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Example of O(n!)?
(16 answers)
Closed 5 years ago.
I can't seem to find any examples that uses O(n!) time complexity.
I can't seem to comprehend how it works. Please help
A trivial example is the random sort algorithm. It randomly shuffles its input until it gets it sorted.
Of course, it has strictly no use in the real world, but still, it is O(n!).
EDIT: As pointed out in the comments, this is actually the average time performance of this algorithm. The best-case time complexity is O(1), which happens when the algorithm finds the right permutation right away, and is unbounded in the worst case, since you have no guarantee that the right permutation will come up.
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Big O, how do you calculate/approximate it?
(24 answers)
Closed 3 years ago.
I have been learning about sorting algorithms and understand the idea of time and space complexity. I was wondering if there was a fairly quick and easy way of working out the complexities given the algorithm (one that may even be quick enough to do in an exam) as opposed to learning all of the complexities for the algorithms.
If this isn't an option, is there an easy way to remember or learn them for some of the more basic sortings algorithms such as merge sort and quicksort.
You have to memorize this:
It depends on the problem. Best option in different situations:
In place and stable: Selection Sort
In place (don't care about stable): Heap Sort
Stable (don't care about in place): Merge Sort
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Which algorithm is faster O(N) or O(2N)?
(6 answers)
Closed 8 years ago.
If an algorithm iterates over a list of numbers two times before returning an answer is the runtime O(2n) or O(n)? Does the runtime of an algorithm always lack a coefficient?
Big-O notation refers to the asymptotic "worst-case" complexity of an algorithm. Any constants are factored out of the analysis. Hence, from a theoretical perspective, O(2n) should always be represented as O(n). However, from the standpoint of practical implementation, if you can cut that down to one iteration over the list of numbers you will see some (small) increase in performance.
It may still be slower than an implementation that doesn't iterate twice, but that is still O(n), as the time complexity scales based only on the size of n.
Convention is that you ignore constant coefficients when reporting Big-O time.
So if an algorithm were O(n), O(2n), or O(3n) for example, you would report O(n).
Your suspicion is correct. You leave off the coefficient. See http://en.wikipedia.org/wiki/Big_O_notation.
From the example,
Now one may apply the second rule: $6x^4$ is a product of 6 and $x^4$
in which the first factor does not depend on x. Omitting this factor
results in the simplified form $x^4$.
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Quicksort vs heapsort
(12 answers)
When is each sorting algorithm used? [closed]
(5 answers)
Closed 8 years ago.
I gave a written round of a company.
I have a doubt in a question can anyone help me?
which is the fastest sorting algorithm among the following?
a - bubble sort
b - shell sort
c - heap sort
d - quick sort
I'm confused b/w quick sort and heap sort both have a time complexity of O(nlogn).
There's no "fastest" sorting algorithm.
Theoretical performance of an algorithm always depends on the input data. In their respective worst cases, Heap sort is faster than Quick sort. In average case, Quick sort is faster. It is probably possible to concoct a custom-tailored best case for each algorithm to make it outperform all the others.
That is actually the reason such "hybrid" algorithms as Introsort exist: Introsort begins with Quick sort and switches to Heap sort if it sees that Quick sort is struggling with this specific input.
On top of that the real-life performance of any algorithm can be significantly affected by how well it works on a specific hardware platform. A theoretically "fast" algorithm can lose miserably to a primitive and "slow" one, if the latter is in better "sync" with the properties of the hardware.
See Wikipedia: Heapsort
Heapsort: Although somewhat slower in practice on most machines than a well-implemented quicksort, it has the advantage of a more favorable worst-case O(n log n) runtime
In the average case: Quick-Sort, since the constant of its O(nlgn) running time is better.
In the worst case: Heap-Sort, since in the worst case, the running time of Quick-Sort is O(n2).
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Closed 11 years ago.
Possible Duplicate:
How to find a binary logarithm very fast? (O(1) at best)
how does the log function work.
How the log of a with base b is calculated.
There are many algorithms for doing this. One of my favorites (WARNING: shameless plug) is this one based on the Fibonacci numbers. The code contains a pretty elaborate comment that goes into how the math works. Given a and ab, it runs in time O(lg b) and O(1) space.
Hope this helps!
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Closed 13 years ago.
Possible Duplicate:
are there any O(1/n) algorithms?
Is it ever possible for your code to be Big O less than O(1)?
O(1) simply means a constant time operation. That time could be 1 nanosecond or 1 million years, the notation is not a measure of absolute time. Unless of course you are working on the OS for a time machine, than perhaps your DoTimeTravel( ) function would have O(-1) complexity :-)
Not really. O(1) is constant time. Whether you express that as O(1) or O(2) or O(.5) really makes little difference as far as purely big O notation goes.
As noted in this question, it is technically possible to have an O(1/n), but that no real-world useful algorithm would satisfy this (though some do algorithm's do have 1/n as part of their algorithmic complexity).
The only thing that would take less than O(1) (constant time) would be an operation that did absolutely nothing, and thus took zero time. But even a NOP usually takes a fixed number of cycles...