迭代计算任意数量集合的笛卡尔积

Question

我想计算 Java 中任意数量的非空集的笛卡尔积。

我已经编写了那个迭代代码...

public static <T> List<Set<T>> cartesianProduct(List<Set<T>> list) {
    List<Iterator<T>> iterators = new ArrayList<Iterator<T>>(list.size());
    List<T> elements = new ArrayList<T>(list.size());
    List<Set<T>> toRet = new ArrayList<Set<T>>();
    for (int i = 0; i < list.size(); i++) {
        iterators.add(list.get(i).iterator());
        elements.add(iterators.get(i).next());
    }
    for (int j = 1; j >= 0;) {
        toRet.add(Sets.newHashSet(elements));
        for (j = iterators.size()-1; j >= 0 && !iterators.get(j).hasNext(); j--) {
            iterators.set(j, list.get(j).iterator());
            elements.set(j, iterators.get(j).next());
        }
        elements.set(Math.abs(j), iterators.get(Math.abs(j)).next());
    }
    return toRet;
}

...但我发现它相当不雅。 有人有更好的迭代解决方案吗？一个使用一些奇妙的函数式方法的解决方案？ 否则......关于如何改进它的建议？错误？

Answer 1

我编写了一个不需要您在内存中填充大量集合的解决方案。不幸的是，所需的代码长达数百行。您可能需要等到它出现在 Guava 项目 (https://github.com/google/guava) 中，我希望它会在今年年底出现。对不起。 :(

请注意，如果您要进行笛卡尔乘积的集合数量是编译时已知的固定数量，则您可能不需要这样的实用程序——您可以只使用该数量的嵌套 for 循环。

编辑：代码现已发布。

Sets.cartesianProduct()

我想你会很高兴的。它只会根据您的要求创建单独的列表；不会用所有 MxNxPxQ 填满内存。

如果你想检查源，它是here。

享受吧！

Answer 2

使用 Google Guava 19 和 Java 8 非常简单：

假设您有要关联的所有数组的列表...

public static void main(String[] args) {
  List<String[]> elements = Arrays.asList(
    new String[]{"John", "Mary"}, 
    new String[]{"Eats", "Works", "Plays"},
    new String[]{"Food", "Computer", "Guitar"}
  );

  // Create a list of immutableLists of strings
  List<ImmutableList<String>> immutableElements = makeListofImmutable(elements);

  // Use Guava's Lists.cartesianProduct, since Guava 19
  List<List<String>> cartesianProduct = Lists.cartesianProduct(immutableElements);

  System.out.println(cartesianProduct);
}

不可变列表的制作方法如下：

/**
 * @param values the list of all profiles provided by the client in matrix.json
 * @return the list of ImmutableList to compute the Cartesian product of values
 */
private static List<ImmutableList<String>> makeListofImmutable(List<String[]> values) {
  List<ImmutableList<String>> converted = new LinkedList<>();
  values.forEach(array -> {
    converted.add(ImmutableList.copyOf(array));
  });
  return converted;
}

输出如下：

[
  [John, Eats, Food], [John, Eats, Computer], [John, Eats, Guitar],
  [John, Works, Food], [John, Works, Computer], [John, Works, Guitar], 
  [John, Plays, Food], [John, Plays, Computer], [John, Plays, Guitar],
  [Mary, Eats, Food], [Mary, Eats, Computer], [Mary, Eats, Guitar],
  [Mary, Works, Food], [Mary, Works, Computer], [Mary, Works, Guitar],
  [Mary, Plays, Food], [Mary, Plays, Computer], [Mary, Plays, Guitar]
]

Answer 3

这是我编写的一个迭代的、惰性的实现。该接口与 Google 的 Sets.cartesianProduct 非常相似，但更灵活一点：它处理的是 Iterables 而不是 Sets。此代码及其单元测试位于https://gist.github.com/1911614。

/* Copyright 2012 LinkedIn Corp.

