Ottenere un set di alimentazione di un set in Java

1. Introduzione

In questo tutorial, studieremo il processo di generazione di un power set di un dato set in Java.

Come promemoria rapido, per ogni set di dimensione n , c'è un set di potenza di dimensione 2n . Impareremo come ottenerlo utilizzando varie tecniche.

2. Definizione di Power Set

L'insieme di potenze di un dato insieme S è l'insieme di tutti i sottoinsiemi di S , incluso S stesso e l'insieme vuoto.

Ad esempio, per un dato insieme:

{"APPLE", "ORANGE", "MANGO"}

il set di alimentazione è:

{ {}, {"APPLE"}, {"ORANGE"}, {"APPLE", "ORANGE"}, {"MANGO"}, {"APPLE", "MANGO"}, {"ORANGE", "MANGO"}, {"APPLE", "ORANGE", "MANGO"} }

Poiché è anche un insieme di sottoinsiemi, l'ordine dei suoi sottoinsiemi interni non è importante e possono apparire in qualsiasi ordine:

{ {}, {"MANGO"}, {"ORANGE"}, {"ORANGE", "MANGO"}, {"APPLE"}, {"APPLE", "MANGO"}, {"APPLE", "ORANGE"}, {"APPLE", "ORANGE", "MANGO"} }

3. Libreria Guava

La libreria Google Guava ha alcune utili utilità Set , come il set di alimentazione. Quindi, possiamo facilmente usarlo per ottenere anche il set di potenza del set dato:

@Test public void givenSet_WhenGuavaLibraryGeneratePowerSet_ThenItContainsAllSubsets() { ImmutableSet set = ImmutableSet.of("APPLE", "ORANGE", "MANGO"); Set
    
      powerSet = Sets.powerSet(set); Assertions.assertEquals((1 << set.size()), powerSet.size()); MatcherAssert.assertThat(powerSet, Matchers.containsInAnyOrder( ImmutableSet.of(), ImmutableSet.of("APPLE"), ImmutableSet.of("ORANGE"), ImmutableSet.of("APPLE", "ORANGE"), ImmutableSet.of("MANGO"), ImmutableSet.of("APPLE", "MANGO"), ImmutableSet.of("ORANGE", "MANGO"), ImmutableSet.of("APPLE", "ORANGE", "MANGO") )); }
    

Guava powerSet opera internamente sull'interfaccia Iterator nel modo in cui quando viene richiesto il sottoinsieme successivo, il sottoinsieme viene calcolato e restituito. Quindi, la complessità dello spazio è ridotta a O (n) invece di O (2n) .

Ma come fa Guava a raggiungere questo obiettivo?

4. Approccio alla generazione del gruppo di alimentazione

4.1. Algoritmo

Discutiamo ora i possibili passaggi per creare un algoritmo per questa operazione.

Il power set di un set vuoto è {{}} in cui contiene solo un set vuoto, quindi questo è il nostro caso più semplice.

Per ogni insieme S diverso dall'insieme vuoto, prima estraiamo un elemento e lo chiamiamo - elemento . Quindi, per il resto degli elementi di un set subsetWithoutElement , calcoliamo il loro power set ricorsivamente - e lo chiamiamo qualcosa come powerSet S ubsetWithoutElement . Quindi, aggiungendo l' elemento estratto a tutti i set in powerSet S ubsetWithoutElement , otteniamo powerSet S ubsetWithElement.

Ora, il power set S è l'unione di un powerSetSubsetWithoutElement e un powerSetSubsetWithElement :

Vediamo un esempio dello stack di power set ricorsivo per l'insieme dato {"APPLE", "ORANGE", "MANGO"} .

Per migliorare la leggibilità dell'immagine utilizziamo nomi brevi: P significa funzione power set e "A", "O", "M" sono rispettivamente forme brevi di "APPLE", "ORANGE" e "MANGO" :

4.2. Implementazione

Quindi, per prima cosa, scriviamo il codice Java per estrarre un elemento e ottenere i restanti sottoinsiemi:

T element = set.iterator().next(); Set subsetWithoutElement = new HashSet(); for (T s : set) { if (!s.equals(element)) { subsetWithoutElement.add(s); } }

Vorremo quindi ottenere il set di potenza di subsetWithoutElement :

Set
    
      powersetSubSetWithoutElement = recursivePowerSet(subsetWithoutElement);
    

Successivamente, dobbiamo aggiungere quel set di potenza all'originale:

