Sets, Maps and Hash Tables

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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

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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

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Sets

public interface Set<T>

{

void add(T element);

void remove();

boolean contains(T element);

Iterator<T> iterator();

}

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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)

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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?)

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Sets – Quality tip

• When using a Set, we must choose a spe-cific implementation (HashSet or TreeSet)

• However, the definition should look like:

Set<Car> cars = new HashSet<Car>();

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Sets – Quality tip

Set<Car> cars = new HashSet<Car>();

• Why…? We should in general only refer to the interface, not the implementation

• Easy to switch implementation!

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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)

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Maps

K1

K2

K3

K4

V1

V3

V2

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Map

public interface Map<K,V>

{

void put(K key,V value);

V get(K key);

void remove(K key);

Set<K> keySet();

}

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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

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Map

Map<String,Car> carMap = new HashMap<String,Car>();

...

Set<String> regNumbers = carMap.keySet();

for (String regNo : regNumbers)

{

Car aCar = carMap.get(regNo);

... // Do something with the Car object

}

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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

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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)

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Hash Tables

0 1 2 3 4

Car1 Car2

Car3f(Car1) = 3

f(Car2) = 0

f(Car3) = 2

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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))

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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?

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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 (???)

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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…)

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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]

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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

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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

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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

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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);

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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

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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;

}

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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

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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…

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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

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Hash Functions

• In general, you must remember to:– Either define the hashCode and the equals method

– Or not define any of them!

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Handling collisions

• Even with a good hash function, we will still experience collisions

• Collision: two different objects t1 and t2 have the same hash code

• We will then try to store both objects in the same position in the array

• Now what…?

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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

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Handling collisions

0 1 2 3 4

Car1Car2 Car3

Car4

Car5

Car6

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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

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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