Algorithms in Java, Parts 1-4 (3rd Edition) (Pts.1-4)
For most applications of priority queues, we want to arrange to have the priority-queue method, instead of returning values for remove the maximum, tell us which of the records has the largest key, and to work in a similar fashion for the other operations. That is, we assign priorities and use priority queues for only the purpose of accessing other information in an appropriate order. This arrangement is akin to use of the indirect-sort or the pointer-sort concepts described in Chapter 6. In particular, this approach is required for operations such as change priority or remove to make sense. We examine an implementation of this idea in detail here, both because we shall be using priority queues in this way later in the book and because this situation is prototypical of the problems we face when we design interfaces and implementations for ADTs.
When we want to remove an item from a priority queue, how do we specify which item? When we want to maintain multiple priority queues, how do we organize the implementations so that we can manipulate priority queues in the same way that we manipulate other types of data? Questions such as these are the topic of Chapter 4. Program 9.8 gives a general interface for priority queues along the lines that we discussed in Section 4.9. It supports a situation where a client has keys and associated information and, while primarily interested in the operation of accessing the information associated with the highest key, may have numerous other data-processing operations to perform on the objects, as we discussed at the beginning of this chapter. All operations refer to a particular priority queue through a handle (a pointer to an object whose class is not specified). The insert operation returns a handle for each object added to the priority queue by the client program. In this arrangement, client programs are responsible for keeping track of handles, which they may later use to specify which objects are to be affected by remove and change priority operations, and which priority queues are to be affected by all of the operations.
Program 9.8 Full priority-queue ADT
This interface for a priority-queue ADT allows client programs to delete items and to change priorities (using object handles provided by the implementation) and to merge priority queues together.
class PQfull // ADT interface { // implementations and private members hidden boolean empty() Object insert(ITEM) ITEM getmax() void change(Object, ITEM) void remove(Object) void join(PQfull) };
This arrangement places restrictions on both the client and the implementation. The client is not given a way to access information through handles except through this interface. It has the responsibility to use the handles properly: for example, there is no good way for an implementation to check for an illegal action such as a client using a handle to an item that is already removed. For its part, the implementation cannot move around information freely, because clients have handles that they may use later. This point will become clearer when we examine details of implementations. As usual, whatever level of detail we choose in our implementations, an abstract interface such as Program 9.8 is a useful starting point for making tradeoffs between the needs of applications and the needs of implementations.
Implementations of the basic priority-queue operations, using an unordered doubly linked-list representation, are given in Programs 9.9 and 9.10. Most of the code is based on elementary linked list operations from Section 3.3, but the implementation of the client handle abstraction is noteworthy in the present context: the insert method returns an Object, which the client can only take to mean "a reference to an object of some unspecified class" since Node, the actual type of the object, is private. So the client can do little else with the reference but keep it in some data structure associated with the item that it provided as a parameter to insert. But if the client needs to change priority the item's key or to remove the item from the priority queue, this object is precisely what the implementation needs to be able to accomplish the task: the appropriate methods can cast the type to Node and make the necessary modifications. It is easy to develop other, similarly straightforward, implementations using other elementary representations (see, for example, Exercise 9.40).
As we discussed in Section 9.1, the implementation given in Programs 9.9 and 9.10 is suitable for applications where the priority queue is small and remove the maximum or find the maximum operations are infrequent; otherwise, heap-based implementations are preferable. Implementing swim and sink for heap-ordered trees with explicit links while maintaining the integrity of the handles is a challenge that we leave for exercises, because we shall be considering two alternative approaches in detail in Sections 9.6 and 9.7.
A full ADT such as Program 9.8 has many virtues, but it is sometimes advantageous to consider other arrangements, with different restrictions on the client programs and on implementations. In Section 9.6 we consider an example where the client program keeps the responsibility for maintaining the records and keys, and the priority-queue routines refer to them indirectly.