Set
    
      powersetSubSetWithElement = new HashSet(); for (Set subsetWithoutElement : powerSetSubSetWithoutElement) { Set subsetWithElement = new HashSet(subsetWithoutElement); subsetWithElement.add(element); powerSetSubSetWithElement.add(subsetWithElement); }
    

Infine l'unione di powerSetSubSetWithoutElement e powerSetSubSetWithElement è il set di alimentazione del set di input specificato:

Set
    
      powerSet = new HashSet(); powerSet.addAll(powerSetSubSetWithoutElement); powerSet.addAll(powerSetSubSetWithElement);
    

Se mettiamo insieme tutti i nostri frammenti di codice, possiamo vedere il nostro prodotto finale:

public Set
    
      recursivePowerSet(Set set) { if (set.isEmpty()) { Set
     
       ret = new HashSet(); ret.add(set); return ret; } T element = set.iterator().next(); Set subSetWithoutElement = getSubSetWithoutElement(set, element); Set
      
        powerSetSubSetWithoutElement = recursivePowerSet(subSetWithoutElement); Set
       
         powerSetSubSetWithElement = addElementToAll(powerSetSubSetWithoutElement, element); Set
        
          powerSet = new HashSet(); powerSet.addAll(powerSetSubSetWithoutElement); powerSet.addAll(powerSetSubSetWithElement); return powerSet; } 
        
       
      
     
    

4.3. Note per i test unitari

Ora proviamo. Abbiamo un po 'di criteri qui per confermare:

  • Innanzitutto, controlliamo la dimensione del set di potenza e deve essere 2n per un set di dimensione n .
  • Quindi, ogni elemento si verificherà solo una volta in un sottoinsieme e 2n-1 sottoinsiemi diversi.
  • Infine, ogni sottoinsieme deve apparire una volta.

If all these conditions passed, we can be sure that our function works. Now, since we've used Set, we already know that there's no repetition. In that case, we only need to check the size of the power set, and the number of occurrences of each element in the subsets.

To check the size of the power set we can use:

MatcherAssert.assertThat(powerSet, IsCollectionWithSize.hasSize((1 << set.size())));

And to check the number of occurrences of each element:

Map counter = new HashMap(); for (Set subset : powerSet) { for (String name : subset) { int num = counter.getOrDefault(name, 0); counter.put(name, num + 1); } } counter.forEach((k, v) -> Assertions.assertEquals((1 << (set.size() - 1)), v.intValue()));

Finally, if we can put all together into one unit test:

@Test public void givenSet_WhenPowerSetIsCalculated_ThenItContainsAllSubsets() { Set set = RandomSetOfStringGenerator.generateRandomSet(); Set
    
      powerSet = new PowerSet().recursivePowerSet(set); MatcherAssert.assertThat(powerSet, IsCollectionWithSize.hasSize((1 << set.size()))); Map counter = new HashMap(); for (Set subset : powerSet) { for (String name : subset) { int num = counter.getOrDefault(name, 0); counter.put(name, num + 1); } } counter.forEach((k, v) -> Assertions.assertEquals((1 << (set.size() - 1)), v.intValue())); }
    

5. Optimization

In this section, we'll try to minimize the space and reduce the number of internal operations to calculate the power set in an optimal way.

5.1. Data Structure

As we can see in the given approach, we need a lot of subtractions in the recursive call, which consumes a large amount of time and memory.

Instead, we can map each set or subset to some other notions to reduce the number of operations.

First, we need to assign an increasing number starting from 0 to each object in the given set S which means we work with an ordered list of numbers.

For example for the given set {“APPLE”, “ORANGE”, “MANGO”} we get:

“APPLE” -> 0

“ORANGE” ->

“MANGO” -> 2

So, from now on, instead of generating subsets of S, we generate them for the ordered list of [0, 1, 2], and as it is ordered we can simulate subtractions by a starting index.

For example, if the starting index is 1 it means that we generate the power set of [1,2].

To retrieve mapped id from the object and vice-versa, we store both sides of mapping. Using our example, we store both (“MANGO” -> 2) and (2 -> “MANGO”). As the mapping of numbers started from zero, so for the reverse map there we can use a simple array to retrieve the respective object.

