1. Data Structures for Java William H. Ford William R. Topp
Chapter 4
Introduction to Algorithms
Bret Ford
© 2005, Prentice Hall
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2. Selection Sort
Proceed through an array position by position, starting with index 0. At the current position, select the element from the remaining elements that is next in order and copy it into the current position.
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3. Selection Sort Example Step 1
The ordering begins by locating the smallest animal, the fish, in the first position. The operation occurs by having the owl that currently occupies the first position exchange places with the fish. The effect is to place the smallest animal at the front of the list.
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4. Selection Sort Example Step 2
The tail of the list, starting at the second position is unordered. The process continues by identifying the smallest animal among the owl, dragon, and dog and arranging it in the second position. The selection is the owl that is already in its proper position and so we can move to the next step.
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5. Selection Sort Example Step 3
With the fish and owl in position, only the last two animals must be ordered. We identify the dog as the next-smallest animal and have it exchange positions with the dragon. After this step, the tail of the list has only one item left, which must be the largest animal. The entire list is ordered.
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6. Selection Sort in Action
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7. selectionSort() public static void selectionSort(int[] arr) {
// index of smallest element in the sublist
int smallIndex;
int pass, j, n = arr.length;
int temp;
// pass has the range 0 to n-2
for (pass = 0; pass < n-1; pass++)
// scan the sublist starting at index pass
smallIndex = pass;
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8. selectionSort() (concluded)
// j traverses the sublist
// arr[pass+1] to arr[n-1]
for (j = pass+1; j < n; j++)
// if smaller element found, assign
// smallIndex to that position
if (arr[j] < arr[smallIndex])
smallIndex = j;
// swap the next smallest
// element into arr[pass]
temp = arr[pass];
arr[pass] = arr[smallIndex];
arr[smallIndex] = temp; }
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9. Selection Sort Example
// declare an integer array
int[] arr = {66, 20, 33, 55, 53, 57, 69, 11, 67, 70};
// call selectionSort() to order the array
Arrays.selectionSort(arr);
System.out.print("Sorted: ");
for (int i=0; i < arr.length; i++)
System.out.print(arr[i] + " ");
Output:
Sorted:
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10. Sublists in an Array
A sublist of an array is a sequence of elements whose range of indices begin at index first and continue up to, but not including last. Indicate a sublist using the notation [first, last).
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11. Sequential Search
Begin with a target value and an index range [first, last). Scan the sublist item by item, looking for the first occurrence of a match with an item named target. Return the index of the match or -1 if target is not in the sublist.
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12. The seqSearch() Method
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13. seqSearch() Implementation
public static int seqSearch(int[] arr,
int first, int last, int target) {
// scan indices in the range first
// <= i < last; return the index
// indicating the position if a match occurs;
// otherwise return -1
for (int i = first; i < last; i++)
if (arr[i] == target)
return i;
// no return yet if match is
// not found; return -1
return -1; }
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14. Binary Search
The binary search exploits the fact that a list is ordered. This allows large sections of the list to be removed from the search.
Select the midpoint of the current sublist [first,last).
If target matches midpoint, the search is successful.
If the target is less than the current midpoint value, look in the lower sublist by selecting its midpoint; otherwise, look in the upper sublist by selecting its midpoint.
Continue until locating the target or the sublist reduces to size 0.
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15. Binary Search Case 1
A match occurs. The search is complete, and mid is the index that locates target.
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16. Binary Search Case 2
The value target is less than midValue, and the search must continue in the lower sublist. The index range for this sublist is [first, mid). Reposition the index last to the end of the sublist (last = mid).
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17. Binary Search Case 3
The value target is greater than midValue, and the search must continue in the upper sublist . The index range for this sublist is [mid+1,last), because the sublist begins immediately to the right of mid . Reposition the index first to the front of the sublist (first = mid+1).
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18. Binary Search Termination
The binary search terminates when a match is found or when the sublist to be searched is empty. An empty sublist occurs when first >= last.
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19. Binary Search Success Example Step 1
target = 23
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20. Binary Search Success Example Step 2
target = 23
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21. Binary Search Success Example Step 3
target = 23
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22. Binary Search Failure Example Step 1
target = 4
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23. Binary Search Failure Example Step 2
target = 4
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24. Binary Search Failure Example Step 3
target = 4
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25. Binary Search Failure Example Step 4
Index range [2,2). first ≥ last, so the search terminates
unsuccessfully. The return value is -1.
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26. binSearch() Method public static int binSearch(int arr[], int first,
int last, int target) {
// index of the midpoint
int mid;
// value that is assigned arr[mid]
int midValue;
// test for nonempty sublist
while (first < last)
mid = (first+last)/2;
midValue = arr[mid];
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27. binSearch() Method (concluded)
if (target == midValue)
// have a match
return mid;
// determine which sublist to search
else if (target < midValue)
// search lower sublist; reset last
last = mid;
else
// search upper sublist; reset first
first = mid+1; }
// target not found
return -1;
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28. System/Memory Efficiency
System efficiency measures how well an algorithm runs on a particular machine.
