The selection sort improves on the bubble sort by reducing the number of swaps necessary from O(N2) to O(N). Unfortunately, the number of comparisons remains O(N2). However, the selection sort can still offer a significant improvement for large records that must be physically moved around in memory, causing the swap time to be much more important than the comparison time. (Typically this isn’t the case in Java, where references are moved around, not entire objects.)
SELECTION SORT ON THE BASEBALL PLAYERS
Let’s consider the baseball players again. In the selection sort, you can no longer compare only players
standing next to each other. Thus you’ll need to remember a certain player’s height; you can use a notebook to write it down. A magenta−colored towel will also come in handy.
A Brief Description
What’s involved is making a pass through all the players and picking (or selecting, hence the name of the sort) the shortest one. This shortest player is then swapped with the player on the left end of the line, at position 0. Now the leftmost player is sorted, and won’t need to be moved again. Notice that in this algorithm the sorted players accumulate on the left (lower indices), while in the bubble sort they accumulated on the right.
The next time you pass down the row of players, you start at position 1, and, finding the minimum, swap with position 1. This continues until all the players are sorted.
A More Detailed Description
In more detail, start at the left end of the line of players. Record the leftmost player’s height in your notebook and throw the magenta towel on the ground in front of this person. Then compare the height of the next player to the right with the height in your notebook. If this player is shorter, cross out the height of the first player, and record the second player’s height instead. Also move the towel, placing it in front of this new “shortest”
(for the time being) player. Continue down the row, comparing each player with the minimum. Change the minimum value in your notebook, and move the towel, whenever you find a shorter player. When you’re done, the magenta towel will be in front of the shortest player.
Swap this shortest player with the player on the left end of the line. You’ve now sorted one player. You’ve made N–1 comparisons, but only one swap.
On the next pass, you do exactly the same thing, except that you can completely ignore the player on the left, because this player has already been sorted. Thus the algorithm starts the second pass at position 1 instead of 0. With each succeeding pass, one more player is sorted and placed on the left, and one less player needs to be considered when finding the new minimum. Figure 3.9 shows how this looks for the first three passes.
Figure 3.9 Selection sort on baseball players
THE SELECTSORT WORKSHOP APPLET
To see how the selection sort looks in action, try out the selectSort Workshop applet. The buttons operate the same way as those in the bubbleSort applet. Use New to create a new array of 10 randomly arranged bars. The red arrow called outer starts on the left; it points to the leftmost unsorted bar. Gradually it will move right as more bars are added to the sorted group on its left.
The magenta min arrow also starts out pointing to the leftmost bar; it will move to record the shortest bar found so far. (The magenta min arrow corresponds to the towel in the baseball analogy.) The blue inner
arrow marks the bar currently being compared with the minimum.
As you repeatedly press Step, inner moves from left to right, examining each bar in turn and comparing it with the bar pointed to by min. If the inner bar is shorter, min jumps over to this new, shorter bar. When
inner reaches the right end of the graph, min points to the shortest of the unsorted bars. This bar is then swapped with outer, the leftmost unsorted bar.
Figure 3.10 shows the situation midway through a sort. The bars to the left of outer are sorted, and inner
has scanned from outer to the right end, looking for the shortest bar. The min arrow has recorded the position of this bar, which will be swapped with outer.
Figure 3.10 The selectSort Workshop applet
Use the Size button to switch to 100 bars, and sort a random arrangement. You’ll see how the magenta min
arrow hangs out with a perspective minimum value for a while, and then jumps to a new one when the blue
inner arrow finds a smaller candidate. The red outer arrow moves slowly but inexorably to the right, as the sorted bars accumulate to its left.
JAVA CODE FOR SELECTION SORT
The listing for the selectSort.java program is similar to that for bubbleSort.java, except that the container class is called ArraySel instead of ArrayBub, and the bubbleSort() method has been replaced by selectSort(). Here’s how this method looks:
public void selectionSort() {
int out, in, min;
for(out=0; out<nElems−1; out++) // outer loop {
min = out; // minimum for(in=out+1; in<nElems; in++) // inner loop if(a[in] < a[min] ) // if min greater, min = in; // we have a new min swap(out, min); // swap them
} // end for(outer) } // end selectionSort()
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MWSS: Data Structures and Algorithms in Java
by Robert Lafore
Waite Group Press, Macmillan Computer Publishing ISBN: 1571690956 Pub Date: 03/20/98
Previous Table of Contents Next
The outer loop, with loop variable out, starts at the beginning of the array (index 0) and proceeds toward higher indices. The inner loop, with loop variable in, begins at out and likewise proceeds to the right. At each new position of in, the elements a[in] and a[min] are compared. If a[in] is smaller, then min
is given the value of in. At the end of the inner loop, min points to the minimum value, and the array elements pointed to by out and min are swapped. Listing 3.2 shows the complete selectSort.java
program.
Listing 3.2 The selectSort.java Program
// selectSort.java
// demonstrates selection sort
// to run this program: C>java SelectSortApp
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− class ArraySel
{
private double[] a; // ref to array a private int nElems; // number of data items //−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− public ArraySel(int max) // constructor
{
a = new double[max]; // create the array nElems = 0; // no items yet }
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− public void insert(double value) // put element into array {
a[nElems] = value; // insert it nElems++; // increment size }
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− public void display() // displays array contents {
for(int j=0; j<nElems; j++) // for each element, System.out.print(a[j] + “ “); // display it
System.out.println(“”); }
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− public void selectionSort()
{
int out, in, min;
for(out=0; out<nElems−1; out++) // outer loop {
min = out; // minimum for(in=out+1; in<nElems; in++) // inner loop if(a[in] < a[min] ) // if min greater, min = in; // we have a new min swap(out, min); // swap them
} // end for(outer) } // end selectionSort()
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− private void swap(int one, int two)
{
double temp = a[one]; a[one] = a[two]; a[two] = temp; }
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− } // end class ArraySel
//////////////////////////////////////////////////////////////// class SelectSortApp
{
public static void main(String[] args) {
int maxSize = 100; // array size
ArraySel arr; // reference to array arr = new ArraySel(maxSize); // create the array arr.insert(77); // insert 10 items arr.insert(99); arr.insert(44); arr.insert(55); arr.insert(22); arr.insert(88); arr.insert(11); arr.insert(00); arr.insert(66); arr.insert(33);
arr.display(); // display items
arr.selectionSort(); // selection−sort them arr.display(); // display them again } // end main()
} // end class SelectSortApp
//−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
The output from selectSort.java is identical to that from bubbleSort.java:
77 99 44 55 22 88 11 0 66 33 0 11 22 33 44 55 66 77 88 99
INVARIANT
In the selectSort.java program, the data items with indices less than or equal to outer are always sorted.
EFFICIENCY OF THE SELECTION SORT
The selection sort performs the same number of comparisons as the bubble sort: N*(N–1)/2. For 10 data items, this is 45 comparisons. However, 10 items require fewer than 10 swaps. With 100 items, 4,950 comparisons are required, but fewer than 100 swaps. For large values of N, the comparison times will dominate, so we would have to say that the selection sort runs in O(N2) time, just as the bubble sort did. However, it is unquestionably faster because there are so few swaps. For smaller values of N, it may in fact be considerably faster, especially if the swap times are much larger than the comparison times.