   Licensed under the Apache License, Version 2.0 (the "License");
   you may not use this file except in compliance with the License.
   You may obtain a copy of the License at

       http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS,
   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   See the License for the specific language governing permissions and
   limitations under the License.
 */

import com.google.common.base.Function;
import com.google.common.collect.Iterables;
import java.lang.reflect.Array;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Iterator;
import java.util.List;
import java.util.NoSuchElementException;

/**
 * Implements the Cartesian product of ordered collections.
 * 
 * @author <a href="mailto:jmkristian@gmail.com">John Kristian</a>
 */
public class Cartesian {
  /**
   * Generate the <a href="http://en.wikipedia.org/wiki/Cartesian_product">Cartesian
   * product</a> of the given axes. For axes [[a1, a2 ...], [b1, b2 ...], [c1, c2 ...]
   * ...] the product is [[a1, b1, c1 ...] ... [a1, b1, c2 ...] ... [a1, b2, c1 ...] ...
   * [aN, bN, cN ...]]. In other words, the results are generated in same order as these
   * nested loops:
   * 
   * <pre>
   * for (T a : [a1, a2 ...])
   *   for (T b : [b1, b2 ...])
   *     for (T c : [c1, c2 ...])
   *       ...
   *         result = new T[]{ a, b, c ... };
   * </pre>
   * 
   * Each result is a new array of T, whose elements refer to the elements of the axes. If
   * you prefer a List, you can call asLists(product(axes)).
   * <p>
   * Don't change the axes while iterating over their product, as a rule. Changes to an
   * axis can affect the product or cause iteration to fail (which is usually bad). To
   * prevent this, you can pass clones of your axes to this method.
   * <p>
   * The implementation is lazy. This method iterates over the axes, and returns an
   * Iterable that contains a reference to each axis. Iterating over the product causes
   * iteration over each axis. Methods of each axis are called as late as practical.
   */
  public static <T> Iterable<T[]> product(Class<T> resultType,
                                          Iterable<? extends Iterable<? extends T>> axes) {
    return new Product<T>(resultType, newArray(Iterable.class, axes));
  }

  /** Works like product(resultType, Arrays.asList(axes)), but slightly more efficient. */
  public static <T> Iterable<T[]> product(Class<T> resultType, Iterable<? extends T>... axes) {
    return new Product<T>(resultType, axes.clone());
  }

  /**
   * Wrap the given arrays in fixed-size lists. Changes to the lists write through to the
   * arrays.
   */
  public static <T> Iterable<List<T>> asLists(Iterable<? extends T[]> arrays) {
    return Iterables.transform(arrays, new AsList<T>());
  }

  /**
   * Arrays.asList, represented as a Function (as used in Google collections).
   */
  public static class AsList<T> implements Function<T[], List<T>> {
    @Override
    public List<T> apply(T[] array) {
      return Arrays.asList(array);
    }
  }

  /** Create a generic array containing references to the given objects. */
  private static <T> T[] newArray(Class<? super T> elementType, Iterable<? extends T> from) {
    List<T> list = new ArrayList<T>();
    for (T f : from)
      list.add(f);
    return list.toArray(newArray(elementType, list.size()));
  }

  /** Create a generic array. */
  @SuppressWarnings("unchecked")
  private static <T> T[] newArray(Class<? super T> elementType, int length) {
    return (T[]) Array.newInstance(elementType, length);
  }

  private static class Product<T> implements Iterable<T[]> {
    private final Class<T> _resultType;
    private final Iterable<? extends T>[] _axes;

    /** Caution: the given array of axes is contained by reference, not cloned. */
    Product(Class<T> resultType, Iterable<? extends T>[] axes) {
      _resultType = resultType;
      _axes = axes;
    }

    @Override
    public Iterator<T[]> iterator() {
      if (_axes.length <= 0) // an edge case
        return Collections.singleton(newArray(_resultType, 0)).iterator();
      return new ProductIterator<T>(_resultType, _axes);
    }

    @Override
    public String toString() {
      return "Cartesian.product(" + Arrays.toString(_axes) + ")";
    }

    private static class ProductIterator<T> implements Iterator<T[]> {
      private final Iterable<? extends T>[] _axes;
      private final Iterator<? extends T>[] _iterators; // one per axis
      private final T[] _result; // a copy of the last result
      /**
       * The minimum index such that this.next() will return an array that contains
       * _iterators[index].next(). There are some special sentinel values: NEW means this
       * is a freshly constructed iterator, DONE means all combinations have been
       * exhausted (so this.hasNext() == false) and _iterators.length means the value is
       * unknown (to be determined by this.hasNext).
       */
      private int _nextIndex = NEW;
      private static final int NEW = -2;
      private static final int DONE = -1;

      /** Caution: the given array of axes is contained by reference, not cloned. */
      ProductIterator(Class<T> resultType, Iterable<? extends T>[] axes) {
        _axes = axes;
        _iterators = Cartesian.<Iterator<? extends T>> newArray(Iterator.class, _axes.length);
        for (int a = 0; a < _axes.length; ++a) {
          _iterators[a] = axes[a].iterator();
        }
        _result = newArray(resultType, _iterators.length);
      }

      private void close() {
        _nextIndex = DONE;
        // Release references, to encourage garbage collection:
        Arrays.fill(_iterators, null);
        Arrays.fill(_result, null);
      }

