1. Stacks and Queues

2. What is a Stack?  Stack is a data structure in which data is added and removed at only one end called the top  Examples of stacks are:  Stack of books  Stack of trays in a cafeteria

3. Stack  A Last In First Out (LIFO) data structure  Primary operations: Push and Pop  Push  Add an element to the top of the stack  Pop  Remove the element from the top of the stack  An example A top empty stack: initial state top push (A)push (B) A B pop A

4. Building Stack Step-by-Step 2 8 1 7 2 7 2 1 7 2 1 7 2 8 1 7 2 8 1 7 2 top Push(8) Push(2) pop()

5. Stack Errors  Stack Overflow  An attempt to add a new element in an already full stack is an error  A common mistake often made in stack implementation  Stack Underflow  An attempt to remove an element from the empty stack is also an error  Again, a common mistake often made in stack implementation

6. Applications of Stacks  Some direct applications:  Page-visited history in a Web browser  Undo sequence in a text editor  Evaluating postfix expressions (e.g., xy+)  Some indirect applications  Auxiliary data structure for some algorithms (e.g., Depth First Search algorithm)  Component of other data structures

7. The Stack Abstract Data Type Stacks are the simplest of all data structures Stacks are the simplest of all data structures Formally, a stack is an abstract data type (ADT) that supports the following two methods: Formally, a stack is an abstract data type (ADT) that supports the following two methods: push(e): Insert element e to the top of the stack push(e): Insert element e to the top of the stack pop(): Remove from the stack and return the top element on the stack; pop(): Remove from the stack and return the top element on the stack; an error occurs if the stack is empty – what error? an error occurs if the stack is empty – what error? Additionally, let us also define the following methods: Additionally, let us also define the following methods: size(): Return the number of elements in the stack size(): Return the number of elements in the stack isEmpty(): Return a Boolean indicating if the stack is empty isEmpty(): Return a Boolean indicating if the stack is empty top(): Return the top element in the stack, without removing it top(): Return the top element in the stack, without removing it an error occurs if the stack is empty an error occurs if the stack is empty

8. The Stack Abstract Data Type Example 5.3: The following table shows a series of stack operations and their effects on an initially empty stack S of integers. Example 5.3: The following table shows a series of stack operations and their effects on an initially empty stack S of integers. OperationOutput Stack Contents push(5)push(3)pop()push(7)pop()top()pop()pop()isEmpty()push(9)push(7)push(3)push(5)size()pop()push(8)pop()pop()--3-755"error“true----45-83(5) (5, 3) (5) (5, 7) (5)(5)()()()(9) (9, 7) (9, 7, 3) (9, 7, 3, 5) (9, 7, 3) (9, 7, 3, 8) (9, 7, 3) (9, 7)

9. A Stack Interface in Java The stack data structure is included as a "built- in" class in the java.util package of Java. The stack data structure is included as a "built- in" class in the java.util package of Java. Class java.util.Stack is a data structure that stores generic Java objects and includes,among others, the following methods: Class java.util.Stack is a data structure that stores generic Java objects and includes,among others, the following methods: push(), push(), pop() pop() peek() (equivalent to top()), peek() (equivalent to top()), size(), and empty() (equivalent to isEmpty()). size(), and empty() (equivalent to isEmpty()). Methods pop() and peek() throw exception EmptyStackException if they are called on an empty stack. Methods pop() and peek() throw exception EmptyStackException if they are called on an empty stack.

