1. Searching Chapter 11

2. 2 Chapter Contents The Problem Searching an Unsorted Array Iterative Sequential Search Recursive Sequential Search Efficiency of Sequential Search Searching a Sorted Array Sequential search Binary Search Java Class Library: the Method binarySearch Searching an Unsorted Chain Iterative Sequential Search Recursive Sequential Search Efficiency of Sequential Search of a Chain Searching a Sorted Chain Sequential Search Binary Search Choosing a Search Method

3. 3 The Problem Searching is an every day occurrence. Searching a particular item among a collection of many items

4. 4 Searching Strategies The sequential search Apply to both unsorted ADT list and sorted ADT list The binary search Sorted ADT list implemented by array Faster than a sequential search Sorting usually takes much more time than searching

5. 5 Iteratively Search an Unsorted Array A method that uses a loop to search an array. Return is boolean type. Compare target entry with each entry until locate the target or look at all entries public boolean contains(Object anEntry) {boolean found = false; for (int index = 0; !found && (index < length); index++) {if (anEntry.equals(entry[index])) found = true; } // end for return found; } // end contains

6. 6 Searching an Unsorted Array Example of two outcomes: An iterative sequential search of an array that (a) finds its target; (b) does not find its target

7. 7 Recursively Search an Unsorted Array Look at the first entry. If that entry is the desired one, we end the search. Otherwise we search the rest of the array( an identical but smaller problem). Base case is an empty array and return false. A recursive algorithm to search an array as a private method for class list Algorithm to search list[first] through list[last] for desiredItem private boolean Search( T desiredItem, int first, int last) { if (there are no elements to search: first > last) return false else if (desiredItem equals list[first]) return true else return the result of searching list[first+1] through list[last] by calling Search ( desiredItem, first+1, last); }

8. 8 Searching an Unsorted Array A recursive sequential search of an array that (a) finds its target; (b) does not find its target.

9. 9 Efficiency of a Sequential Search Best caseO(1) Locate desired item first Worst caseO(n) Must look at all the items Average caseO(n) Must look at half the items O(n/2) is still O(n) Easy to understand and implement sequential search algorithm. However, if problem size is big, it is time consuming.

10. 10 Searching a Sorted Array A sequential search can be more efficient if the data is sorted Coins sorted by their mint dates.

11. 11 Binary Search of Sorted Array Ignoring one-half of the data when the data is sorted.

12. 12 Binary Search of Sorted Array Algorithm for a binary search Algorithm binarySearch(a, first, last, desiredItem) mid = (first + last)/2 // approximate midpoint if (first > last) return false else if (desiredItem equals a[mid]) return true else if (desiredItem a[mid]) return binarySearch(a, mid+1, last, desiredItem)

13. 13 Binary Search of Sorted Array A recursive binary search of a sorted array that (a) finds its target;

14. 14 Binary Search of Sorted Array A recursive binary search of a sorted array that (b) does not find its target.

15. 15 Java Class Library: The Method binarySearch The class Arrays in java.util defines versions of a static method with following specification: /** Task: Searches an entire array for a given item. * @param array the array to be searched * @param desiredItem the item to be found in the array * @return index of the array element that equals desiredItem; * otherwise returns -belongsAt-1, where belongsAt is * the index of the array element that should contain * desiredItem */ public static int binarySearch(type[] array, type desiredItem);

16. 16 Efficiency of a Binary Search Eliminates about half of the array from consideration after examining the middle element once. Counting the maximum number of comparisons under the worst-case Comparison is made each time the array is divided in half. First, begin with n items, first division left us with n/2 items, then second division left us with n/4 items, then third division left with n/8 items, until we get one element. n/2^k = 1 what is k for the total number of division? If n is a power of 2, k = log 2 n. If not, find a k such that: 2^(k-1) <n< 2^k; (k-1)<log 2 n <k

17. 17 Efficiency of a Binary Search Best caseO(1) Locate desired item first Worst caseO(log n) Must look at all the items Average caseO(log n)

18. 18 Iterative Sequential Search of an Unsorted Chain A chain of linked nodes that contain the entries in a list. Consecutive entries in the list beginning from the first. Moving from node to node is not as simple as moving from one array location to another. Use next to point to the next node.

19. 19 Iterative Sequential Search of an Unsorted Chain Implementation of iterative sequential search public boolean contains(Object anEntry) {boolean found = false; Node currentNode = firstNode; while (!found && (currentNode != null)) {if (anEntry.equals(currentNode.getData())) found = true; else currentNode = currentNode.getNextNode(); } // end while return found; } // end contains

20. 20 Recursive Sequential Search of an Unsorted Chain Recursive search method /** Task: Recursively searches a chain of nodes for desiredItem, * beginning with the node that current references. */ private boolean search(Node current, Object desiredItem) {boolean found; if (current = = null) found = false; else if (desiredItem.equals(current.getData())) found = true; else found = search(current.getNextNode(), desiredItem); return found; } // end search

21. 21 Efficiency of a Sequential Search of a Chain Best caseO(1) Locate desired item first Worst caseO(n) Must look at all the items Average caseO(n) Must look at half the items O(n/2) is just O(n)

22. 22 Searching a Sorted Chain Method to search a sorted chain public boolean contains(Object anEntry) {boolean found = false; Comparable entry = (Comparable)anEntry; Node currentNode = firstNode; while ( (currentNode != null) && (entry.compareTo(currentNode.getData()) > 0) ) {currentNode = currentNode.getNextNode(); } // end while if ( (currentNode != null) && entry.equals(currentNode.getData()) ) {found = true; } // end if return found; } // end contains Note: Binary search of a chain of linked nodes is impractical. It is less efficient than sequential search

23. 23 Choosing a Search Method Use a sequential search to search a chain of linked nodes. For array of objects, need to know which is applicable. Sequential search, must have a method equals Binary search, must have a method compareTo and array must be sorted.

24. 24 Choosing a Search Method Iterative search saves time, memory over recursive search. Usually, we use iterative version for the sequential search. For binary search, coding the binary search recursively is easier than coding it iteratively. Since binary search is fast, recursive will not require much additional space for recursive calls. Best Case Average Case Worst Case Sequential search (unsorted data) O(1)O(n) Sequential search (sorted data) O(1)O(n) Binary Search (sorted array) O(1)O(log n)