1. 1 Hashing Techniques: Implementation Implementing Hash Functions Implementing Hash Tables Implementing Chained Hash Tables Implementing Open Hash Tables

2. 2 Implementing Hash Functions Recall that search keys can be letters, strings or structures like records/containers. Thus we make the search key as general as possible, of type Object, in our implementation. Then, can we simultaneously represent both basic values and objects? Define wrappers for the basic values to make the hash function generic. Java’s wrapper classes are not adequate for our purpose.

3. 3 Implementing Hash Functions (cont'd) For convenience, we represent our hash function h as the composition of two functions f and g, i.e., h = g o f. That is, given f and g and a key k, the hash value of k is: h(k) = g(f(k)). Decomposing h provides a good separation of concern. Ideally, f and g should be independent of each other. With care, we can design f and g in such a way that the composition h = g o f is a good hash function.

4. 4 Implementing Hash Functions (cont'd) For a set K of keys and the set Z of non-negative integers, we define f : f: K -> Z The function g maps Z into {0,1,…,n-1}: g: Z -> {0, 1, …, n-1} Since Java’s Object class defines a method hashCode(), we can carefully define f to correspond to it. Note that the choice of f depends on the characteristics of its domain, the key set K.

5. 5 Hash Functions with Integral Keys The Integral data types in Java are byte, short, int, long and char Their underlying implementation can be viewed as integer, and we have: f (k) = k The wrapper class for these types can be written: 1 public class Int extends AbstractObject{ 2 protected int value; 3 public int hashCode(){ 4 return value; 5 } 6} In this case the hashCode method simply returns the content of the value field.

6. 6 Hash Functions with String Keys A string in Java is a sequence of characters. There are many ways of defining functions of strings that return integers. One way is the folding technique achieved by summing the ASCII values of the characters in a string. That is, given a string s = c 0 c 1 c 2 …c n-1, we have f(s) = c 0 + c 1 + c 2 + … + c n-1 mod n Since f can easily overflow, we use the alternative: char[0] + (27 * char[1]) + (729 * char[2])mod n

7. 7 Hash Functions with String Keys (cont'd) 1public class Str extends AbstractObject { 2 protected String value; 3 private size = value.length(); 4 public int hashCode() { 5int hashVal = 0; 6return (value.charAt(0) + 7 (27 * value.charAt(1)) + 8 (729 * value.charAt(2))) % size; 9 } 10}

8. 8 Implementing Hash Tables: The Hierarchy Tree MyComparable Container SearchableContainer HashTable AbstractObject AbstractContainer AbstractSearchableContainer AbstractHashTable ChainedHashTable OpenScatterTable

9. 9 The HashTable Interface From the introductory sessions, the interface HashTable extends the SearchableContainer interface. It has the following definition: 1public interface HashTable 2 extends SearchableContainer { 3 double getLoadFactor(); 4 } HashTable introduces getLoadFactor() which returns the load factor of the hash table. An AbstractHashTable is defined from which concrete hash table implementations will result.

10. 10 The AbstractHashTable Class 1 public abstract class AbstractHashTable 2 extends AbstractSearchableContainer 3 implements HashTable { 5 public abstract int getLength(); 7 protected final int f(Object obj) { 8 return obj.hashCode(); 9 } 10 protected final int g(int i) { 11 return Math.abs(i) % getLength(); 12 } 13 protected final int h(Object obj) { 14 return g(f(obj)); 15 } 16 //... 17 }

11. 11 The ChainedHashTable 1public class ChainedHashTable 2extends AbstractHashTable { 3protected LinkedList [] array; 4public ChainedHashTable(int size) { 5array = new LinkedList[size]; 6for(int j = 0; j < size; j++) 7array[j] = new LinkedList(); 8} 9public int getLength() { 10return array.length; 11} 12public void purge() { 13for(int i = 0; i < getLength(); i++) 14array[i].purge(); 15super.count = 0; 16} 17}

12. 12 Chained Hash Tables: Insertion & Deletion 1public class ChainedHashTable 2 extends AbstractHashTable { 3 protected LinkedList [] array; 4 public void insert(Comparable comparable) { 5 array[h(comparable)].append(comparable); 6 super.count++; 7 } 8 public void withdraw(Comparable comparable) { 9 array[h(comparable)].extract(comparable); 10 super.count--; 11 } 12}

13. 13 Chained Hash Tables: Retrieval 1public class ChainedHashTable extends 2AbstractHashTable { 3 protected LinkedList [] array; 4 public Comparable find(Comparable comparable){ 5 for(LinkedList.Element chain = 6 array[h(comparable)].getHead(); 7 chain != null; chain = 8 chain.getNext()){ 9 Comparable comparable1 = 10 (Comparable)chain.getDatum(); 11 12 if(comparable.isEQ(comparable1)) 13 return comparable1; 14 } 15 return null; 16 } 17}

14. 14 Open Hash Table Implementation 1 public class OpenScatterTable 2 extends AbstractHashTable { 3 4 protected Entry array[]; 5 static final int empty = 0; 6 static final int occupied = 1; 7 static final int deleted = 2; 8 9 protected static final class Entry { 10 int state; 11 Comparable object; 12 void purge(){ state = 0; object = null;} 13 14 Entry() {state = 0;} 15 } 16 }

15. 15 Open Hash Table: Initialization 1 public class OpenScatterTable 2 extends AbstractHashTable { 3 protected Entry array[]; 4 5 public OpenScatterTable(int size) { 6 7 array = new Entry[size]; 8 9 for(int j = 0; j < size; j++) 10 array[j] = new Entry(); 11 } 12 public void purge() { 13 for(int i = 0; i < getLength(); i++) 14 array[i].purge(); 15 super.count = 0; 16 } 17 }

16. 16 Open Hash Tables: Finding Empty Cells 1 public class OpenScatterTable 2 extends AbstractHashTable { 3 4protected static int c(int i) { 5 return i; 6} 7 protected int findUnoccupied(Object obj) { 8 9 int i = h(obj); 10 11 for(int j = 0; j < super.count + 1; j++) { 12 int k = (i + c(j)) % getLength(); 13 if(array[k].state != occupied) 14 return k; 15 } 16 throw new ContainerFullException(); 17 } 18}

17. 17 Open Hash Tables: Insertion 1public class OpenScatterTable 2 extends AbstractHashTable { 3 4 public void insert(Comparable comparable) { 5 6 if(super.count == getLength()) { 7 throw new ContainerFullException(); 8 } else { 9 int i = findUnoccupied(comparable); 10 array[i].state = occupied; 11 array[i].object = comparable; 12 super.count++; 13 return; 14 } 15 } 16}

18. 18 Open Hash Tables: Retrieval 1 public class OpenScatterTable 2 extends AbstractHashTable { 3 protected int findMatch(Comparable comparable) { 4 int i = h(comparable); 5 for(int j = 0; j < getLength(); j++) { 6 int k = (i + c(j)) % getLength(); 7 if(array[k].state == empty) break; 8 if(array[k].state == occupied && 9 comparable.isEQ(array[k].object)) 10 return k; 11 } 11 return -1; 12 } 13 public Comparable find(Comparable comparable) { 14 int i = findMatch(comparable); 15 if(i >= 0) return array[i].object; 16 else return null; 17 } 18 }

19. 19 Open Hash Tables: Deletion 1 public class OpenScatterTable 2 extends AbstractHashTable { 3 4 public void withdraw(Comparable comparable) { 5 if(super.count == 0) 6 throw new ContainerEmptyException(); 7 int i = findInstance(comparable); 8 if(i < 0) { 9 throw 10new IllegalArgumentException( 11 "object not found"); 12 } else { 13 array[i].state = 2; 14 array[i].object = null; 15 super.count--; 16 return; 17 } 18 } 19 }

20. 20 Review Questions Why are the methods f, g and h on Page 9 made final? Would someone not like to override them? Compare the insert() method of the ChainedHashTable class and that of OpenScatterTable. Are they both correct? How do you modify the findUnoccupied() method and ensure that it is efficient? Why is the isEQ() method used in Line 12 of find() on Page 13 instead of Java Object's equals() method? The method withdraw() of OpenScatterTable will be very inefficient if there are many deletions. Why?

21. 21 Exercises 1. Why does the Int wrapper class, in particular, extends the AbstractObject class? 2. Are the wrapper classes Int and Str concrete classes or abstract classes? Why? 3. What does the hashCode() method of Object class return by default? Why do we have to redefine it in our hash table implementation? 4. If the cost of inserting a record r in a hash table is O(1), what is the cost of deleting r? 5. Can the load factor in an open addressed hash table that is efficiently implemented be 1? Explain your answer. 6. In some hashing schemes, table size has no effect on collisions. Explain.