10. The Stack Abstract Data Type Implementing an abstract data type in Java involves two steps. The first step is the definition of a Java Application Programming Interface (API), or simply interface, which describes the names of the methods that the ADT supports and how they are to be declared and used. Implementing an abstract data type in Java involves two steps. The first step is the definition of a Java Application Programming Interface (API), or simply interface, which describes the names of the methods that the ADT supports and how they are to be declared and used. In addition, we must define exceptions for any error conditions that can arise. For instance, the error condition that occurs when calling method pop() or top() on an empty stack is signaled by throwing an exception of type EmptyStackException, In addition, we must define exceptions for any error conditions that can arise. For instance, the error condition that occurs when calling method pop() or top() on an empty stack is signaled by throwing an exception of type EmptyStackException,

11. The Stack Abstract Data Type Code Fragment 5.2: Interface Stack documented with comments in Javadoc style (Section 1.9.3). Note also the use of the generic parameterized type, E, which implies that a stack can contain elements of any specified class.

12. A Simple Array-Based Stack Implementation We can implement a stack by storing its elements in an array. We can implement a stack by storing its elements in an array. Specifically, the stack in this implementation consists of Specifically, the stack in this implementation consists of an N-element array S an N-element array S plus an integer variable t that gives the index of the top element in array S. plus an integer variable t that gives the index of the top element in array S. Figure 5.2: Implementing a stack with an array S. The top element in the stack is stored in the cell S[t]. Recalling that arrays start at index 0 in Java, Recalling that arrays start at index 0 in Java, we initialize t to −1, and we use this value for t to identify an empty stack. we initialize t to −1, and we use this value for t to identify an empty stack. Likewise, we can use t to determine the number of elements (t + 1). Likewise, we can use t to determine the number of elements (t + 1). FullStackException, to signal the error that arises if we try to insert a new element into a full array. FullStackException, to signal the error that arises if we try to insert a new element into a full array. Exception FullStackException is specific to this implementation and is not defined in the stack ADT. Exception FullStackException is specific to this implementation and is not defined in the stack ADT.

13. Code Fragment 5.3: Implementing a stack using an array of a given size, N.

14. Analyzing the Array-Based Stack Implementation The correctness of the methods in the array-based implementation follows immediately from the definition of the methods themselves. The correctness of the methods in the array-based implementation follows immediately from the definition of the methods themselves. Note that we could have avoided resetting the old S[t] to null and we would still have a correct method. Note that we could have avoided resetting the old S[t] to null and we would still have a correct method. There is a trade-off in being able to avoid this assignment should we be thinking about implementing these algorithms in Java. There is a trade-off in being able to avoid this assignment should we be thinking about implementing these algorithms in Java. The trade-off involves the Java garbage collection mechanism that searches memory for objects that are no longer referenced by active objects, and reclaims their space for future use. Let e = S[t] be the top element before the pop method is called. The trade-off involves the Java garbage collection mechanism that searches memory for objects that are no longer referenced by active objects, and reclaims their space for future use. Let e = S[t] be the top element before the pop method is called. By making S[t] a null reference, we indicate that the stack no longer needs to hold a reference to object e. Indeed, if there are no other active references to e, then the memory space taken by e will be reclaimed by the garbage collector. By making S[t] a null reference, we indicate that the stack no longer needs to hold a reference to object e. Indeed, if there are no other active references to e, then the memory space taken by e will be reclaimed by the garbage collector. Table 5.1 shows the running times for methods in a realization of a stack by an array. Each of the stack methods in the array realization executes a constant number of statements involving arithmetic operations, comparisons, and assignments. Table 5.1 shows the running times for methods in a realization of a stack by an array. Each of the stack methods in the array realization executes a constant number of statements involving arithmetic operations, comparisons, and assignments. In addition, pop also calls isEmpty, which itself runs in constant time. Thus, in this implementation of the Stack ADT, each method runs in constant time, that is, they each run in O(1) time. In addition, pop also calls isEmpty, which itself runs in constant time. Thus, in this implementation of the Stack ADT, each method runs in constant time, that is, they each run in O(1) time.