      @Override
      public boolean hasNext() {
        if (_nextIndex == NEW) { // This is the first call to hasNext().
          _nextIndex = 0; // start here
          for (Iterator<? extends T> iter : _iterators) {
            if (!iter.hasNext()) {
              close(); // no combinations
              break;
            }
          }
        } else if (_nextIndex >= _iterators.length) {
          // This is the first call to hasNext() after next() returned a result.
          // Determine the _nextIndex to be used by next():
          for (_nextIndex = _iterators.length - 1; _nextIndex >= 0; --_nextIndex) {
            Iterator<? extends T> iter = _iterators[_nextIndex];
            if (iter.hasNext()) {
              break; // start here
            }
            if (_nextIndex == 0) { // All combinations have been generated.
              close();
              break;
            }
            // Repeat this axis, with the next value from the previous axis.
            iter = _axes[_nextIndex].iterator();
            _iterators[_nextIndex] = iter;
            if (!iter.hasNext()) { // Oops; this axis can't be repeated.
              close(); // no more combinations
              break;
            }
          }
        }
        return _nextIndex >= 0;
      }

      @Override
      public T[] next() {
        if (!hasNext())
          throw new NoSuchElementException("!hasNext");
        for (; _nextIndex < _iterators.length; ++_nextIndex) {
          _result[_nextIndex] = _iterators[_nextIndex].next();
        }
        return _result.clone();
      }

      @Override
      public void remove() {
        for (Iterator<? extends T> iter : _iterators) {
          iter.remove();
        }
      }

      @Override
      public String toString() {
        return "Cartesian.product(" + Arrays.toString(_axes) + ").iterator()";
      }
    }
  }
}

Answer 4

基于索引的解决方案

使用索引是一种简单的替代方法，它快速且节省内存，并且可以处理任意数量的集合。实现 Iterable 允许在 for-each 循环中轻松使用。有关用法示例，请参见 #main 方法。

public class CartesianProduct implements Iterable<int[]>, Iterator<int[]> {
    private final int[] _lengths;
    private final int[] _indices;
    private boolean _hasNext = true;

    public CartesianProduct(int[] lengths) {
        _lengths = lengths;
        _indices = new int[lengths.length];
    }

    public boolean hasNext() {
        return _hasNext;
    }

    public int[] next() {
        int[] result = Arrays.copyOf(_indices, _indices.length);
        for (int i = _indices.length - 1; i >= 0; i--) {
            if (_indices[i] == _lengths[i] - 1) {
                _indices[i] = 0;
                if (i == 0) {
                    _hasNext = false;
                }
            } else {
                _indices[i]++;
                break;
            }
        }
        return result;
    }

    public Iterator<int[]> iterator() {
        return this;
    }

    public void remove() {
        throw new UnsupportedOperationException();
    }

    /**
     * Usage example. Prints out
     *
     * <pre>
     * [0, 0, 0] a, NANOSECONDS, 1
     * [0, 0, 1] a, NANOSECONDS, 2
     * [0, 0, 2] a, NANOSECONDS, 3
     * [0, 0, 3] a, NANOSECONDS, 4
     * [0, 1, 0] a, MICROSECONDS, 1
     * [0, 1, 1] a, MICROSECONDS, 2
     * [0, 1, 2] a, MICROSECONDS, 3
     * [0, 1, 3] a, MICROSECONDS, 4
     * [0, 2, 0] a, MILLISECONDS, 1
     * [0, 2, 1] a, MILLISECONDS, 2
     * [0, 2, 2] a, MILLISECONDS, 3
     * [0, 2, 3] a, MILLISECONDS, 4
     * [0, 3, 0] a, SECONDS, 1
     * [0, 3, 1] a, SECONDS, 2
     * [0, 3, 2] a, SECONDS, 3
     * [0, 3, 3] a, SECONDS, 4
     * [0, 4, 0] a, MINUTES, 1
     * [0, 4, 1] a, MINUTES, 2
     * ...
     * </pre>
     */
    public static void main(String[] args) {
        String[] list1 = {"a", "b", "c",};
        TimeUnit[] list2 = TimeUnit.values();
        int[] list3 = new int[]{1, 2, 3, 4};

        int[] lengths = new int[]{list1.length, list2.length, list3.length};
        for (int[] indices : new CartesianProduct(lengths)) {
            System.out.println(Arrays.toString(indices) //
                    + " " + list1[indices[0]] //
                    + ", " + list2[indices[1]] //
                    + ", " + list3[indices[2]]);
        }
    }
}

