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How to find a pair of indices $1\le i \le n$,$1\le j \le n$, $i\neq j$ in array $A[1..n]$ such that $0\le A[i]-A[j]\le\frac{max(A)-min(A)}{n-1}$ in $\Theta(n)$ time? max(A) is the max number in the array while min(A) is the min number.

Let $avg=\frac{max(A)-min(A)}{n-1}$. If we sort the array $A$ and calculate the differences of adjacent members then $avg$ is the average difference.

I feel like the question has to do with order statistics (select algorithm) but I don't quite know how exactly.

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  • $\begingroup$ Sorting is already $O(nlog(n))$. $\endgroup$ – fade2black Jul 16 '17 at 14:04
  • $\begingroup$ @fade2black I think there's another solution which doesn't rely on sorting $\endgroup$ – Yos Jul 16 '17 at 14:06
  • $\begingroup$ Are numbers bounded? $\endgroup$ – rus9384 Jul 16 '17 at 14:42
  • $\begingroup$ @rus9384 if you mean that the numbers are within a given range, then no we're not given any range $\endgroup$ – Yos Jul 16 '17 at 14:44
  • $\begingroup$ @rus9384 if we call select algorithm (p.216 CLRS) on the array we can find out the median element in $\Theta(n)$ expected time (although the problem doesn't state that expected time is allowed). Then wouldn't those elements $i,j$ be any elements to the left of the median? (because select uses partition routine which shuffles every element lesser than the partition to the left) $\endgroup$ – Yos Jul 16 '17 at 15:53
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Hint: to build intuition, suppose I told you that you want to find a pair of elements that differ by at most $1000$. Can you think of any way to find such a pair of elements? Would looking at the decimal representation of the numbers help? Could you characterize what their decimal representation would have to look like?

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  • $\begingroup$ Does your solution result in linear time complexity? I think I know the solution to the problem if sorting is involved: we sort the numbers, the numbers to the left of the median of the sorted array should satisfy the condition. But this results in $\Theta(n\log n)$ complexity $\endgroup$ – Yos Jul 16 '17 at 18:36
  • $\begingroup$ @Yos, try it and find out! $\endgroup$ – D.W. Jul 16 '17 at 18:41
  • $\begingroup$ I don't know how it helps since even sorting for array which has it's elements bounded can be done in linear time. $\endgroup$ – rus9384 Jul 16 '17 at 20:50
  • $\begingroup$ @rus9384, in this case the elements aren't bounded, so you can't necessarily sort in linear time (for instance, counting sort doesn't run in linear time) -- but there are other ways to solve this without sorting. $\endgroup$ – D.W. Jul 17 '17 at 0:30

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