Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

As in the examples above from an airport, the key used for comparisons is often
more than a single numerical value, such as price, length, weight, or speed. That is,
a key can sometimes be a more complex property that cannot be quantified with a
single number. For example, the priority of standby passengers is usually
determined by taking into account a host of different factors, including frequentflyer
status, the fare paid, and check-in time. In some applications, the key for an
object is part of the object itself (for example, it might be an instance variable
storing the list price of a book, or the weight of a car). In other applications, the key
is not part of the object but the object gets assigned its key by the application (for
example, the quality rating given to a stock by a financial analyst, or the priority
assigned to a standby passenger by a gate agent).
Comparing Keys with Total Orders
A priority queue needs a comparison rule that will never contradict itself. In order
for a comparison rule, which we denote by ≤, to be robust in this way, it must
define a total order relation, which is to say that the comparison rule is defined
for every pair of keys and it must satisfy the following properties:
• Reflexive property: k ≤ k.
• Antisymmetric property: if k 1 ≤ k 2 and k 2 ≤ k 1 , then k 1 = k 2 .
• Transitive property: if k 1 ≤ k 2 and k 2 ≤ k3, then k 1 ≤ k 3 .
Any comparison rule, ≤, that satisfies these three properties will never lead to a
comparison contradiction. In fact, such a rule defines a linear ordering
relationship among a set of keys; hence, if a (finite) collection of elements has a
total order defined for it, then the notion of a smallest key, k min , is well defined, as
a key in which k min ≤ k, for any other key k in our collection.
A priority queue is a collection of elements, called values, each having an
associated key that is provided at the time the element is inserted. A key-value
pair inserted into a priority queue is called an entry of the priority queue. The
name "priority queue" comes from the fact that keys determine the "priority" used
to pick entries to be removed. The two fundamental methods of a priority queue P
are as follows:
• insert(k,x): Insert a value x with key k into P.
• removeMin(): Return and remove from P an entry with the smallest
key, that is, an entry whose key is less than or equal to that of every other entry
in P.
By the way, some people refer to the removeMin method as the
"extractMin" method, so as to stress that this method simultaneously removes
and returns an entry P. There are many applications where the insert and
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