Memory efficiency is a measure of the amount of memory an algorithm uses. If an algorithm uses too much memory, it can be too slow or may not execute at all on a particular system.
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29. Running Time Analysis
Machine independent measures of efficiency involve counting the number of comparisons, assignment statements, etc.
To analyze the complexity of an algorithm, identify the dominant operation and measure the number of times it occurs.
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30. Running Time: min() Method
Count the number of comparisons, T(n), required to locate the minimum of the elements in an n-element array. T(n) = n-1
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31. Running Time: Selection Sort
Count the number of comparisons in the n-1 passes of the algorithm. T(n) = (n-1) + (n-2) T(n) = n(n-1)/2 = n2/2 - n/2
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32. Running Time: Sequential Search
Best case: Locating the target at index 0. T(n) = 1
Worst caset: Locating the target at index n-1 or finding no match. T(n) = n
Average case: Average of the number of comparisons to locate a target at each position. T(n)=( n)/n=n(n+1)/2 (1/n) = (n+1)/2
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33. Running Time: Binary Search
Best case: Target found at first midpoint. T(n) = 1
Worst case: Length of sublists decrease by a factor of two at each iteration. T(n) = (int)log2n + 1
Average case: A sophisticated analysis shows that the average number of iterations is one less than the number in the worst case.
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34. Big-O Notation
To get an approximate measure for T(n), define a notation that involves the dominant term of T(n). O(<term>) is called the Big-O measure for the algorithm.
Selection sort is O(n2).
The average case for sequential search is O(n).
The worst case for binary search is O(log2n).
If T(n) = 8n3+5n2-11n+1, then T(n) is O(n3).
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35. Common Orders of Magnitude
Constant time: Algorithm is O(1) when its running time is independent of the number of items.
Examples: Find the minimum of an ordered n-element array.
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36. Common Orders of Magnitude (continued)
Linear: Algorithm is O(n) when its running time is proportional to n, the size of the list. If n doubles, number of operations doubles.
Example: Find the minimum of an unordered n-element array.
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37. Common Orders of Magnitude (continued)
Quadratic: An algorithm is quadratic if its running time is O(n2). If n doubles, the number of operations increases by a factor of 4.
Example: Selection sort.
Cubic: An algorithm is cubic if its running time is O(n3). Doubling n increases the number of operations by a factor of 8.
Example: The number of products in matrix multiplication is cubic.
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38. Common Orders of Magnitude (continued)
Logarithmic: An algorithm is logarithmic if its running time is O(log2) or O(n log2). Occurs when the algorithm repeatedly subdivides the data into sublists whose sizes are 1/2, 1/4, 1/8, ... of the original size n.
Example: Binary search is O(log2), and quicksort is O(n log2).
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39. Common Orders of Magnitude (concluded)
Exponential: An algorithm is exponential if its running time is O(an). These algorithms deal with problems that require searching through a large number of potential solutions before finding an answer.
Example: The traveling salesperson problem.
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40. Comparison of Graphs
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41. Orders of Magnitude for Selected Values of n
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42. The Timing Class
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43. Timing Class Example Timing sortTimer = new Timing();
double timeInSec;
// flank the process with calls to start() and stop()
sortTimer.start(); // start timing
Sort.selectionSort(arr); // the sorting process
timeInSec = sortTimer.stop(); // get time in seconds
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44. Program 4.1 import java.util.Random; import java.text.DecimalFormat;
import ds.util.Arrays;
import ds.time.Timing;
public class Program4_1 {
public static void main(String[] args)
final int ARRAY_SIZE = ,
TARGET_SIZE = 50000;
// arrays for the search
int[] listSeq = new int[ARRAY_SIZE],
listBin = new int[ARRAY_SIZE],
targetList = new int[TARGET_SIZE];
int i;
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45. Program 4.1 (continued) // use Timing object t to compute
// times for each process
Timing t = new Timing();
double seqTime, binTime;
// random number object
Random rnd = new Random();
// format real numbers with
// three decimal places
DecimalFormat fmt =
new DecimalFormat("#.000");
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46. Program 4.1 (continued) // initialize the arrays with
// random numbers in the
// range 0 to 999,999
for (i = 0; i < ARRAY_SIZE; i++)
listSeq[i] =
listBin[i] =
rnd.nextInt( );
// initialize targetList with
// same range 0 to 999,999
for (i=0;i < TARGET_SIZE; i++)
targetList[i] = rnd.nextInt( );
// time the sequential search
// with elements from listSeq
t.start();
for (i = 0; i < TARGET_SIZE; i++)
Arrays.seqSearch(listSeq,0,ARRAY_SIZE,
targetList[i]);
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47. Program 4.1 (concluded) binTime = t.stop(); System.out.println(
"Binary Search takes " +
fmt.format(binTime) + " seconds.");
"Ratio of sequential to" +
"binary search time is " +
fmt.format(seqTime/binTime)); } /*
Run:
Sequential Search takes seconds.
Binary Search takes .016 seconds.
Ratio of sequential to binary search time is */
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