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Wondering if there is a better solution
Автор Polar_, история, 21 месяц назад, По-английски

I found this on leetcode someone posted the problem . I am wondering if there is a solution for followup .
Given a calculation class where you initialize values for $$$a,b,c$$$ and the following function $$$f(x) = ax^2 + bx + c$$$, and a list of numbers, apply this function to all values in the list. 

class Calculation {
int a;
int b;
int c;

int[] calculate(int[] arr)
}

Followup: We want the output array to be sorted. Can we do better than $$$O(nlogn)$$$?

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21 месяц назад, # |
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Your problem is simply the same as sorting an array, which is not faster than n log n if it is a comparison-based sort.

You can use Radix sort or Count sort to break the lower bound based on comparison sorting.

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21 месяц назад, # |
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If the initial list of numbers is already sorted, then you can solve this in $$$O(n)$$$ time.

Since $$$ax^2 + bx + c$$$ is a quadratic function, it either increases until some point and then decreases (if $$$a < 0$$$) or it decreases until some point and then increases (if $$$a > 0$$$). This critical point can be found through the first derivative, i.e., $$$2ax + b = 0$$$, which solves to $$$x = \frac{-b}{2a}$$$.

So the algorithm works like this, assuming the array of numbers is sorted in non-decreasing order:

  • Calculate the critical point $$$x = \frac{-b}{2a}$$$, and implicitly partition the array based on this point, e.g., save the index in which all elements before this index are smaller than $$$\frac{-b}{2a}$$$ and all elements after or equal to this index are greater than or equal to $$$\frac{-b}{2a}$$$.
  • Apply the function to each element of the array (this can be done while looking for the critical point too).
  • If $$$a > 0$$$, then the right partition will be in increasing order and the left partition will be in decreasing order. For $$$a < 0$$$, it's the opposite. Now reverse the appropriate partition so that both are in increasing order.
  • Perform the merge function (cannot be done efficiently in-place, so you'll need another array) to merge these two sorted array partitions.

Complexity: $$$O(n)$$$ runtime, $$$O(n)$$$ auxiliary space

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