Lab exercise #9: Sorting

Sorting with a priority queue

Today we'll examine another sorting method with a running time that is O(n lg n). This method uses a priority queue as an intermediate data structure.

The idea is simple: Given an unsorted array, create an empty priority queue that can hold elements of that array. Traverse the array, adding each element in turn to the priority queue. Then remove the elements one at a time from the priority queue, storing them back into the array, from left to right. Since at each step the element that can be extracted from the priority queue is the least of the remaining items, it goes into the leftmost position of the array that has not already been filled.

Since this algorithm involves n invocations of the priority queue's add method and n invocations of its remove method to sort an array of size n, and since each of this invocations (according to the documentation for java.util.PriorityQueue) requires no more than O(lg n) time, the entire algorithm is O(n lg n).

Implementing the algorithm

The file /home/stone/courses/java/examples/SortTester.java defines a class SortTester that runs a sorting method on a randomly constructed array of Doubles and reports how long it takes to execute that method. It also contains a stub for the priorityQueueSort sorting method.

Implement the priorityQueueSort method. The priority queue that it uses as an intermediate data structure should be stored in a local variable of the method.

Timing the sort

Revise the SortTester program so that it calls priorityQueueSort repeatedly, with array sizes of 10000, 20000, 30000, and so on up to 160000, and reports the running times. What pattern do you expect to see if the priorityQueueSort method is really O(n lg n)? What pattern would you expect to see if the method were O(n²)?

Comparison with the Java library sort

Revise the main method so that it invokes the sorting method provided by java.util.Arrays instead of priorityQueueSort and re-run it. Do the running times exhibit the O(n lg n) pattern? Are they less than the corresponding times for priorityQueueSort? Is this consistent with both sorting algorithms having running times of the same order?

Extras: Using the priority queue sort in an application

Revise your SparseFinder program from the lab on demographics so that it stores the County objects in an array rather than a list. Revise the County class so that it implements the Comparable interface. Then, returning to SparseFinder, use the priority queue sort to arrange the elements of the array of County in order of increasing population density immediately after you read it in. Revise findSparsest so that it simply returns the County that is in position 0 of the sorted array.

Modify SparseFinder so that it prints out the names and population densities of the ten most sparsely populated counties in the state of Iowa, in order of increasing population density.