Overview of Comparison Sorting Algorithms

Key Insight

Comparison sorts determine the sorted order of an input array by comparing elements. Examples include insertion sort, merge sort, heapsort, and quicksort. Insertion sort has a worst-case running time of Theta(n^2) but is fast for small inputs due to tight inner loops and sorts in-place, meaning it uses only a constant number of additional array elements. Merge sort offers a better asymptotic running time of Theta(n log n) but typically does not operate in-place, requiring additional memory for its merge procedure.

Heapsort, another comparison sort, sorts 'n' numbers in-place in O(n log n) time. It leverages an important data structure called a 'heap', which can also be used to implement a priority queue. Quicksort, while having a worst-case running time of Theta(n^2), has an expected running time of Theta(n log n). It is also an in-place algorithm and often outperforms heapsort in practice due to small hidden constant factors in its running time, making it popular for large input arrays.

All these algorithms, insertion sort, merge sort, heapsort, and quicksort, are classified as comparison sorts because they rely on comparisons between elements to establish their relative order. This characteristic is central to understanding their theoretical performance limitations, as studied through the decision-tree model, which reveals a fundamental lower bound on their running times.