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[Answers] Why is the time complexity of insertion sort not brought down even if we use binary sort for the comparisons?

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Problem Detail: 

There are two factors that decide the running time of the insertion sort algorithm : the number of comparisons, and the number of movements. In the case of number of comparisons, the sorted part (left side of $j$) of the array is searched linearly for the right place of the $j^{th}$ element. If instead, we use a binary search, then the time complexity of finding a place for the $j^{th}$ element comes down from $\operatorname{O}(n)$ to $\operatorname{O}(\log n)$. So, for all the $n$ elements, the time complexity for comparisons becomes $\operatorname{O}(n \log n)$. Even so, the number of movements is still going to take $\operatorname{O}(n)$ time, and the total time complexity isn't brought down and remains $\operatorname{O}(n^2)$. Why is that?

Are any of my statements wrong assumptions?

Edit Can a possible explanation be: the total time complexity isn't brought down and remains $\operatorname{O}(n^2)$. This is because to search an element (using binary search) it takes $\operatorname{O}(\log n)$ time, and to move the elements it takes $\operatorname{O}(n)$ time. Total cost is $\operatorname{O}(\log n)+\operatorname{O}(n)=\operatorname{O}(n)$ time. To do this for $n-1$ elements, it takes $n(n-1)=\operatorname{O}(n^2)$ time.?

Asked By : Somenath Sinha

Answered By : aelguindy

For the $j^{th}$ element, you would do ~ $\log j$ comparisons and (in the worst case) ~$j$ shifts.

Summing over $j$, you get

$$ \sum_{j = 1}^{n} (j + \log j) = \frac{n(n+1)}{2} + \log (n!) = O(n^2 + n \log n) = O(n^2) $$

The idea is that the linear work of shifting trumps the logarithmic work of comparing. You end up doing less comparisons, but still a linear amount of work per iteration. So the complexity does not change.

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