15. Table 5.1: Performance of a stack realized by an array. The space usage is O(N), where N is the size of the array, determined at the time the stack is instantiated. Note that the space usage is independent from the number n ≤ N of elements that are actually in the stack. Table 5.1: Performance of a stack realized by an array. The space usage is O(N), where N is the size of the array, determined at the time the stack is instantiated. Note that the space usage is independent from the number n ≤ N of elements that are actually in the stack. MethodTime size O(1) is Empty O(1) top push pop

16. A Drawback with the Array-Based Stack Implementation The array implementation of a stack is simple and efficient. This implementation has one negative aspect it must assume a fixed upper bound, CAPACITY, on the ultimate size of the stack. In Code Fragment 5.4, we chose the capacity value 1,000 more or less arbitrarily. An application may actually need much less space than this, which would waste memory. An application may need more space than this, which would cause our stack implementation to generate an exception as soon as a client program tries to push its 1,001st object on the stack. Thus, even with its simplicity and efficiency, the array-based stack implementation is not necessarily ideal. Fortunately, there is another implementation, which we discuss next, that does not have a size limitation and use space proportional to the actual number of elements stored in the stack. Still, in cases where we have a good estimate on the number of items needing to go in the stack, the array-based implementation is hard to beat. Stacks serve a vital role in a number of computing applications, so it is helpful to have a fast stack ADT implementation such as the simple array-based implementation.

17. Implementing a Stack with a Generic Linked List Using a singly linked list to implement the stack ADT. Using a singly linked list to implement the stack ADT. In designing such an implementation, we need to decide if In designing such an implementation, we need to decide if the top of the stack is at the head the top of the stack is at the head or at the tail of the list. or at the tail of the list. Since we can insert and delete elements in constant time only at the head. Since we can insert and delete elements in constant time only at the head. Thus, it is more efficient to have the top of the stack at the head of our list. Thus, it is more efficient to have the top of the stack at the head of our list. in order to perform operation size in constant time, in order to perform operation size in constant time, we keep track of the current number of elements in an instance variable. we keep track of the current number of elements in an instance variable. Rather than use a linked list that can only store objects of a certain type, we would like, in this case, to implement a generic stack using a generic linked list. Rather than use a linked list that can only store objects of a certain type, we would like, in this case, to implement a generic stack using a generic linked list. Thus, we need to use a generic kind of node to implement this linked list. We show such a Node class in Code Fragment 5.6. Thus, we need to use a generic kind of node to implement this linked list. We show such a Node class in Code Fragment 5.6.

18. Implementing a Stack with a Generic Linked List Code Fragment 5.6: Class Node, which implements a generic node for a singly linked list. Code Fragment 5.6: Class Node, which implements a generic node for a singly linked list.

19. A Generic NodeStack Class A Java implementation of a stack, by means of a generic singly linked list, is given in Code Fragment 5.7. A Java implementation of a stack, by means of a generic singly linked list, is given in Code Fragment 5.7. All the methods of the Stack interface are executed in constant time. All the methods of the Stack interface are executed in constant time. In addition to being time efficient, this linked list implementation has a space requirement that is O(n), where n is the current number of elements in the stack. In addition to being time efficient, this linked list implementation has a space requirement that is O(n), where n is the current number of elements in the stack. Thus, this implementation does not require that a new exception be created to handle size overflow problems. We use an instance variable top to refer to the head of the list (which points to the null object if the list is empty). Thus, this implementation does not require that a new exception be created to handle size overflow problems. We use an instance variable top to refer to the head of the list (which points to the null object if the list is empty). When we push a new element e on the stack, we simply create a new node v for e, reference e from v, and insert v at the head of the list. When we push a new element e on the stack, we simply create a new node v for e, reference e from v, and insert v at the head of the list. Likewise, when we pop an element from the stack, we simply remove the node at the head of the list and return its element. Thus, we perform all insertions and removals of elements at the head of the list. Likewise, when we pop an element from the stack, we simply remove the node at the head of the list and return its element. Thus, we perform all insertions and removals of elements at the head of the list.

20. Code Fragment 5.7: Class NodeStack, which implements the Stack interface using a singly linked list, whose nodes are objects of class Node from Code Fragment 5.6.

21. Reversing an Array Using a Stack We can use a stack to reverse the elements in an array, We can use a stack to reverse the elements in an array, thereby producing a nonrecursive algorithm for the array-reversal problem introduced in Section 3.5.1. thereby producing a nonrecursive algorithm for the array-reversal problem introduced in Section 3.5.1. The basic idea is simply to push all the elements of the array in order into a stack and then fill the array back up again by popping the elements off of the stack. The basic idea is simply to push all the elements of the array in order into a stack and then fill the array back up again by popping the elements off of the stack. In Code Fragment 5.8, we give a Java implementation of this algorithm. Incidentally, this method also illustrates how we can use generic types in a simple application that uses a generic stack. In Code Fragment 5.8, we give a Java implementation of this algorithm. Incidentally, this method also illustrates how we can use generic types in a simple application that uses a generic stack. In particular, when the elements are popped off the stack in this example, they are automatically returned as elements of the E type; hence, they can be immediately returned to the input array. We show an example use of this method in Code Fragment 5.9. In particular, when the elements are popped off the stack in this example, they are automatically returned as elements of the E type; hence, they can be immediately returned to the input array. We show an example use of this method in Code Fragment 5.9.

22. Code Fragment 5.8: A generic method that reverses the elements in an array of type E objects, using a stack declared using the Stack interface.

23. Code Fragment 5.9: A test of the reverse method using two arrays.

24. Code Fragment 5.8: A generic method that reverses the elements in an array of type E objects, using a stack declared using the Stack interface.

25. Code Fragment 5.9: A test of the reverse method using two arrays.

26. Matching Parentheses and HTML Tags Two related applications of stacks, Two related applications of stacks, the first of which is for matching parentheses the first of which is for matching parentheses and grouping symbols in arithmetic expressions. and grouping symbols in arithmetic expressions. Arithmetic expressions can contain various pairs of grouping symbols, such as Arithmetic expressions can contain various pairs of grouping symbols, such as Parentheses: "(" and ")"Parentheses: "(" and ")" Braces: "{" and "}"Braces: "{" and "}" Brackets: "[" and "]"Brackets: "[" and "]" Floor function symbols: " ⌊ " and " ⌋ "Floor function symbols: " ⌊ " and " ⌋ " Ceiling function symbols: " ⌈ " and " ⌉,"Ceiling function symbols: " ⌈ " and " ⌉," and each opening symbol must match with its corresponding closing symbol. : and each opening symbol must match with its corresponding closing symbol. : [(5 + x) − (y + z)]. [(5 + x) − (y + z)].

27. Matching Parentheses and HTML Tags The following examples further illustrate this concept: The following examples further illustrate this concept: Correct: ( )(( )){([( )])}Correct: ( )(( )){([( )])} Correct: ((( )(( )){([( )])}))Correct: ((( )(( )){([( )])})) Incorrect: )(( )){([( )])}Incorrect: )(( )){([( )])} Incorrect: ({[])}Incorrect: ({[])} Incorrect: (.Incorrect: (.

28. An Algorithm for Parentheses Matching An important problem in processing arithmetic expressions is to make sure their grouping symbols match up correctly. An important problem in processing arithmetic expressions is to make sure their grouping symbols match up correctly. We can use a stack S to perform the matching of grouping symbols in an arithmetic expression with a single left-to-right scan. We can use a stack S to perform the matching of grouping symbols in an arithmetic expression with a single left-to-right scan. The algorithm tests that left and right symbols match up and also that the left and right symbols are both of the same type. The algorithm tests that left and right symbols match up and also that the left and right symbols are both of the same type. Suppose we are given a sequence X = x0x1x2…xn−1, where each xi is a token that can be a grouping symbol, a variable name, an arithmetic operator, or a number. Suppose we are given a sequence X = x0x1x2…xn−1, where each xi is a token that can be a grouping symbol, a variable name, an arithmetic operator, or a number. The basic idea behind checking that the grouping symbols in S match correctly, The basic idea behind checking that the grouping symbols in S match correctly, is to process the tokens in X in order. is to process the tokens in X in order. Each time we encounter an opening symbol, we push that symbol onto S, Each time we encounter an opening symbol, we push that symbol onto S, and each time we encounter a closing symbol, we pop the top symbol from the stack S (assuming S is not empty) and we check that these two symbols are of the same type. and each time we encounter a closing symbol, we pop the top symbol from the stack S (assuming S is not empty) and we check that these two symbols are of the same type. If the stack is empty after we have processed the whole sequence, then the symbols in X match. If the stack is empty after we have processed the whole sequence, then the symbols in X match. Assuming that the push and pop operations are implemented to run in constant time, this algorithm runs in O(n), that is linear, time. We give a pseudo-code description of this algorithm in Code Fragment 5.10. Assuming that the push and pop operations are implemented to run in constant time, this algorithm runs in O(n), that is linear, time. We give a pseudo-code description of this algorithm in Code Fragment 5.10.

29. Code Fragment 5.10: Algorithm for matching grouping symbols in an arithmetic expression.

30. Matching Tags in an HTML Document Another application in which matching is important is in the validation of HTML documents. Another application in which matching is important is in the validation of HTML documents. HTML is the standard format for hyperlinked documents on the Internet. I HTML is the standard format for hyperlinked documents on the Internet. I n an HTML document, portions of text are delimited by HTML tags. n an HTML document, portions of text are delimited by HTML tags. A simple opening HTML tag has the form " " and the corresponding closing tag has the form "." Commonly used HTML tags include A simple opening HTML tag has the form " " and the corresponding closing tag has the form "." Commonly used HTML tags include body: document bodybody: document body h1: section headerh1: section header center: center justifycenter: center justify p: paragraphp: paragraph ol: numbered (ordered) listol: numbered (ordered) list li: list item.li: list item. We show a sample HTML document and a possible rendering in Figure 5.3. We show a sample HTML document and a possible rendering in Figure 5.3.

31. Figure 5.3: Illustrating HTML tags. (a) An HTML document; (b) its rendering.

32. Code Fragment 5.11: A complete Java program for testing if an HTML document has fully matching tags. (Continues in Code Fragment 5.12.)

33. What is Queue?  Queue is a data structure in which insertion is done at one end (Front), while deletion is performed at the other end (Rear)  Contrast with stack, where insertion and deletion at one and the same end  It is First In, First Out (FIFO) structure  For example, customers standing in a check-out line in a store, the first customer in is the first customer served.

34. Queue operations  Enqueue: insert an element at the rear of the list  Dequeue: delete the element at the front of the list Insert (Enqueue) Remove (Dequeue) rearfront

35. Building a Queue Step-by-Step  There are several different algorithms to implement Enqueue and Dequeue  Enqueuing  The front index is always fixed  The rear index moves forward in the array front rear Enqueue(3) 3 front rear Enqueue(6) 3 6 front rear Enqueue(9) 3 6 9

36.  Dequeuing  The element at the front of the queue is removed  Move all the elements after it by one position Dequeue() front rear 6 9 Dequeue() front rear 9 rear = -1 front Building a Queue Step-by-Step

37. The Queue Abstract Data Type Formally, the queue abstract data type defines a collection that keeps objects in a sequence, where Formally, the queue abstract data type defines a collection that keeps objects in a sequence, where element access and deletion are restricted to the first element in the sequence, the front of the queue, element access and deletion are restricted to the first element in the sequence, the front of the queue, and element insertion is restricted to the end of the sequence, the rear of the queue. and element insertion is restricted to the end of the sequence, the rear of the queue. This restriction enforces the rule that items are inserted and deleted in a queue according to the first-in first-out (FIFO) principle. This restriction enforces the rule that items are inserted and deleted in a queue according to the first-in first-out (FIFO) principle. The queue abstract data type (ADT) supports the following two fundamental methods: The queue abstract data type (ADT) supports the following two fundamental methods: enqueue(e): Insert element e at the rear of the queue. enqueue(e): Insert element e at the rear of the queue. dequeue(): Remove and return from the queue the object at the front; dequeue(): Remove and return from the queue the object at the front; an error occurs if the queue is empty. an error occurs if the queue is empty. Additionally, similar to the case with the Stack ADT, the queue ADT includes the following supporting methods: Additionally, similar to the case with the Stack ADT, the queue ADT includes the following supporting methods: size(): Return the number of objects in the queue. size(): Return the number of objects in the queue. isEmpty(): Return a Boolean value that indicates whether the queue is empty. isEmpty(): Return a Boolean value that indicates whether the queue is empty. front(): Return, but do not remove, the front object in the queue; front(): Return, but do not remove, the front object in the queue; an error occurs if the queue is empty. an error occurs if the queue is empty.

38. The Queue Abstract Data Type Example 5.4: The following table shows a series of queue operations and their effects on an initially empty queue Q of integer objects. For simplicity, we use integers instead of integer objects as arguments of the operations. Example 5.4: The following table shows a series of queue operations and their effects on an initially empty queue Q of integer objects. For simplicity, we use integers instead of integer objects as arguments of the operations. OperationOutput front ← Q ← rear enqueue(5)enqueue(3) dequeue( ) enqueue(7) front( ) dequeue( ) isEmpty( ) enqueue(9)enqueue(7)size()enqueue(3)enqueue(5) dequeue( ) --5-377"error"true--2--9(5) (5, 3) (3) (3, 7) (7)(7) ( ) (9) (9, 7) (9, 7, 3) (9, 7, 3, 5) (7, 3, 5)

39. The Queue Abstract Data Type Example Applications Example Applications There are several possible applications for queues. There are several possible applications for queues. Stores, Stores, theaters, theaters, reservation centers, reservation centers, and other similar services typically process customer requests according to the FIFO principle. and other similar services typically process customer requests according to the FIFO principle. A queue would therefore be a logical choice for a data structure to handle transaction processing for such applications. A queue would therefore be a logical choice for a data structure to handle transaction processing for such applications. For example, it would be a natural choice for handling calls to the reservation center of an airline or to the box office of a theater. For example, it would be a natural choice for handling calls to the reservation center of an airline or to the box office of a theater.

40. A Queue Interface in Java A Java interface for the queue ADT is given in Code Fragment 5.13. A Java interface for the queue ADT is given in Code Fragment 5.13. This generic interface specifies that objects of arbitrary object types can be inserted into the queue. This generic interface specifies that objects of arbitrary object types can be inserted into the queue. Thus, we don't have to use explicit casting when removing elements. Thus, we don't have to use explicit casting when removing elements. Note that the size and isEmpty methods have the same meaning as their counterparts in the Stack ADT. Note that the size and isEmpty methods have the same meaning as their counterparts in the Stack ADT. These two methods, as well as the front method, are known as accessor methods, for they return a value and do not change the contents of the data structure. These two methods, as well as the front method, are known as accessor methods, for they return a value and do not change the contents of the data structure.