One of the possible implementations of this function would be:

private Map map = new HashMap(); private List reverseMap = new ArrayList(); private void initializeMap(Collection collection) { int mapId = 0; for (T c : collection) { map.put(c, mapId++); reverseMap.add(c); } }

Now, to represent subsets there are two well-known ideas:

  1. Index representation
  2. Binary representation

5.2. Index Representation

Each subset is represented by the index of its values. For example, the index mapping of the given set {“APPLE”, “ORANGE”, “MANGO”} would be:

{ {} -> {} [0] -> {"APPLE"} [1] -> {"ORANGE"} [0,1] -> {"APPLE", "ORANGE"} [2] -> {"MANGO"} [0,2] -> {"APPLE", "MANGO"} [1,2] -> {"ORANGE", "MANGO"} [0,1,2] -> {"APPLE", "ORANGE", "MANGO"} }

So, we can retrieve the respective set from a subset of indices with the given mapping:

private Set
    
      unMapIndex(Set
     
       sets) { Set
      
        ret = new HashSet(); for (Set s : sets) { HashSet subset = new HashSet(); for (Integer i : s) { subset.add(reverseMap.get(i)); } ret.add(subset); } return ret; }
      
     
    

5.3. Binary Representation

Or, we can represent each subset using binary. If an element of the actual set exists in this subset its respective value is 1; otherwise it is 0.

For our fruit example, the power set would be:

{ [0,0,0] -> {} [1,0,0] -> {"APPLE"} [0,1,0] -> {"ORANGE"} [1,1,0] -> {"APPLE", "ORANGE"} [0,0,1] -> {"MANGO"} [1,0,1] -> {"APPLE", "MANGO"} [0,1,1] -> {"ORANGE", "MANGO"} [1,1,1] -> {"APPLE", "ORANGE", "MANGO"} }

So, we can retrieve the respective set from a binary subset with the given mapping:

private Set
    
      unMapBinary(Collection
     
       sets) { Set
      
        ret = new HashSet(); for (List s : sets) { HashSet subset = new HashSet(); for (int i = 0; i < s.size(); i++) { if (s.get(i)) { subset.add(reverseMap.get(i)); } } ret.add(subset); } return ret; }
      
     
    

5.4. Recursive Algorithm Implementation

In this step, we'll try to implement the previous code using both data structures.

Before calling one of these functions, we need to call the initializeMap method to get the ordered list. Also, after creating our data structure, we need to call the respective unMap function to retrieve the actual objects:

public Set
    
      recursivePowerSetIndexRepresentation(Collection set) { initializeMap(set); Set
     
       powerSetIndices = recursivePowerSetIndexRepresentation(0, set.size()); return unMapIndex(powerSetIndices); }
     
    

So, let's try our hand at the index representation:

private Set
    
      recursivePowerSetIndexRepresentation(int idx, int n) { if (idx == n) { Set
     
       empty = new HashSet(); empty.add(new HashSet()); return empty; } Set
      
        powerSetSubset = recursivePowerSetIndexRepresentation(idx + 1, n); Set
       
         powerSet = new HashSet(powerSetSubset); for (Set s : powerSetSubset) { HashSet subSetIdxInclusive = new HashSet(s); subSetIdxInclusive.add(idx); powerSet.add(subSetIdxInclusive); } return powerSet; }
       
      
     
    

Now, let's see the binary approach:

private Set
    
      recursivePowerSetBinaryRepresentation(int idx, int n) { if (idx == n) { Set
     
       powerSetOfEmptySet = new HashSet(); powerSetOfEmptySet.add(Arrays.asList(new Boolean[n])); return powerSetOfEmptySet; } Set
      
        powerSetSubset = recursivePowerSetBinaryRepresentation(idx + 1, n); Set
       
         powerSet = new HashSet(); for (List s : powerSetSubset) { List subSetIdxExclusive = new ArrayList(s); subSetIdxExclusive.set(idx, false); powerSet.add(subSetIdxExclusive); List subSetIdxInclusive = new ArrayList(s); subSetIdxInclusive.set(idx, true); powerSet.add(subSetIdxInclusive); } return powerSet; }
       
      
     
    

5.5. Iterate Through [0, 2n)

Now, there is a nice optimization we can do with the binary representation. If we look at it, we can see that each row is equivalent to the binary format of a number in [0, 2n).

