764:
261:
36:
172:, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements.
1872:
elements. In the absolute worst case, the search must visit every element once. This happens when the value being searched for is either the last element in the list, or is not in the list. However, on average, assuming the value searched for is in the list and each list element is equally likely to
870:
has O(n) time when the elements are sorted on the first iteration. In each iteration all elements are checked if in order. There are n! possible permutations; with a balanced random number generator, almost each permutation of the array is yielded in n! iterations. Computers have limited memory, so
361:
In some situations it may be necessary to use a pessimistic analysis in order to guarantee safety. Often however, a pessimistic analysis may be too pessimistic, so an analysis that gets closer to the real value but may be optimistic (perhaps with some known low probability of failure) can be a much
416:
have very poor worst-case behaviors, but a well written hash table of sufficient size will statistically never give the worst case; the average number of operations performed follows an exponential decay curve, and so the run time of an operation is statistically bounded.
346:
of text). Similarly, even when a sensible description of a particular "average case" (which will probably only be applicable for some uses of the algorithm) is possible, they tend to result in more difficult analysis of equations.
235:
is used in computer science to describe an algorithm's behavior under optimal conditions. For example, the best case for a simple linear search on a list occurs when the desired element is the first element of the list.
863:
Heapsort has O(n) time when all elements are the same. Heapify takes O(n) time and then removing elements from the heap is O(1) time for each of the n elements. The run time grows to O(nlog(n)) if all elements must be
400:
Many algorithms with bad worst-case performance have good average-case performance. For problems we want to solve, this is a good thing: we can hope that the particular instances we care about are average. For
216:
The terms are used in other contexts; for example the worst- and best-case outcome of an epidemic, worst-case temperature to which an electronic circuit element is exposed, etc. Where components of specified
342:
means is difficult, and often that average input has properties which make it difficult to characterise mathematically (consider, for instance, algorithms that are designed to operate on
409:
can be used for some specific problems to show that the worst case is no harder than the average case, or, equivalently, that the average case is no easier than the worst case.
247:. Algorithms may also be trivially modified to have good best-case running time by hard-coding solutions to a finite set of inputs, making the measure almost meaningless.
201:, but it does have uses: for example, where the best cases of individual tasks are known, they can be used to improve the accuracy of an overall worst-case analysis.
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Worst-case performance analysis and average-case performance analysis have some similarities, but in practice usually require different tools and approaches.
239:
Development and choice of algorithms is rarely based on best-case performance: most academic and commercial enterprises are more interested in improving
1951:
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the generated numbers cycle; it might not be possible to reach each permutation. In the worst case this leads to O(∞) time, an infinite loop.
830:/2. Working out the resulting average-case running time yields a quadratic function of the input size, just like the worst-case running time.
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more practical approach. One modern approach in academic theory to bridge the gap between worst-case and average-case analysis is called
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852:)), which contributes to making it a very fast algorithm in practice. But given a worst-case input, its performance degrades to O(
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856:). Also, when implemented with the "shortest first" policy, the worst-case space complexity is instead bounded by O(log(
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are used, devices must be designed to work properly with the worst-case combination of tolerances and external conditions.
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cost can be much closer to the average cost, while still providing a guaranteed upper limit on the running time. So e.g.
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When analyzing algorithms which often take a small time to complete, but periodically require a much larger time,
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elements, assumed to be all different and initially in random order. On average, half the elements in a list
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405:, this is very bad: we want typical instances of a cryptographic problem to be hard. Here methods like
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Introduction to
Algorithms (Cormen, Leiserson, Rivest, and Stein) 2001, Chapter 2 "Getting Started".In
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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations
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is often of particular concern since it is important to know how much time might be needed
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can be used to determine the worst-case running time over a (possibly infinite) series of
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are the most used in algorithm analysis. Less widely found is
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Search data structure § Asymptotic worst-case analysis
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Worst-case versus amortized versus average-case performance
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A measure of how efficiently algorithms use resources
60:. Unsourced material may be challenged and removed.
1873:be the value searched for, the search visits only
432:Sorting algorithm § Comparison of algorithms
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412:On the other hand, some data structures like
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385:are frequently based on amortized analysis.
289:. Unsourced material may be challenged and
388:The worst-case analysis is related to the
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309:Learn how and when to remove this message
120:Learn how and when to remove this message
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213:, to determine expected running times.
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1935:Best-case complexity
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283:improve this section
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20:. For the case, see
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1969:(10), ACM: 76-84,
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