Ordering Object and Priority Queue

Summary: The module describes comparator objects for determining a binary order and builds a generalized priority queue with it.

Order Relation

When comparing two objects, there are two ways to look at the comparison behavior, which creates a notion of ordering between the two objects:

1. Each object knows how to compare itself -- Comparison is considered intrinsic to the object and is part of its behavior.
2. Comparison is done by an external object -- A third object is introduced that contains the ability to compare two objects.

These two different outlooks on binary object ordering will be explored below in terms of the two Java interfaces that are used to model them.

java.lang.Comparable

There are many computing tasks that require performing some sort of comparison between data objects. A few data types are endowed with a "natural" ordering of their values. The integers have a natural ordering "less or equal to", labeled "<=", defined as follows.

n <= m iff m = n + k, for some non-negative integer k

(Note: a rigorous mathematical definition of the set of non-negative integers is beyond the scope of this lecture).

The above natural order of the integers is a concrete instance of an abstract concept called an order relation. An order relation on a set S is a boolean function R on S x S that is

• reflexive: R(x, x) is true for all x in S,
• anti-symmetric: R(x,y) and R(y,x) implies x = y, for all x, y in S, and
• transitive: R(x, y) and R(y, z) implies R(x, z), for all x, y, z in S.

To model order relations that are naturally endowed in certain types of data objects, Java provides an interface called Comparable, which has exactly one method called

int compareTo(Object rhs), defined abstractly as

• x.compareTo(y) < 0 means x is "less than" y,
• x.compareTo(y) == 0 means x is "equal to" y, and
• x.compareTo(y) > 0 means y is "less than" x.

For example, the Integer class implements the Comparable interface as follows. If x and y are Integer objects then,

• x.compareTo(y) < 0 means x < y,
• x.compareTo(y) == 0 means x == y, and
• x.compareTo(y) > 0 means y < x.

Common data types that have a natural ordering among their values, such as Double, String, Character, all implement Comparable.

Advantages and disadvantages

• Advantages
  • Objects carry their ability to be compared with them and thus can be involved in comparison operations anywhere.
  • The comparison operation is invariant and thus consistent in all situations.
• Disadvantages
  • Comparison operation cannot be changed to match different situations.

java.util.Comparator

Most of the time, the ordering among the data objects is an extrinsic operation imposed on the object by the user of the objects. For example, the Pizza objects in homework 2 have no concepts of comparing among themselves, however, the user can impose an ordering on them by comparing their price/area ratios or their profits. To model extrinsic ordering relation, Java provides an interface in the java.util package called Comparator, which has exactly two methods:

• int compare(Object x, Object y), to model the ordering
  • compare(x, y) < 0 means x is "less than" y,
  • compare(x, y) == 0 means x is "equal to" y, and
  • compare(x, y) > 0 means yis "less than" x, and
• boolean equals(Object x), to model equality of Comparators. Unlike the equals method of most objects, equality of Comparators also requires that their comparison behavior be identical.

In most applications, when we implement a Comparator interface, we only need to override the compare(...) method and simply inherit the equals(...) method fromObject.

Advantages and disadvantages

• Advantages
  • Object ordering can be considered variant and be redefined for different situations.
  • Creating composite comparison behavior can be easily acheived by decorating or other composite techniques.
• Disadvantages
  • Objects are not intrinsically comparable, requiring the inclusion of an otherwise disconnected entity in the system.

EXERCISE 1

Implement a Comparator class that will compare Integer objects.

In-Order Insertion Example

Recall the problem of inserting an Integer object in order into a sorted list of Integers. Instead of writing an algorithm that only works for Integer, we can write an algorithm that will work for any Object that can be compared by some Comparator. The table below contrasts the insert in order algorithm for Integer and the insert in order algorithm that uses aComparator as a strategy (as in strategy pattern) for comparison. Both are algorithms for LRStruct, the mutable list.

