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While answering to this question a debate began in comments about complexity of QuickSort. What I remember from my university time is that QuickSort is O(n^2) in worst case, O(n log(n)) in average case and O(n log(n)) (but with tighter bound) in best case.
What I need is a correct mathematical explanation of the meaning of average complexity to explain clearly what it is about to someone who believe the big-O notation can only be used for worst-case.
What I remember if that to define average complexity you should consider complexity of algorithm for all possible inputs, count how many degenerating and normal cases. If the number of degenerating cases divided by n tend towards 0 when n get big, then you can speak of average complexity of the overall function for normal cases.
Is this definition right or is definition of average complexity different ? And if it's correct can someone state it more rigorously than I ?
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Average case — represented as Big Theta or Θ(n) Theta, commonly written as Θ, is an Asymptotic Notation to denote the asymptotically tight bound on the growth rate of runtime of an algorithm.
To understand what Big O notation is, we can take a look at a typical example, O(n²), which is usually pronounced “Big O squared”. The letter “n” here represents the input size, and the function “g(n) = n²” inside the “O()” gives us an idea of how complex the algorithm is with respect to the input size.
So, In binary search, the best case is O(1), average and worst case is O(logn). In short, there is no kind of relationship of the type “big O is used for worst case, Theta for average case”.
Big O Notation is a tool used to describe the time complexity of algorithms. It calculates the time taken to run an algorithm as the input grows. In other words, it calculates the worst-case time complexity of an algorithm. Big O Notation in Data Structure describes the upper bound of an algorithm's runtime.
Let's refer Big O Notation in Wikipedia:
Let f and g be two functions defined on some subset of the real numbers. One writes
f(x)=O(g(x)) as x --> infinityif ...
So what the premise of the definition states is that the function f should take a number as an input and yield a number as an output. What input number are we talking about? It's supposedly a number of elements in the sequence to be sorted. What output number could we be talking about? It could be a number of operations done to order the sequence. But stop. What is a function? Function in Wikipedia:
a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Are we producing exacly one output with our prior defition? No, we don't. For a given size of a sequence we can get a wide variation of number of operations. So to ensure the definition is applicable to our case we need to reduce a set possible outcomes (number of operations) to a single value. It can be a maximum ("the worse case"), a minimum ("the best case") or an average.
The conclusion is that talking about best/worst/average case is mathematically correct and using big O notation without those in context of sorting complexity is somewhat sloppy.
On the other hand, we could be more precise and use big Theta notation instead of big O notation.
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