Slight changes in the interface also might be appropriate. For example, we might want a method that returns the value of the highest priority key in the queue, rather than just a way to reference that key and its associated information. Also, the issues that we considered in Sections 4.9 and 4.10 associated with memory management and copy semantics come into play. We are not considering the copy operation and have chosen just one out of several possibilities for join (see Exercises 9.44 and 9.45).
Program 9.9 Unordered doubly linked list priority queue
This implementation includes the construct, test if empty, and insert methods from the interface of Program 9.8 (see Program 9.10 for implementations of the other four methods). It maintains a simple unordered list, with head and tail nodes. We specify the class Node to be a doubly linked list node (with an item and two links). The private data fields are just the list's head and tail links.
class PQfull { private static class Node { ITEM key; Node prev, next; Node(ITEM v) { key = v; prev = null; next = null; } } private Node head, tail; PQfull() { head = new Node(null); tail = new Node(null); head.prev = tail; head.next = tail; tail.prev = head; tail.next = head; } boolean empty() { return head.next.next == head; } Object insert(ITEM v) { Node t = new Node(v); t.next = head.next; t.next.prev = t; t.prev = head; head.next = t; return t; } ITEM getmax() // See Program 9.10 void change(Object x, ITEM v) // See Program 9.10 void remove(Object x) // See Program 9.10 void join(PQfull p) // See Program 9.10 }
It is easy to add such procedures to the interface in Program 9.8, but it is much more challenging to develop an implementation where logarithmic performance for all operations is guaranteed. In applications where the priority queue does not grow to be large, or where the mix of insert and remove the maximum operations has some special properties, a fully flexible interface might be desirable. But in applications where the queue will grow to be large, and where a tenfold or a hundredfold increase in performance might be noticed or appreciated, it might be worthwhile to restrict to the set of operations where efficient performance is assured. A great deal of research has gone into the design of priority-queue algorithms for different mixes of operations; the binomial queue described in Section 9.7 is an important example.
Exercises
9.39 Which priority-queue implementation would you use to find the 100 smallest of a set of 106 random numbers? Justify your answer.
9.40 Provide implementations similar to Programs 9.9 and 9.10 that use ordered doubly linked lists. Note: Because the client has handles into the data structure, your programs can change only links (rather than keys) in nodes.
9.41 Provide implementations for insert and remove the maximum (the priority-queue interface in Program 9.1) using complete heap-ordered trees represented with explicit nodes and links. Note: Because the client has no handles into the data structure, you can take advantage of the fact that it is easier to exchange information fields in nodes than to exchange the nodes themselves.
9.43 Add a (brute-force) implementation of the join operation to your implementation from Exercise 9.42.
These method implementations complete the priority queue implementation of Program 9.9. The remove the maximum operation requires scanning through the whole list, but the overhead of maintaining doubly linked lists is justified by the fact that the change priority, remove, and join operations all are implemented in constant time, using only elementary operations on the lists (see Chapter 3 for more details on doubly linked lists).
The change and remove methods take an Object reference as a parameter, which must reference an object of (private) type Node a client can only get such a reference from insert.
We might make this class Cloneable and implement a clone method that makes a copy of the whole list (see Section 4.9), but client object handles would be invalid for the copy. The join implementation appropriates the list nodes from the parameter to be included in the result, but it does not make copies of them, so client handles remain valid.
ITEM getmax() { ITEM max; Node x = head.next; for (Node t = x; t.next != head; t = t.next) if (Sort.less(x.key, t.key)) x = t; max = x.key; remove(x); return max; } void change(Object x, ITEM v) { ((Node) x).key = v; } void remove(Object x) { Node t = (Node) x; t.next.prev = t.prev; t.prev.next = t.next; } void join(PQfull p) { tail.prev.next = p.head.next; p.head.next.prev = tail.prev; head.prev = p.tail; p.tail.next = head; tail = p.tail; }
9.46 Provide a priority-queue interface and implementation that supports construct and remove the maximum, using tournaments (see Section 5.7). Program 5.19 will provide you with the basis for construct.
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