一种递归计算的高效方法

Posted by abloz on March 5, 2013

周海汉/文 http://abloz.com 递归计算,有时能非常直观的解决问题,但是非常耗资源,计算很慢,还可能导致堆栈耗尽,计算失败。所以很多时候具体实现时不提倡采用递归,而是将递归转为循环的方式来实现。但这种方式又不直观,容易出错。

有没有一种方法可以即递归,又快速实现,减少运算资源消耗呢? python中的生成器是一种解决方案。以斐波纳契数列为例:

def fibonacci(n):
    a,b=0,1
    for i in xrange(n):
        yield a
        a,b=b,a+b

x=fibonacci(100)
print list(x)

$ python fib.py [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738L, 19740274219868223167L, 31940434634990099905L, 51680708854858323072L, 83621143489848422977L, 135301852344706746049L, 218922995834555169026L]

另一种方法,更具有通用性,适合任何语言。这是本文重点。 思想精髓是存储递归结果到map映射表中。

本文用python来作为脚本语言:

r={}
def recusive(n):
  # init
  # return exit result

  if n in r:
    return r[n]
  else:
    r[n]=f(n-1,n-2 ... )

对于fibonacci,实现如下:

$ cat fib1.py
f={}
def fibonacci(n):
    if n<=0:
        return 0
    if n<=2:
        return 1
    if n in f:
        return f[n]
    f[n]=fibonacci(n-1)+fibonacci(n-2)
    return f[n]



for i in xrange(100):
    print fibonacci(i)

zhouhh@ubuntu:~$ python fib1.py 0 1 1 2 3 5 8 13 21 34 55 … 83621143489848422977 135301852344706746049 218922995834555169026

参考 让递归计算速度快一万倍