Answer 5

以下答案使用迭代而不是递归。它使用与我之前的答案相同的 Tuple 类。

这是一个单独的答案，因为恕我直言，两者都是有效的不同方法。

这是新的主类：

public class Example {
    public static <T> List<Tuple<T>> cartesianProduct(List<Set<T>> sets) {
        List<Tuple<T>> tuples = new ArrayList<Tuple<T>>();
        for (Set<T> set : sets) {
            if (tuples.isEmpty()) {
                for (T t : set) {
                    Tuple<T> tuple = new Tuple<T>();
                    tuple.add(t);
                    tuples.add(tuple);
                }
            } else {
                List<Tuple<T>> newTuples = new ArrayList<Tuple<T>>();
                for (Tuple<T> subTuple : tuples) {
                    for (T t : set) {
                        Tuple<T> tuple = new Tuple<T>();
                        tuple.addAll(subTuple);
                        tuple.add(t);
                        newTuples.add(tuple);
                    }
                }
                tuples = newTuples;
            }
        }
        return tuples;
    }
}

Answer 6

您可能对有关笛卡尔积的另一个问题感兴趣（编辑：删除以保存超链接，搜索标签笛卡尔积）。这个答案有一个很好的递归解决方案，我很难改进。您是否特别想要迭代解决方案而不是递归解决方案？

---

在 perl 和 a clean explanation 中查看了关于堆栈溢出的另一个迭代解决方案之后，这里是另一个解决方案：

public static <T> List<Set<T>> uglyCartesianProduct(List<Set<T>> list) {
    List<Iterator<T>> iterators = new ArrayList<Iterator<T>>(list.size());
    List<T> elements = new ArrayList<T>(list.size());
    List<Set<T>> toRet = new ArrayList<Set<T>>();

    for (int i = 0; i < list.size(); i++) {
        iterators.add(list.get(i).iterator());
        elements.add(iterators.get(i).next());
    }

    for (int i = 0; i < numberOfTuples(list); i++) {
        toRet.add(new HashSet<T>());
    }

    int setIndex = 0;
    for (Set<T> set : list) {
        int index = 0;
        for (int i = 0; i < numberOfTuples(list); i++) {
            toRet.get(index).add((T) set.toArray()[index % set.size()]);
            index++;
        }
        setIndex++;
    }
    return toRet;
}

private static <T> int numberOfTuples(List<Set<T>> list) {
    int product = 1;
    for (Set<T> set : list) {
        product *= set.size();
    }
    return product;
}

Answer 7

这是一种惰性迭代器方法，它使用函数来产生适当的输出类型。

public static <T> Iterable<T> cartesianProduct(
        final Function<Object[], T> fn, Object[]... options) {
    final Object[][] opts = new Object[options.length][];
    for (int i = opts.length; --i >= 0; ) {
        // NPE on null input collections, and handle the empty output case here
        // since the iterator code below assumes that it is not exhausted the
        // first time through fetch.
        if (options[i].length == 0) {
            return Collections.emptySet();
        }
        opts[i] = options[i].clone();
    }
    return new Iterable<T>() {
        public Iterator<T> iterator() {
            return new Iterator<T>() {
                final int[] pos = new int[opts.length];
                boolean hasPending;
                T pending;
                boolean exhausted;

                public boolean hasNext() {
                    fetch();
                    return hasPending;
                }

                public T next() {
                    fetch();
                    if (!hasPending) {
                        throw new NoSuchElementException();
                    }
                    T out = pending;
                    pending = null;  // release for GC
                    hasPending = false;
                    return out;
                }

                public void remove() {
                    throw new UnsupportedOperationException();
                }

                private void fetch() {
                    if (hasPending || exhausted) {
                        return;
                    }
                    // Produce a result.
                    int n = pos.length;
                    Object[] args = new Object[n];
                    for (int j = n; --j >= 0; ) {
                        args[j] = opts[j][pos[j]];
                    }
                    pending = fn.apply(args);
                    hasPending = true;
                    // Increment to next.
                    for (int i = n; --i >= 0; ) {
                        if (++pos[i] < opts[i].length) {
                            for (int j = n; --j > i; ) {
                                pos[j] = 0;
                            }
                            return;
                        }
                    }
                    exhausted = true;
                }
            };
        }
    };
}

