简体   繁体   English

求解类似的递归:T(n) = 3T(n/3) + n/3

[英]Solving the similar recurrence: T(n) = 3T(n/3) + n/3

Given..鉴于...

T(0) = 3 for n <= 1

T(n) = 3T(n/3) + n/3 for n > 1

So the answer's suppose to be O(nlogn) .. Here's how I did it and it's not giving me the right answer:所以答案应该是O(nlogn) .. 这就是我是怎么做的,它没有给我正确的答案:

T(n) = 3T(n/3) + n/3

T(n/3) = 3T(n/3^2) + n/3^2

Subbing this into T(n) gives..将其代入 T(n) 给出..

T(n) = 3(3T(n/3^2) + n/3^2) + n/3

T(n/3^2) = 3(3(3T(n/3^3) + n/3^3) + n/3^2) + n/3

Eventually it'll look like..最终它会看起来像..

T(n) = 3^k (T(n/3^k)) + cn/3^k

Setting k = lgn..设置k = lgn..

T(n) = 3^lgn * (T(n/3^lgn)) + cn/3^lgn

T(n) = n * T(0) + c

T(n) = 3n + c

The answer's O(n) though..What is wrong with my steps?答案是O(n) ..我的步骤有什么问题?

Eventually it'll look like.. T(n) = 3^k (T(n/3^k)) + cn/3^k最终它看起来像.. T(n) = 3^k (T(n/3^k)) + cn/3^k

No. Eventually it'll look like不。最终它会看起来像

T(n) = 3^k * T(n/3^k) + k*n/3

You've opened the parenthesis inaccurately.你打开的括号不准确。

T(n) = 3T(n/3) + n/3
T(n/3) = 3T(n/9) + n/9

T(n) = 3(3T(n/9) + n/9) + n/3
     = 9T(n/9) + 2*n/3      //statement 1

T(n/9)= 3T(n/27) + n/27
T(n)  = 9 (3T(n/27)+n/27) + 2*n/3 // replacing T(n/9) in statement 1
      =  27 T (n/27) + 3*(n/3)

T(n)  = 3^k* T(n/3^k) + k* (n/3) // eventually

replace k with log n to the base 3.用 log n 替换 k 到基数 3。

T(n)  = n T(1) + (log n) (n/3);
// T(1) = 3
T(n)  = 3*n + (log n) (n/3);
Hence , O (n* logn)

These types of problems are easily solved using the masters theorem .使用大师定理很容易解决这些类型的问题。 In your case a = b = 3 , c = log3(3) = 1 and because n^c grows with the same rate as your f(n) = n/3 , you fall in the second case.在您的情况下a = b = 3c = log3(3) = 1并且因为n^c以与f(n) = n/3相同的速率增长,所以您属于第二种情况。

Here you have your k=1 and therefore the answer is O(n log(n))这里你有你的k=1 ,因此答案是O(n log(n))

This question can be solved by Master Theorem:这个问题可以通过主定理来解决:
In a recursion form :以递归形式:
在此处输入图片说明

where a>=1, b>1, k >=0 and p is a real number, then:其中 a>=1, b>1, k >=0 且 p 为实数,则:

  1. if a > b k , then如果 a > b k ,则
    主定理的第一种情况

  2. if a = b k如果 a = b k

     a.) if p >-1, then

2.主定理的一个例子

       b.) if p = -1, then 

在此处输入图片说明

       c.) if p < -1, then 

在此处输入图片说明
3. if a < b k 3. 如果 a < b k
a.) if p >=0, then a.) 如果 p >=0,则

在此处输入图片说明
b.) if p<0, then T(n) = O(n k ) b.) 如果 p<0,则 T(n) = O(n k )

So, the above equation所以,上面的方程

   T(n) = 3T(n/3) + n/3  

a = 3, b = 3, k =1, p =0    

so it fall into 2.a case, where a = b k所以它属于 2.a 情况,其中 a = b k
So answer will be所以答案将是

O(n⋅log(n)) 

声明:本站的技术帖子网页,遵循CC BY-SA 4.0协议,如果您需要转载,请注明本站网址或者原文地址。任何问题请咨询:yoyou2525@163.com.

 
粤ICP备18138465号  © 2020-2024 STACKOOM.COM