对比Fibonacci两种算法效率

看数据结构看到Fibonacci，而且说如果用递归的方法来实现的话会很低效，尤其是Index比较大的情况下，遂决定试一下。

代码和输出在下面，观察输出，Index为50的时候，递归时间就已经好久了，再大一点我等不及了。调用次数也是指数级的上升，而与之对比的是，迭代的调用次数和时间始终在一个可控的范围内。有趣的是观察Recurse的调用次数也形成了一个Fibonacci序列的变体，F(N)=F(N-1)+F(N-2)+N-1。

造成递归调用如此低效的原因在于求解F(N)的时候需要算一遍F(N-1)和F(N-2)，求解F(N-1)的时候需要算一遍F(N-2)和F(N-3)，可以看出F(N-2)在求解相邻的两个值的时候被算了两遍，如此不断垒乘下去，会造成很大的浪费，这也违背了递归调用的第四条原则：“

Compound interest rule

Never duplicate work by solving the same instance of a problem in separate recursive calls.

”

</pre><p><pre name="code" class="cpp">//============================================================================
// Name        : myAlgorithm.cpp
// Author      : 
// Version     :
// Copyright   : Your copyright notice
// Description : Hello World in C++, Ansi-style
//============================================================================

#include <iostream>
using namespace std;

#include <time.h>

long iRecurseCalledTimes=0;
long iIterationCalledTimes=0;

long funcRecurse(long n)
{
	iRecurseCalledTimes++;
	if(2==n||1==n)
		return 1;
	else
		return funcRecurse(n-1)+funcRecurse(n-2);
}

long funcIteration(long n)
{
	iIterationCalledTimes++;
	long i0=0,i1=1,result=1;
	if(2==n||1==n)
		return 1;
	else
	for(long i=2;i!=n+1;i++)
	{
		result=i0+i1;
		i0=i1;
		i1=result;
	}
	return result;
}

int main() {
	cout << "!!!Hello World!!!" << endl; // prints !!!Hello World!!!

	unsigned long iTestMax=0;
	cin>>iTestMax;
	clock_t start, finish;
	double  duration[2];
	long lRtn[2];

	cout<<"Index\tR-Time\tI-Time\tR-Calls\tI-Calls\tR-Rtn\tI-Rtn\t"<<endl;
	for(long i=1;i!=iTestMax+1;++i)
	{
		start = clock();
		lRtn[0]=funcRecurse(i);
		finish = clock();
		duration[0] = (double)(finish - start) / CLOCKS_PER_SEC;

		start = clock();
		lRtn[1]=funcIteration(i);
		finish = clock();
		duration[1] = (double)(finish - start) / CLOCKS_PER_SEC;
		cout << i << "\t"<< duration[0] << "\t" << duration[1] << "\t" << iRecurseCalledTimes << "\t" << iIterationCalledTimes << "\t" << lRtn[0] <<"\t" << lRtn[1] <<endl;
	}
	return 0;
}

Console Output:

!!!Hello World!!!
50
Index	R-Time	I-Time	R-Calls	I-Calls	R-Rtn	I-Rtn	
1	0	0	1	1	1	1
2	0	0	2	2	1	1
3	0	0	5	3	2	2
4	0	0	10	4	3	3
5	0	0	19	5	5	5
6	0	0	34	6	8	8
7	0	0	59	7	13	13
8	0	0	100	8	21	21
9	0	0	167	9	34	34
10	0	0	276	10	55	55
11	0	0	453	11	89	89
12	0	0	740	12	144	144
13	0	0	1205	13	233	233
14	0	0	1958	14	377	377
15	0	0	3177	15	610	610
16	0	0	5150	16	987	987
17	0	0	8343	17	1597	1597
18	0	0	13510	18	2584	2584
19	0	0	21871	19	4181	4181
20	0	0	35400	20	6765	6765
21	0	0	57291	21	10946	10946
22	0	0	92712	22	17711	17711
23	0	0	150025	23	28657	28657
24	0	0	242760	24	46368	46368
25	0	0	392809	25	75025	75025
26	0	0	635594	26	121393	121393
27	0	0	1028429	27	196418	196418
28	0	0	1664050	28	317811	317811
29	0	0	2692507	29	514229	514229
30	0.01	0	4356586	30	832040	832040
31	0.01	0	7049123	31	1346269	1346269
32	0.01	0	11405740	32	2178309	2178309
33	0.03	0	18454895	33	3524578	3524578
34	0.04	0	29860668	34	5702887	5702887
35	0.06	0	48315597	35	9227465	9227465
36	0.1	0	78176300	36	14930352	14930352
37	0.17	0	126491933	37	24157817	24157817
38	0.26	0	204668270	38	39088169	39088169
39	0.44	0	331160241	39	63245986	63245986
40	0.71	0	535828550	40	102334155	102334155
41	1.14	0	866988831	41	165580141	165580141
42	1.89	0	1402817422	42	267914296	267914296
43	2.97	0	2269806295	43	433494437	433494437
44	4.79	0	3672623760	44	701408733	701408733
45	7.78	0	5942430099	45	1134903170	1134903170
46	12.66	0	9615053904	46	1836311903	1836311903
47	20.61	0	15557484049	47	2971215073	2971215073
48	33.24	0	25172538000	48	4807526976	4807526976
49	53.99	0	40730022097	49	7778742049	7778742049
50	88.66	0	65902560146	50	12586269025	12586269025