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The best case input is an array that is already sorted. In this case insertion sort has a linear running time (i.e., O(n)). During each iteration, the first remaining element of the input is only compared with the right-most element of the sorted subsection of the array. The simplest worst case input is an array sorted in reverse order.
Worst-case performance analysis and average-case performance analysis have some similarities, but in practice usually require different tools and approaches. Determining what typical input means is difficult, and often that average input has properties which make it difficult to characterise mathematically (consider, for instance, algorithms ...
The best-case for the algorithm is when the numbers are already sorted, which takes () steps to perform the task. However, the input in the worst-case for the algorithm is when the numbers are reverse sorted and it takes O ( n 2 ) {\displaystyle O(n^{2})} steps to sort them; therefore the worst-case time-complexity of insertion sort is of O ( n ...
For example, since the run-time of insertion sort grows quadratically as its input size increases, insertion sort can be said to be of order O(n 2). Big O notation is a convenient way to express the worst-case scenario for a given algorithm, although it can also be used to express the average-case — for example, the worst-case scenario for ...
Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort quadratic: Multiplying two n-digit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort , selection sort or insertion sort (), >
Insertion sort is widely used for small data sets, while for large data sets an asymptotically efficient sort is used, primarily heapsort, merge sort, or quicksort. Efficient implementations generally use a hybrid algorithm , combining an asymptotically efficient algorithm for the overall sort with insertion sort for small lists at the bottom ...
The next pass, 3-sorting, performs insertion sort on the three subarrays (a 1, a 4, a 7, a 10), (a 2, a 5, a 8, a 11), (a 3, a 6, a 9, a 12). The last pass, 1-sorting, is an ordinary insertion sort of the entire array (a 1,..., a 12). As the example illustrates, the subarrays that Shellsort operates on are initially short; later they are longer ...
Among quadratic sorting algorithms (sorting algorithms with a simple average-case of Θ(n 2)), selection sort almost always outperforms bubble sort and gnome sort. Insertion sort is very similar in that after the k th iteration, the first k {\displaystyle k} elements in the array are in sorted order.