13.1. Chapter Introduction: Sorting¶
We sort many things in our everyday lives: A handful of cards when playing Bridge; bills and other piles of paper; jars of spices; and so on. And we have many intuitive strategies that we can use to do the sorting, depending on how many objects we have to sort and how hard they are to move around. Sorting is also one of the most frequently performed computing tasks. We might sort the records in a database so that we can search the collection efficiently. We might sort customer records by zip code so that when we print an advertisement we can then mail them more cheaply. We might use sorting to help an algorithm to solve some other problem. For example, Kruskal's algorithm to find a minimal-cost spanning tree must sort the edges of a graph by their lengths before it can process them.
Because sorting is so important, naturally it has been studied intensively and many algorithms have been devised. Some of these algorithms are straightforward adaptations of schemes we use in everyday life. For example, a natural way to sort your cards in a bridge hand is to go from left to right, and place each card in turn in its correct position relative to the other cards that you have already sorted. This is the idea behind Insertion Sort. Other sorting algorithms are totally alien to how humans do things, having been invented to sort thousands or even millions of records stored on the computer. For example, no normal person would use Quicksort to order a pile of bills by date, even though Quicksort is the standard sorting algorithm of choice for most software libraries. After years of study, there are still unsolved problems related to sorting. New algorithms are still being developed and refined for special-purpose applications.
Along with introducing this central problem in computer science, studying sorting algorithms helps us to understand issues in algorithm design and analysis. For example, the sorting algorithms in this chapter show multiple approaches to using divide and conquer. In particular, there are multiple ways to do the dividing. Mergesort divides a list in half. Quicksort divides a list into big values and small values. Radix Sort divides the problem by working on one digit of the key at a time. Sorting algorithms can also illustrate a wide variety of algorithm analysis techniques. Quicksort illustrates that it is possible for an algorithm to have an average case whose growth rate is significantly smaller than its worst case. It is possible to speed up one sorting algorithm (such as Shellsort or Quicksort) by taking advantage of the best case behavior of another algorithm (Insertion Sort). Special case behavior by some sorting algorithms makes them a good solution for special niche applications (Heapsort). Sorting provides an example of an important technique for analyzing the lower bound for a problem. External Sorting refers to the process of sorting large files stored on disk.
This chapter covers several standard algorithms appropriate for sorting a collection of records that fit into the computer's main memory. It begins with a discussion of three simple, but relatively slow, algorithms that require \(\Theta(n^2)\) time in the average and worst cases to sort \(n\) records. Several algorithms with considerably better performance are then presented, some with \(\Theta(n \log n)\) worst-case running time. The final sorting method presented requires only \(\Theta(n)\) worst-case time under special conditions (but it cannot run that fast in the general case). The chapter concludes with a proof that sorting in general requires \(\Omega(n \log n)\) time in the worst case.