Memperoleh Kuasa Set dari Java

1. Pengenalan

Dalam tutorial ini, kita akan mengkaji proses menghasilkan set kuasa dari set tertentu di Java.

Sebagai peringatan ringkas, untuk setiap set ukuran n , ada set kekuatan ukuran 2n . Kami akan belajar bagaimana menggunakannya dengan pelbagai teknik.

2. Definisi Set Kuasa

Kekuatan set S tertentu adalah set semua subset S , termasuk S itu sendiri dan set kosong.

Contohnya, untuk set tertentu:

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

set kuasa adalah:

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

Oleh kerana ia juga merupakan set subset, susunan subset dalamannya tidak penting dan mereka boleh muncul dalam urutan apa pun:

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

3. Perpustakaan Jambu Batu

Perpustakaan Jambu Batu Google mempunyai beberapa utiliti Set yang berguna , seperti set kuasa. Oleh itu, kita dapat menggunakannya dengan mudah untuk mendapatkan kekuatan dari set yang diberikan juga:

@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 beroperasi secara dalaman melalui antara muka Iterator dengan cara apabila subset berikutnya diminta, subset dikira dan dikembalikan. Jadi, kerumitan ruang dikurangkan menjadi O (n) dan bukannya O (2n) .

Tetapi, bagaimana Jambu dapat mencapainya?

4. Pendekatan Penjanaan Set Kuasa

4.1. Algoritma

Sekarang mari kita bincangkan langkah-langkah yang mungkin untuk membuat algoritma untuk operasi ini.

Kekuatan set kosong adalah {{}} di mana ia hanya mengandungi satu set kosong, jadi begitulah kesederhanaan kami.

Untuk setiap set S selain set kosong, pertama-tama kita mengekstrak satu elemen dan menamakannya - elemen . Kemudian, untuk sisa elemen subsetWithoutElement , kami mengira set kuasa mereka secara berulang - dan menamakannya seperti powerSet S ubsetWithoutElement . Kemudian, dengan menambahkan elemen yang diekstrak ke semua set di powerSet S ubsetWithoutElement , kita mendapat powerSet S ubsetWithElement.

Sekarang, set kuasa S adalah penyatuan powerSetSubsetWithoutElement dan powerSetSubsetWithElement :

Mari kita lihat contoh tumpukan set daya rekursif untuk set yang diberikan {“APPLE”, “ORANGE”, “MANGO”} .

Untuk meningkatkan keterbacaan gambar, kami menggunakan bentuk nama pendek: P bermaksud fungsi set daya dan "A", "O", "M" adalah bentuk pendek dari "APPLE", "ORANGE", dan "MANGO" , masing-masing:

4.2. Pelaksanaan

Jadi, pertama, mari tulis kod Java untuk mengekstrak satu elemen dan dapatkan subset yang tinggal:

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

Kami kemudian mahu mendapatkan set kuasa subsetWithoutElement :

Set
    
      powersetSubSetWithoutElement = recursivePowerSet(subsetWithoutElement);
    

Seterusnya, kita perlu menambahkan kuasa yang kembali ke asal:

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

Akhirnya penyatuan powerSetSubSetWithoutElement dan powerSetSubSetWithElement adalah set kuasa dari set input yang diberikan:

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

Sekiranya kami mengumpulkan semua coretan kod kami, kami dapat melihat produk akhir kami:

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. Catatan untuk Ujian Unit

Sekarang mari kita uji. Kami mempunyai sedikit kriteria di sini untuk mengesahkan:

  • Pertama, kita periksa ukuran set kuasa dan mestilah 2n untuk satu set ukuran n .
  • Kemudian, setiap elemen hanya akan berlaku satu kali dalam subset dan 2n-1 subset yang berbeza.
  • Akhirnya, setiap subset mesti muncul sekali.

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.

Seperti biasa kod sumber tersedia di GitHub.