100
354224848179261915075
1ms

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当我们都把n运行35时，所用时间相差上千倍；

我们做一个更大的测试，当N=10000时，第二种方法所需的时间：

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10000
336447648764317832666216120051075433103021484606800639065647699746800814421666623681555955136337340255820653326808361593737347904838652682630408924630564318873545443695598274916066020998841839338646527

313000888302692356736131351175792974378544137521305205043477016022647583189065278908551543661595829872796829875106312005754287834532155151038708182989697916131278562650331954871402142875326981879620469

360978799003509623022910263681314931952756302278376284415403605844025721143349611800230912082870460889239623288354615057765832712525460935911282039252853934346209042452489294039017062338889910858410651

831733604374707379085526317643257339937128719375877468974799263058370657428301616374089691784263786242128352581128205163702980893320999057079200643674262023897831114700540749984592503606335609338838319

233867830561364353518921332797329081337326426526339897639227234078829281779535805709936910491754708089318410561463223382174656373212482263830921032977016480547262438423748624114530938122065649140327510

866433945175121615265453613331113140424368548051067658434935238369596534280717687753283482343455573667197313927462736291082106792807847180353291311767789246590899386354593278945237776744061922403376386

740040213303432974969020283281459334188268176838930720036347956231171031012919531697946076327375892535307725523759437884345040677155557790564504430166401194625809722167297586150269684431469520346149322

911059706762432685159928347098912847067408620085871350162603120719031720860940812983215810772820763531866246112782455372085323653057759564300725177443150515396009051686032203491632226408852488524331580

515348496224348482993809050704834824493274537326245677558790891871908036620580095947431500524025327097469953187707243768259074199396322659841474981936092852239450397071654431564213281576889080587831834

0491743455627052022356484649519611246026831397097506938264870661326450766507461151267752274862159864253071129844118262266105716351506926002986170494542504749137811515413994155067125627119713325276363193

9606902895650288268608362241082050562430701794976171121233066073310059947366875

56ms

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运算结果非常庞大，但运算时间却也是非常小。