## A linear time algorithm to find the array of indices of bigger numbers

Suppose you have an array nums of integers, unsorted, containing the values from 1 to n. You have a second array, call it firstBigger, initially empty, which you want to populate with integers such that firstBigger[i] contains j if j is the least index such that i < j and nums[j] > nums[i]. A brute force search of nums runs in n^2 time. I need to find a linear time search.

For example, if nums = [4,6,3,2,1,7,5] and we use 1-indexing, then firstBiggest = [2,6,6,6,6,None,None].

I have considered first computing the array of differences in linear time. Certainly anywhere in the array of differences with a positive value, this indicates a place in firstBigger where it should store i+1. But I’m not seeing how to fill any other coordinates efficiently.

I might have gotten close when I started thinking of analyzing the array end-to-start. The last nth coordinate of firstBigger is going to be None, the n-1th has to be directly compared to the nth. As we proceed backward, if the number at i is smaller than at i+1 we make this assignment. Otherwise we look up the first number bigger than the one at i+1. If that’s still too small, again look up the first number bigger than that.

On average this does better than the naive algorithm, but in the worst case it’s still n^2. I can’t see any room to optimize this.

## Find a threshold such that one function is always bigger than the other

Given the recursively defined function $$c$$, which I believe is a variant of the Péter-Ackermann function: $$c(m,n)=\begin{cases}0&\text{for }m=0\ n^2+n+1&\text{for }m = 1\text{ and }n\ge 0\ c(m-1, 1)&\text{for }m>1\text{ and }n=0\ c(m-1,c(m,n-1))&\text{for }m>1\text{ and }n>0\ \end{cases}$$

and the function $$d$$ that builds a “tower” of powers of $$2$$ of size $$n+2$$:

$$d(n) = \underbrace{2^{2^{.^{.^{.^{.^{2}}}}}}}_{\text{ n+2 }}$$

The inputs $$m$$ and $$n$$ are both natural numbers. I’m asked to find an $$x$$, such that for all numbers $$y \ge x$$, $$c(y,y) > d(y)$$.

I rewrote the two functions using Python in order to calculate some values:

 c(m, n):     if m == 0:         return 0     else if m == 1 and n >= 0:         return n**2+n+1              # return n^2+n+1     else if m > 1 and n == 0:         return c(m-1, 1)     else if m > 1 and n > 0:         return c(m-1, c(m, n-1))  
 d(n):     exp_num = n-1     result = 2     while exp_num != -1:         result = result**2           # result = result^2         exp_num = exp_num - 1     final_result = 2**result         # final_result = 2^result     return final_result  

Some inputs and outputs:

c(1, 1) = 3

c(2, 2) = 183

d(1) = 16

d(2) = 65536

d(3) = 115792089237316195423570985008687907853269984665640564039457584007913129639936

I believe the threshold is $$3$$, as c(3, 3) doesn’t seem to be realistically computable considering there are over 19K digit in A(4, 2) (where A is the Péter-Ackermann function). Unfortunately I have no idea how to prove this. Any help would be much appreciated.

## Given two functions, find a threshold such that one is always bigger than the other

I’ve written the following two functions in Python (they should also be fairly easy to understand as Pseudo):

 c(m, n):     if m == 0:         return 0     else if m == 1 and n >= 0:         return n**2+n+1              # return n^2+n+1     else if m > 1 and n == 0:         return c(m-1, 1)     else if m > 1 and n > 0:         return c(m-1, c(m, n-1))  
 d(n):     exp_num = n-1     result = 2     while exp_num != -1:         result = result**2           # result = result^2         exp_num -= 1     final_result = 2**result         # final_result = 2^result     return final_result  

Where the inputs m and n are both natural numbers. I’m asked to find an x, such that for all numbers y >= x, c(y,y) > d(y).

Some inputs and outputs:

c(1, 1) = 3

c(2, 2) = 183

d(1) = 16

d(2) = 65536

d(3) = 115792089237316195423570985008687907853269984665640564039457584007913129639936

As you can see d grows extremely faster than c, which I believe is a variant of the Péter-Ackermann function. How can I approach this problem? Any help would be much appreciated.

## How to Make boot directory bigger?

I have plenty space on my HD 420 GB but very little for boot directory & i can’t upgrade to ubuntu 16.04 what is the eseyist way to inlarge the boot directory?

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## product image getting bigger as the gallery image number increases magento 2 and vise versa

product image getting bigger as the gallery image number increases magento 2 and vise Versa How to make the product image load in big size no matter what number is the product gallery number??

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## Copy an OS drive with all it’s partitions, to a bigger hard drive

I have a windows machine, and it’s OS is alingside with the a few more partitions on one hard drive.

I want to use Ubuntu live cd, in Order to copy that OS drive to another, bigger hard drive and utilize that space.

I managed to copy the drive using ddrescue and it works fine, but then I have a lot of un utilized space on the new bigger drive.

I would like to os partition To use all the extra space

I tried to copy the partitions using Gparted and resize the OS partition, the system did not boot afterwords

I tried to use ddrescue to copy the drive, and then Gparted to change the partition’s size, But it’s not the last partition on the disk, so I could not, When I tried to resize the other later partitions and then “make room” for that partition and then resizing it, the machine did not boot afterwords.

How can I copy the drive, and keep it bootable, allowing one partition use all the extra space I have on the larger disk?

Thanks

## (JAVA) Bigger number than allowed as “int”. No error message

I´ve written the following code to test the limits of an int variable:

package com.company;

public class Main {

public static void main(String[] args) {      //minimun int value     int minValue = -2_147_483_648;       //2 times minimum int value     int doubleMinValue = 2*minValue;      //print both     System.out.println(minValue);     System.out.println((2*minValue));  } 

}

And I got the following result:

-2147483648

0

Process finished with exit code 0

I was expecting to receive a ERROR MESSAGE, but I received a WRONG ANSWER. What is the Java logic in this case ?

Thanks.