So, if we iterate through numbers from 0 to 2n, we can convert that index to binary, and use it to create a boolean representation of each subset:

private List
    
      iterativePowerSetByLoopOverNumbers(int n) { List
     
       powerSet = new ArrayList(); for (int i = 0; i < (1 << n); i++) { List subset = new ArrayList(n); for (int j = 0; j < n; j++) subset.add(((1 < 0); powerSet.add(subset); } return powerSet; }
     
    

5.6. Minimal Change Subsets by Gray Code

Now, if we define any bijective function from binary representation of length n to a number in [0, 2n), we can generate subsets in any order that we want.

Gray Code is a well-known function that is used to generate binary representations of numbers so that the binary representation of consecutive numbers differ by only one bit (even the difference of the last and first numbers is one).

We can thus optimize this just a bit further:

private List
    
      iterativePowerSetByLoopOverNumbersWithGrayCodeOrder(int n) { List
     
       powerSet = new ArrayList(); for (int i = 0; i < (1 << n); i++) { List subset = new ArrayList(n); for (int j = 0; j 
      
       > 1); subset.add(((1 < 0); } powerSet.add(subset); } return powerSet; }
      
     
    

6. Lazy Loading

To minimize the space usage of power set, which is O(2n), we can utilize the Iterator interface to fetch every subset, and also every element in each subset lazily.

6.1. ListIterator

First, to be able to iterate from 0 to 2n, we should have a special Iterator that loops over this range but not consuming the whole range beforehand.

To solve this problem, we'll use two variables; one for the size, which is 2n, and another for the current subset index. Our hasNext() function will check that position is less than size:

abstract class ListIterator implements Iterator { protected int position = 0; private int size; public ListIterator(int size) { this.size = size; } @Override public boolean hasNext() { return position < size; } }

And our next() function returns the subset for the current position and increases the value of position by one:

@Override public Set next() { return new Subset(map, reverseMap, position++); }

6.2. Subset

To have a lazy load Subset, we define a class that extends AbstractSet, and we override some of its functions.

By looping over all bits that are 1 in the receiving mask (or position) of the Subset, we can implement the Iterator and other methods in AbstractSet.

For example, the size() is the number of 1s in the receiving mask:

@Override public int size() { return Integer.bitCount(mask); }

And the contains() function is just whether the respective bit in the mask is 1 or not:

@Override public boolean contains(@Nullable Object o) { Integer index = map.get(o); return index != null && (mask & (1 << index)) != 0; }

We use another variable – remainingSetBits – to modify it whenever we retrieve its respective element in the subset we change that bit to 0. Then, the hasNext() checks if remainingSetBits is not zero (that is, it has at least one bit with a value of 1):

@Override public boolean hasNext() { return remainingSetBits != 0; }

And the next() function uses the right-most 1 in the remainingSetBits, then converts it to 0, and also returns the respective element:

@Override public E next() { int index = Integer.numberOfTrailingZeros(remainingSetBits); if (index == 32) { throw new NoSuchElementException(); } remainingSetBits &= ~(1 << index); return reverseMap.get(index); }

6.3. PowerSet

To have a lazy-load PowerSet class, we need a class that extends AbstractSet .

The size() function is simply 2 to the power of the set's size:

@Override public int size() { return (1 << this.set.size()); }

As the power set will contain all possible subsets of the input set, so contains(Object o) function checks if all elements of the object o are existing in the reverseMap (or in the input set):

@Override public boolean contains(@Nullable Object obj) { if (obj instanceof Set) { Set set = (Set) obj; return reverseMap.containsAll(set); } return false; }

To check equality of a given Object with this class, we can only check if the input set is equal to the given Object:

@Override public boolean equals(@Nullable Object obj) { if (obj instanceof PowerSet) { PowerSet that = (PowerSet) obj; return set.equals(that.set); } return super.equals(obj); }

The iterator() function returns an instance of ListIterator that we defined already:

@Override public Iterator
    
      iterator() { return new ListIterator
     
      (this.size()) { @Override public Set next() { return new Subset(map, reverseMap, position++); } }; }
     
    

The Guava library uses this lazy-load idea and these PowerSet and Subset are the equivalent implementations of the Guava library.

For more information, check their source code and documentation.

Furthermore, if we want to do parallel operation over subsets in PowerSet, we can call Subset for different values in a ThreadPool.

7. Summary

To sum up, first, we studied what is a power set. Then, we generated it by using the Guava Library. After that, we studied the approach and how we should implement it, and also how to write a unit test for it.

Finally, we utilized the Iterator interface to optimize the space of generation of subsets and also their internal elements.

Come sempre il codice sorgente è disponibile su GitHub.