1. Sets, Maps and Hash Tables

2. RHS – SOC 2 Sets We have learned that different data struc- tures have different advantages – and drawbacks Choosing the proper data structure depends on typical usage patterns Array- and list-oriented data structures are appropriate when the order of elements matter – but that is not always the case

3. RHS – SOC 3 Sets A Set is a data structure which can hold an unordered collection of elements Not having to worry about ordering can improve performance of other operations On a Set, we want to be able to –Insert an element –Delete an element –Check if a given element is in the Set

4. RHS – SOC 4 Sets public interface Set { void add(T element); void remove(); boolean contains(T element); Iterator iterator(); }

5. RHS – SOC 5 Sets It turns out that insertion, deletion and check for containment can be done in O(log(n)), or even faster! Depends on the underlying implemen- tation of the interface In Java, implementation is either –HashSet (based on Hash Tables) –TreeSet (based on Trees)

6. RHS – SOC 6 Sets A Set iterator is ”simpler” than e.g. a List iterator –Elements will occur in ”random” order –No add method – we just call add on the Set itself –No previous method – does not make sense The Set iterator does however have a delete method (why?)

7. RHS – SOC 7 Sets – Quality tip When using a Set, we must choose a spe- cific implementation (HashSet or TreeSet) However, the definition should look like: Set cars = new HashSet ();

8. RHS – SOC 8 Sets – Quality tip Set cars = new HashSet (); Why…? We should in general only refer to the interface, not the implementation Easy to switch implementation!

9. RHS – SOC 9 Maps A Map is a data structure which stores associations between –A collection of keys –A collection of values All keys map to a value Keys are unique (values are not)

10. RHS – SOC 10 Maps K1K1 K2K2 K3K3 K4K4 V1V1 V3V3 V2V2

11. RHS – SOC 11 Map public interface Map { void put(K key,V value); V get(K key); void remove(K key); Set keySet(); }

12. RHS – SOC 12 Map The keySet method returns a Set containing all keys in the Map You must then iterate through this Set, in order to get all values stored in the Map

13. RHS – SOC 13 Map Map carMap = new HashMap ();... Set regNumbers = carMap.keySet(); for (String regNo : regNumbers) { Car aCar = carMap.get(regNo);... // Do something with the Car object }

14. RHS – SOC 14 Exercises Review: R16.1, R16.4, R16.6 Programming: P16.4, P16.12

15. RHS – SOC 15 Hash Tables A Set and a Map are both abstract data types – we need a concrete implemen- tation in order to use them In the Java library, two implementations are available: –Sets: HashSet, TreeSet –Maps: HashMap, TreeMap

16. RHS – SOC 16 Hash Tables The implementations HashSet and HashMap are based on a Hash Table A Hash Table is based on the below ideas: –Create an array of length N, which can store objects of some type T –Find a mapping from T to the interval [0; N-1] (a Hash Function f) –Store an object t of type T in the position f(t)

17. RHS – SOC 17 Hash Tables 01234 Car 1 Car 2 Car 3 f(Car 1 ) = 3 f(Car 2 ) = 0 f(Car 3 ) = 2

18. RHS – SOC 18 Hash Tables A Hash Table is thus ”almost” an array Instead of having an index directly available, we must calculate it If calculation can be done in constant time, then all basic operations (insert, delete, lookup) can be done in constant time! Better than tree-based implementations, which have O(log(N))

19. RHS – SOC 19 Hash Tables However, there are some issues: –How do we define a good mapping from the objects to [0; N-1]? –What happens if we try to store two objects at the same position?

20. RHS – SOC 20 Hash Functions Before finding a good mapping – i.e. a good hash function – we must consider the size of the array For good performance, the array should at least be as large as the maximal number of objects stored Rule of thumb is about 30 % larger Size should be a prime number (???)

21. RHS – SOC 21 Hash Functions What if the expected number of objects is unknown in advance? We can expand a hash table dynamically If the hash table in running out of space, double the capacity Start out with a reasonably large array (space is cheap…)

22. RHS – SOC 22 Hash Functions Having handled the choice of N, how do we define a proper hash function? Properties of a hash function: –Must map all objects of type T to the interval [0; N-1] –Should map objects as uniformly as possible to the interval [0; N-1]

23. RHS – SOC 23 Hash Functions We can enforce the mapping to [0;N-1] by using the modulo operator: f(t) = g(t) % N g(t) can then produce any integer value How do we achieve a uniform distribution? Theory for this is complicated, but there are some general rules to follow

24. RHS – SOC 24 Hash Functions A good hash function should be ”almost ran- dom”, but deterministic –”Almost random” – values are well distri- buted in the interval –Deterministic – always produce the same output for the same input

25. RHS – SOC 25 Hash Functions In Java, all objects have a hashCode method –Defined in Object class –Can be overrided –Returns an integer (the Hash Code) –We must use modulo on the value ourselves

26. RHS – SOC 26 Hash Functions Hash function for integers: –The number itself… Hash function for strings: final int HASH_MULTIPLIER = 31; int h = 0; for (int i = 0; i < s.length; i++) h = (HASH_MULTIPLIER * h) + s.charAt(i);

27. RHS – SOC 27 Hash Functions Hash code for an object can be calculated by combining hash codes for instance fields Combine values in a way similar to the algorithm used to find string hash codes

28. RHS – SOC 28 Hash Functions public int hashCode() { final int MULTIPLIER = 31; int h1 = regNo.hashCode(); int h2 = mileage; int h3 = model.hashCode(); int h = h1*MULTIPLIER + h2; h = h*MULTIPLIER + h3; return h; }

29. RHS – SOC 29 Hash Functions But wait…what about numeric overflow? We multiply a ”random” integer value with a number…? Does not really matter… As long as the algorithm is deterministic, overflow is not a problem Just helps ”scrambling” the value

30. RHS – SOC 30 Hash Functions Common pitfalls: –Remember to define a hashCode function –If you forget, the hashCode implementation in Object is used –Based solely on memory location of object –Two objects with the same value of instance fields will produce different hash codes…

31. RHS – SOC 31 Hash Functions Common pitfalls: –The hashCode function must be ”compatible” with your equals function –If a.equals(b) it must hold that a.hashCode() == b.hashCode() –If not, duplicates are allowed! –The reverse condition is not required; two different objects may have the same hash code

32. RHS – SOC 32 Hash Functions In general, you must remember to: –Either define the hashCode and the equals method –Or not define any of them!

33. RHS – SOC 33 Handling collisions Even with a good hash function, we will still experience collisions Collision: two different objects t 1 and t 2 have the same hash code We will then try to store both objects in the same position in the array Now what…?

34. RHS – SOC 34 Handling collisions What we store in each position in the array is not the objects themselves, but a linked list of objects Objects with the same hash code h are stored in the linked list in position h With a good hash function, the average length of non-empty lists is less than 2

35. RHS – SOC 35 Handling collisions 01234 Car 1 Car 2 Car 3 Car 4 Car 5 Car 6

36. RHS – SOC 36 Handling collisions Basic operations (insert, delete, lookup) follow this structure: –Calculate hash code for the object –Find the corresponding position in the array Insert: Insert element at the end of list Delete/Lookup: Iterate through list until element is found, or end of list is reached

37. RHS – SOC 37 Handling collisions Basic operations are thus not done in truly constant time However, if a proper hash function is used, running time is constant in practice Use hash-based implementations unless special circumstances apply –Hard to define hash/equals function –More functionality required

38. RHS – SOC 38 Exercises Review: R16.8, R16.10 Programming: P16.6