Answer 8

我为字符串表编写了一个递归笛卡尔积算法。您可以将其修改为具有集合。下面是算法。我的article中也有说明

public class Main {
    public static void main(String[] args) {
        String[] A = new String[]{"a1", "a2", "a3"};
        String[] B = new String[]{"b1", "b2", "b3"};
        String[] C = new String[]{"c1"};

        String[] cp = CartesianProduct(0, A, B, C);

        for (String s : cp) {
            System.out.println(s);
        }
    }

    public static String[] CartesianProduct(int prodLevel, String[] res, String[]... s) {
        if (prodLevel < s.length) {
            int cProdLen = res.length * s[prodLevel].length;
            String[] tmpRes = new String[cProdLen];

            for (int i = 0; i < res.length; i++) {
                for (int j = 0; j < s[prodLevel].length; j++) {
                    tmpRes[i * res.length + j] = res[i] + s[prodLevel][j];
                }
            }
            res = Main.CartesianProduct(prodLevel + 1, tmpRes, s);
        }
        return res;
    }
}

Answer 9

我相信这是正确的。不是追求效率，而是通过递归和抽象的简洁风格。

关键的抽象是引入一个简单的Tuple 类。这有助于以后的泛型：

class Tuple<T> {
    private List<T> list = new ArrayList<T>();

    public void add(T t) { list.add(t); }

    public void addAll(Tuple<T> subT) {
        for (T t : subT.list) {
            list.add(t);
        }
    }

    public String toString() {
        String result = "(";

        for (T t : list) { result += t + ", "; }

        result = result.substring(0, result.length() - 2);
        result += " )";

        return result;
    }
}

有了这个类，我们可以这样写一个类：

public class Example {
    public static <T> List<Tuple<T>> cartesianProduct(List<Set<T>> sets) {
        List<Tuple<T>> tuples = new ArrayList<Tuple<T>>();

        if (sets.size() == 1) {
            Set<T> set = sets.get(0);
            for (T t : set) {
                Tuple<T> tuple = new Tuple<T>();
                tuple.add(t);
                tuples.add(tuple);
            }
        } else {
            Set<T> set = sets.remove(0);
            List<Tuple<T>> subTuples = cartesianProduct(sets);
            System.out.println("TRACER size = " + tuples.size());
            for (Tuple<T> subTuple : subTuples) {
                for (T t : set) {
                    Tuple<T> tuple = new Tuple<T>();
                    tuple.addAll(subTuple);
                    tuple.add(t);
                    tuples.add(tuple);
                }
            }
        }
        return tuples;
    }
}

我有一个很好的例子，但为了简洁起见，省略了。

Answer 10

您可以使用Stream.reduce 方法。

Java 9 没有额外的库。

public static <U> List<Set<U>> cartesianProduct(List<Set<? extends U>> sets) {
    // incorrect incoming data
    if (sets == null) return Collections.emptyList();
    return sets.stream()
            // non-null and non-empty sets
            .filter(set -> set != null && set.size() > 0)
            // represent each set element as Set<U>
            .map(set -> set.stream().map(Set::<U>of)
                    // Stream<List<Set<U>>>
                    .collect(Collectors.toList()))
            // summation of pairs of inner sets
            .reduce((set1, set2) -> set1.stream()
                    // combinations of inner sets
                    .flatMap(inner1 -> set2.stream()
                            // merge two inner sets into one
                            .map(inner2 -> Stream.of(inner1, inner2)
                                    .flatMap(Set::stream)
                                    .collect(Collectors.toSet())))
                    // list of combinations
                    .collect(Collectors.toList()))
            // List<Set<U>>
            .orElse(Collections.emptyList());
}

public static void main(String[] args) {
    Set<Integer> set1 = Set.of(1, 2);
    Set<Double> set2 = Set.of(3.0, 4.0);
    Set<Long> set3 = Set.of(5L, 6L);

    List<Set<Number>> sets = cartesianProduct(List.of(set1, set2, set3));
    // output
    sets.forEach(System.out::println);
}

输出（元素的顺序可能不同）：

[1, 3.0, 5]
[1, 3.0, 6]
[1, 4.0, 5]
[1, 4.0, 6]
[2, 3.0, 5]
[2, 3.0, 6]
[2, 4.0, 5]
[2, 4.0, 6]

---

^{另见：Cartesian product of an arbitrary number of sets}