First, let's look at a regular in-order insertion algorithm that works only on Integer objects because the comparison technique is hard-coded in:

InsertInOrderLRS Visitor class

import lrs.*;

public class InsertInOrderLRS implements IAlgo {

  public static final InsertInOrderLRS Singleton = new InsertInOrderLRS();

  private InsertInOrderLRS() {
  }

  /**
  * Simply inserts the given parameter n at the front.
  * @param host an empty LRStruct.
  * @param n an Integer to be inserted in order into host.
  * @return LRStruct
  */
  public Object emptyCase(LRStruct host, Object... n) {
    return host.insertFront(n[0]);
  }

  /**
  * Based on the comparison between first and n,
  * inserts at the front or recurs!
  * @param host a non-empty LRStruct.
  * @param n an Integer to be inserted in order into host.
  * @return LRStruct
  */
  public Object nonEmptyCase(LRStruct host, Object... n) {

    if (n[0] < (Integer)host.getFirst();) {  // could use Integer.compareTo() 
      return host.insertFront(n[0]);
    }
    else {
      return host.getRest().execute(this, n[0]);
    }
  }
}

A visitor to an LRStruct that performs an in-order insertion, but only for Integer objects.

Now let's look at how the simple substitution of a Comparator for the hard-coded comparison allows the otherwise identical code to be used for any objects. All one has to do is to supply the desired Comparator instance that matches the type of objects stored in the LRStruct:

InsertInOrder Visitor class

import lrs.*;
import java.util.*;

public class InsertInOrder implements IAlgo {

  private Comparator _order;

  public InsertInOrder(Comparator ord) {
      _order = ord;
  }

  /**
  * Simply inserts the given parameter n at the front.
  * @param host an empty LRStruct.
  * @param n an Object to be inserted in order into host,
  *   based on the given Comparator.
  * @return LRStruct
  */
  public Object emptyCase(LRStruct host, Object... n) {
    return host.insertFront(n[0]);
  }

  /**
  * Based on the comparison between first and n,
  * inserts at the front or recurs!
  * @param host a non-empty LRStruct.
  * @param n an Object to be inserted in order into host,
  *   based on the given Comparator.
  * @return LRStruct
  */
  public Object nonEmptyCase(LRStruct host, Object... n) {
    if (_order.compare(n[0], host.getFirst()) < 0) {
      return host.insertFront(n[0]);
    }
    else {
      return host.getRest().execute(this, n[0]);
    }
  }
}

A visitor to an LRStruct that performs an in-order insertion, for any objects that are comparable with the given Comparator.

3. Priority Queue

The InsertInOrder algorithm given in the above can be used as part of a strategy for a Restricted Access Container ("RAC") called priority queue. In the code below, it is assumed that "smaller" objects have higher priority, i.e. will be retrieved from the RAC before "larger" objects. The code also shows how, if the RAC elements are Comparable objects, that aComparator could be used to provide their ordering in the RAC, even to the point of reversing their order if desired.

PQComparatorRACFactory priority queue implementation

package rac;

import lrs.*;
import lrs.visitor.*;
import java.util.*;

/*
 * Implements a factory for restricted access containers that
 * return the ``highest priority'' item.
 */
public class PQComparatorRACFactory extends ALRSRACFactory {

    private Comparator _comp;

    /**
     * Used when the items in the container are Comparable objects.
     */
    public PQComparatorRACFactory() {
        _comp = new Comparator() {
            public int compare(Object x, Object y) {
                /*
                 * Intentionally reverse the ordering so that the
                 * largest item will be first, just to show that it can be done.
                 */
                return ((Comparable)y).compareTo(x);
            }
        };
    }

    /**
     * Used when we want to prioritize the items according to a given Comparator.
     * @param comp the item that is smallest according to comp has the highest
     * priority.
     */
    public PQComparatorRACFactory(Comparator comp) {
        _comp = comp;
    }

    /**
     * Create a container that returns the item with the highest priority
     * according to a given Comparator.
     */
    public IRAContainer makeRAC() {
        return new LRSRAContainer(new InsertInOrder(_comp));
    }
}

A priority queue can easily be implemented from a RAC by using a Comparator.