在O(n)时间内，用n位数字对0…n的整数数组进行排序

I'm attempting to create an algorithm that can sort an array of integers in O(N) time.

我试图创建一个算法，可以在O(N)时间内对整数数组进行排序。

• The number of digits in all of the intergers is N
• 所有格中的位数都是N
• Each element has an unknown number of digits
• 每个元素都有未知的数字
• The algorithm should sort the array in O(N) time regardless of how the digits are distributed
• 算法应该在O(N)时间内对数组进行排序，而不考虑数字的分布

I have a working solution for this problem, that runs in O(N) time, I'm just having trouble trying to prove that it does so.

对于这个问题，我有一个可行的解决方案，它在O(N)时间内运行，我只是很难证明它是这样的。

Create a set of N buckets and add items to their corresponding bucket based off how
many digits are in the integer -O(N)

Radix sort each bucket, and then concatenate the buckets back together. 
Sum k=0 to N of O(k*n)
k = Number of digits
n = number of items with k digits

The solution that I have come up with is that the ∑k*∑n will always equal N.

我想出了解决方案是,∑k *∑n总是等于n。

Attempt at a proof

试图证明

Base case: Array has 1 item.
T(N)= k*1. k=N = O(N)

I'm unsure how to do the inductive step (if it is even required).

我不知道如何做归纳步骤(如果需要的话)。

1 个解决方案

#1

The following screenshot explains it:

下面的截图解释了这一点: