1. Sets and Maps (and Hashing)
Chapter 9

2. Chapter Objectives
To understand the Java Map and Set interfaces and how to use them
To learn about hash codes and how they are used to facilitate efficient search and retrieval
To study two forms of hash tables—open addressing and chaining—and to understand their relative benefits and performance tradeoffs
Chapter 9: Sets and Maps

3. Chapter Objectives To learn how to implement both hash table forms
To be introduced to the implementation of Maps and Sets
To see how two earlier applications can be more easily implemented using Map objects for data storage
Chapter 9: Sets and Maps

4. Review of Sets Set is unordered, and has no duplicate elements
Suppose A = {1,3,5,7,9,11}, B = {2,3,5,7,11,13}
Then
A  B = {1,2,3,5,7,9,11,13}
A  B = {3,5,7,11}
A  B = {1,9}
B  A = {2,13}
If C = {3,5,9}, then C  A
Chapter 9: Sets and Maps

5. Sets and the Set Interface
The part of the Collection hierarchy that relates to sets
Includes three interfaces, two abstract classes, and two actual classes
Chapter 9: Sets and Maps

6. The Set Abstraction
A set is a collection that contains no duplicate elements
And at most, one null element
In a set, index of an element is meaningless
If s is a set,
s.contains(“apple”) returns true or false
s.indexOf(“apple”) makes no sense
s.get(i) is also nonsensical
Chapter 9: Sets and Maps

7. The Set Abstraction Operations on sets include: Testing for membership
Adding (inserting) elements
Removing elements
Union
Intersection
Difference
Subset
Chapter 9: Sets and Maps

8. The Set Interface and Methods
Has required methods for …
Testing set membership
Testing for an empty set
Determining set size
Creating an iterator over the set
Two optional methods for …
To add an element
To remove an element
Constructors enforce no duplicate members, and…
…add method does not allow duplicate item
Chapter 9: Sets and Maps

9. The Set Interface and Methods
Chapter 9: Sets and Maps

10. Comparison of Lists and Sets
Duplicate elements
OK in a list
Not allowed in sets: Set.add returns false if you try to insert a duplicate element
Get method
List has a get method
A set has no get method (index is meaningless)
Iterators
Lists have iterators
Can also iterate thru elements in a set
Chapter 9: Sets and Maps

11. Maps A map relates one set to another set
Map is a set of ordered pairs (x,y)
Where x == key and y == value (element)
For example
This map is:
{(J,Jane), (B,Bill), (B2,Bill), (S,Sam), (B1,Bob)}
Chapter 9: Sets and Maps

12. Maps Map is a set of ordered pairs (x,y)
Where x == key and y == value (element)
Keys must be unique
But values need not be unique (onto, not 1-to-1)
Each key “maps” to a particular value (element)
Or, you might say it “corresponds” to
Maps used for very efficient storage and retrieval of information in tables
Key is used like index into a list
But key does not need to be integer
Chapter 9: Sets and Maps

13. Maps Suppose we have the map:
{(J,Jane), (B,Bill), (B2,Bill), (S,Sam), (B1,Bob)}
And it is stored in “aMap”
Then
What does aMap.get(“B2”) return?
“Bill”
What does aMap.get(“Bill”) return?
Null, since nothing in aMap has key == “Bill”
Chapter 9: Sets and Maps

14. Map Interface
Chapter 9: Sets and Maps

15. Hash Tables For maps, want to access entry by its key, not its value
A hash table is used for such access
For efficiency, want to access element directly by its key
As opposed to searching for key value in an array
Using a hash table we can retrieve an item in constant time, on average, and linear time in worst case
That is, O(1) is expected, but O(n) is worst case
Chapter 9: Sets and Maps

16. Hash Codes and Index Calculation
Hashing idea
Transform an item’s key value into an integer
Then use this integer as a numeric index
Chapter 9: Sets and Maps

17. Hash Code Index Example
Suppose we want to store number of occurrences of each Unicode characters in a file
There are 65,536 Unicode characters
What to do?
Could create an array of size 65,536 and store count of character i in array element i
This will work, but…
…very inefficient for a small file
Suppose file only has 100 characters!
Is there a better way?
Chapter 9: Sets and Maps

18. Hash Code Index Calculation
Suppose we want to store number of occurrences of each Unicode characters in a file
There are 65,536 Unicode characters
File of 100 characters
Use a hash code for each character
But how to compute hash code?
Could do the following:
Create an array of size 200 and compute index as index = uniChar % 200
Good since it uses less space
Bad if there are collisions
2 or more characters in file “hash” to same value
Chapter 9: Sets and Maps

19. Methods for Generating Hash Codes
Usually, keys consist of strings of letters and/or digits
The number of possible key values is much larger than the table size
Generating a good hash code is something of an art
Some experimentation, trial-and-error may be required
Desirable properties of a “hash function”?
A “random” (uniform) distribution of values
Relatively simple function
Efficient to compute
Collisions can always occur---what to do?
Chapter 9: Sets and Maps

20. Java HashCode Method
For strings, could simply sum int values of all characters
Will return the same hash code for sign and sing
The Java API algorithm accounts for position of the characters as follows…
The String.hashCode() returns the integer calculated by the formula: s0 x 31(n-1) + s1 x 31(n-2) + … + sn-1 where si is the ith character of the string, and n is the length of the string
“Cat” will have a hash code of: ‘C’ x ‘a’ x 31 + ‘t’
Since 31 is a prime number, fewer collisions
Chapter 9: Sets and Maps

21. Open Addressing We consider two ways to organize hash tables
Chaining
For open addressing, linear probing can be used to deal with collisions
If that element contains an item with a different key, increment the index by one
Keep incrementing until you find the key or null entry
Null indicates element is not in the table
Chapter 9: Sets and Maps

22. Open Addressing Algorithm
Chapter 9: Sets and Maps

23. Table Wraparound and Search Termination
As index increases, must wrap around (circular array)
Leads to the potential of an infinite loop
How do you know when to stop searching if the table is full and you have not found the correct value?
Stop when the index value for the next probe is the same as the hash code value for the object, or…
Ensure that the table is never full by increasing its size after an insertion if its occupancy rate exceeds a specified threshold (sparser table has fewer collisions)
Chapter 9: Sets and Maps

24. Open Addressing Example
Suppose we have the following values and hash codes
Name
hashCode
hashCode % 5
hashCode %11
“Tom”
84274 4 3
“Dick” 5
“Harry” 10
“Sam”
82879
“Pete” 7
Chapter 9: Sets and Maps

25. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
null 1 2 3 4
Chapter 9: Sets and Maps

26. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
null 1 2 3 4
“Tom”
Chapter 9: Sets and Maps

27. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
“Dick” 1
null 2 3 4
“Tom”
Chapter 9: Sets and Maps

28. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
“Dick” 1
null 2 3
“Harry” 4
“Tom”
Chapter 9: Sets and Maps

29. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
“Dick” 1
“Sam” 2
null 3
“Harry” 4
“Tom”
Chapter 9: Sets and Maps

30. Open Addressing Example
Suppose we use hashCode % 5 to create hash table
Using open addressing
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
index
data
“Dick” 1
“Sam” 2
“Pete” 3
“Harry” 4
“Tom”
Chapter 9: Sets and Maps

31. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3 4 5 6 7 8 9 10
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

32. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3
“Tom” 4 5 6 7 8 9 10
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

33. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3
“Tom” 4 5
“Dick” 6 7 8 9 10
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

34. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3
“Tom” 4 5
“Dick” 6 7 8 9 10
“Harry”
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

35. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3
“Tom” 4 5
“Dick” 6
“Sam” 7 8 9 10
“Harry”
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

36. Open Addressing Example
Suppose we use hashCode % 11 to create hash table
Using open addressing
Index
data
null 1 2 3
“Tom” 4 5
“Dick” 6
“Sam” 7
“Pete” 8 9 10
“Harry”
Name
hashCode % 5
“Tom” 3
“Dick” 5
“Harry” 10
“Sam”
“Pete” 7
Chapter 9: Sets and Maps

37. Hash Table Operations
Iterating thru hash table gives entries in “arbitrary” order
Deleting from hash table
Cannot just insert a null --- why not?
Null used for stopping/not found condition
Can insert a “dummy value”
So, removing does not improve search time
Reducing collisions
Expand size of hash table, and rehash elements
Tradeoff between table size and search efficiency
Chapter 9: Sets and Maps

38. Reducing Collisions by Quadratic Probing
Linear probing tends to form clusters of keys in the table, causing longer search chains
Quadratic probing can reduce the effect of clustering
Increments form a quadratic series
Disadvantages?
More work to calculate next index (multiplication, addition, and modular division)
Not all table elements are examined when looking for an insertion index
Chapter 9: Sets and Maps

39. Chaining Chaining is an alternative to open addressing
Each table element references a linked list that contains all of the items that hash to the same table index
The linked list is often called a bucket
The approach sometimes called bucket hashing
Only items that have the same value for their hash codes will be examined when looking for an object
Chapter 9: Sets and Maps

40. Chaining Recall hashCode % 5
Chaining creates linked list for each collision
In this example
Linked list for Tom, Dick, Sam
Another linked list for Harry and Pete
Name
hashCode % 5
“Tom” 4
“Dick”
“Harry” 3
“Sam”
“Pete”
Chapter 9: Sets and Maps

41. Chaining
Chapter 9: Sets and Maps

42. Chaining Plusses? Conceptually simple Minimizes table size
Good search efficiency
Minuses?
Overhead of linked lists (more storage)
More complex (perhaps)
Chapter 9: Sets and Maps

43. Performance of Hash Tables
Load factor is number of filled cells divided by table size
Load factor has greatest effect on performance
The lower the load factor, the better the performance
Why?
Less chance of collision in a sparsely populated table
But, smaller the load factor, more wasted space…
Chapter 9: Sets and Maps

44. Performance of Hash Tables
Chapter 9: Sets and Maps

45. Maps and Hashing Maps use hash tables!
Hashing converts the key into an index
Index is place where corresponding value stored
Makes it possible to search efficiently
Recall, O(1), on average
Without having an (explicit) index
Of course, there is some additional overhead
Chapter 9: Sets and Maps

46. Implementing a Hash Table
Chapter 9: Sets and Maps

47. Implementing a Hash Table
Chapter 9: Sets and Maps

48. Implementation of Maps and Sets
Class Object implements methods hashCode and equals, so every class can access these methods unless it overrides them
Object.equals compares two objects based on their addresses, not their contents
Object.hashCode calculates an object’s hash code based on its address, not its contents
Java recommends that if you override the equals method, then you should also override the hashCode method
Chapter 9: Sets and Maps

49. Implementing HashSetOpen
Chapter 9: Sets and Maps

50. Implementing Java Map and Set Interfaces
The Java API uses a hash table to implement both the Map and Set interfaces
The task of implementing the two interfaces is simplified by the inclusion of abstract classes AbstractMap and AbstractSet in the Collection hierarchy
Chapter 9: Sets and Maps

51. Nested Interface Map.Entry
One requirement on the key-value pairs for a Map object is that they implement the interface Map.Entry<K, V>, which is an inner interface of interface Map
An implementer of the Map interface must contain an inner class that provides code for the methods in the table below
Chapter 9: Sets and Maps

52. Additional Applications of Maps
Can implement the phone directory using a map
Chapter 9: Sets and Maps

53. Additional Applications of Maps
Huffman Coding Problem
Use a map for creating an array of elements and replacing each input character by its bit string code in the output file
Frequency table
The key will be the input character
The value is the character code string
Chapter 9: Sets and Maps

54. Chapter Review
The Set interface describes an abstract data type that supports the same operations as a mathematical set
The Map interface describes an abstract data type that enables a user to access information corresponding to a specified key
A hash table uses hashing to transform an item’s key into a table index so that insertions, retrievals, and deletions can be performed in expected O(1) time
A collision occurs when two keys map to the same table index
In open addressing, linear probing is often used to resolve collisions
Chapter 9: Sets and Maps

55. Chapter Review
The best way to avoid collisions is to keep the table load factor relatively low by rehashing when the load factor reaches a value such as 0.75
In open addressing, you can’t remove an element from the table when you delete it, but you must mark it as deleted
A set view of a hash table can be obtained through method entrySet
Two Java API implementations of the Map (Set) interface are HashMap (HashSet) and TreeMap (TreeSet)
Chapter 9: Sets and Maps