Each step of choosing a pivot takes O n time, and so we should spend roughly O n time before getting each good pivot. Since we get at most O lg n good pivots, the overall runtime is O n lg n on expectation. You'll get that quicksort takes, on average, O n lg n time. The reason that I've mentioned this proof is that it gives you a good framework for thinking about how to choose a pivot in quicksort.
If you can pick a pivot that's pretty close to the middle on each iteartion, you can guarantee O n lg n runtime. If you can't guarantee that you'll get a good pivot on any iteration, but can say that on expectation it takes only a constant number of iterations before you get a good pivot, then you can also guarantee O n lg n expected runtime.
Given this, let's take a look at your proposed pivot schemes. For a , if the array is random, picking the first element as the pivot is essentially the same as picking a random pivot at each step, and so by the above analysis you'll get O n lg n runtime on expectation. For b , if you know that the array is mostly sorted, then picking the median is a good strategy.
The reason is that if we can say that each element is "pretty close" to where it should be in the sorted sequence, then you can make an argument that every pivot you choose is a good pivot, giving you the O n lg n runtime you want. The term "pretty close" isn't very mathematically precise, but I think you could formalize this without too much difficulty if you wanted to.
As for c and d , of the two, d is the only one guaranteed to get O n lg n on expectation. Heapsort is also not a stable sorting algorithm. Systems that exhibit strong locality of reference are great candidates for performance optimization. Heap sort, however, deals with larger leaps. This makes quicksort more favorable for smaller inputs.
To me there is a very fundamental difference between heapsort and quicksort: the latter uses a recursion. In recursive algorithms the heap grows with the number of recursions. The program takes almost 10 GB of ram and any extra memory will make my computer to start swapping to virtual disk memory.
My disk is a RAM disk, but still swapping to it make a huge difference in speed. I just compared implementations of selection, quick, merge, and heap sort to see how they'd stack up against each other. The answer is that they all have their downsides. TL;DR: Quick is the best general purpose sort reasonably fast, stable, and mostly in-place Personally I prefer heap sort though unless I need a stable sort. Unless your data is already sorted, or very, very nearly so. Quick, in my experience, is not actually that quick all the time.
Bonuses for using quick sort as a general sort though are that it's reasonably fast and it's stable. It's also an in-place algorithm, but as it's generally implemented recursively, it will take up additional stack space.
Timing on some sorts seem to confirm this, especially when the values fall within a tight range. It's way faster than selection sort on 10,, items, but slower than merge or heap. Merge sort is guaranteed O n log n since its sort is not data dependent.
It just does what it does, regardless of what values you've given it. It's also stable, but very large sorts can blow out your stack if you're not careful about implementation. There are some complex in-place merge sort implementations, but generally you need another array in each level to merge your values into. If those arrays live on the stack you can run into issues. Article Contributed By :. Easy Normal Medium Hard Expert. Writing code in comment? Please use ide. Load Comments.
What's New. Most popular in Difference Between. Collectives on Stack Overflow. Learn more. Quicksort superiority over Heap Sort Ask Question. Asked 11 years, 11 months ago. Active 10 months ago. Viewed 21k times. Nitish Upreti Nitish Upreti 6, 9 9 gold badges 47 47 silver badges 88 88 bronze badges.
The worst case occurs when the elements are already sorted - a relative rare case - and one that can be easily avoided by doing a simple shuffle first if this use case could occur in your system. Locality of reference is the key for QR's fast runtime performance. Paul Simple shuffle won't solve the problem of duplicate values in array for Quicksort. Add a comment. Active Oldest Votes. John Feminella John Feminella k 42 42 gold badges silver badges bronze badges. For small working sets the locality of reference issue is critical to avoiding unwanted page faults.
It is a strong argument to end the function with a call to sort the left hand most partition, followed by a tail recursive optimization for the right hand partition. But not strong enough to do it in practice. Always sort the smallest partition first to avoid blowing the stack — Stephan Eggermont. StephanEggermont: If the left partition holds millions of items and the right partition holds two, clearly the right partition should be sorted first.
Would there by any problem, however, with sorting the left partition first unless it's e. Worst-case stack depth would be increased, but only by a constant factor. Locality of reference is not practically influenced by doing left or right partition first — Stephan Eggermont.
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