41. Code Fragment 5.13: Interface Queue documented with comments in Javadoc style.

42. A Simple Array-Based Queue Implementation We present a simple realization of a queue by means of an array, Q, of fixed capacity, storing its elements. We present a simple realization of a queue by means of an array, Q, of fixed capacity, storing its elements. Since the main rule with the queue ADT is that we insert and delete objects according to the FIFO principle, we must decide how we are going to keep track of the front and rear of the queue. Since the main rule with the queue ADT is that we insert and delete objects according to the FIFO principle, we must decide how we are going to keep track of the front and rear of the queue. One possibility is to adapt the approach we used for the stack implementation, One possibility is to adapt the approach we used for the stack implementation, letting Q[0] be the front of the queue and then letting the queue grow from there. letting Q[0] be the front of the queue and then letting the queue grow from there. This is not an efficient solution, however, for it requires that we move all the elements forward one array cell each time we perform a dequeue operation. This is not an efficient solution, however, for it requires that we move all the elements forward one array cell each time we perform a dequeue operation. Such an implementation would therefore take O(n) time to perform the dequeue method, where n is the current number of objects in the queue. Such an implementation would therefore take O(n) time to perform the dequeue method, where n is the current number of objects in the queue. If we want to achieve constant time for each queue method, we need a different approach, but that approach may have its own demerits. If we want to achieve constant time for each queue method, we need a different approach, but that approach may have its own demerits.

43. Using an Array in a Circular Way To avoid moving objects once they are placed in Q, To avoid moving objects once they are placed in Q, we define two variables f and r, which have the following meanings: we define two variables f and r, which have the following meanings: f is an index to the cell of Q storing the first element of the queue (next candidate to be removed by a dequeue operation), f is an index to the cell of Q storing the first element of the queue (next candidate to be removed by a dequeue operation), f = r queue is empty. f = r queue is empty. r is an index to the next available array cell in Q. r is an index to the next available array cell in Q. Initially, we assign f = r = 0, which indicates that the queue is empty. Initially, we assign f = r = 0, which indicates that the queue is empty. When we remove an element from the front of the queue, increment f to index the next cell. When we remove an element from the front of the queue, increment f to index the next cell. Adding an element, we store it in cell Q[r] and increment r to index the next available cell in Q. Adding an element, we store it in cell Q[r] and increment r to index the next available cell in Q. This scheme allows us to implement methods front, enqueue, and dequeue in constant time, that is, O(1) time. This scheme allows us to implement methods front, enqueue, and dequeue in constant time, that is, O(1) time. However, there is still a problem with this approach. However, there is still a problem with this approach. what happens if we repeatedly enqueue and dequeue a single element N different times. We would have f = r = N. If we were then to try to insert the element just one more time, we would get an array-out-of-bounds error (since the N valid locations in Q are from Q[0] to Q[N − 1]), even though there is plenty of room in the queue in this case. We would have f = r = N. If we were then to try to insert the element just one more time, we would get an array-out-of-bounds error (since the N valid locations in Q are from Q[0] to Q[N − 1]), even though there is plenty of room in the queue in this case. To avoid this problem and be able to utilize all of the array Q, we let the f and r indices "wrap around" the end of Q. To avoid this problem and be able to utilize all of the array Q, we let the f and r indices "wrap around" the end of Q. That is, we now view Q as a "circular array" that goes from Q[0] to Q[N − 1] and then immediately back to Q[0] again. (See Figure 5.4.) That is, we now view Q as a "circular array" that goes from Q[0] to Q[N − 1] and then immediately back to Q[0] again. (See Figure 5.4.)

44. Figure 5.4: Using array Q in a circular fashion: (a)the "normal" configuration with f ≤ r; (b) the "wrapped around" configuration (b)with r < f. The cells storing queue elements are highlighted.

45. Using the Modulo Operator to Implement a Circular Array Implementing this circular view of Q is actually pretty easy. Implementing this circular view of Q is actually pretty easy. Each time we increment f or r, we compute this increment as "(f + 1) mod N" or "(r + 1) mod N," respectively. Each time we increment f or r, we compute this increment as "(f + 1) mod N" or "(r + 1) mod N," respectively. "mod" is the modulo operator, which is computed by taking the remainder after an integral division. "mod" is the modulo operator, which is computed by taking the remainder after an integral division. For example, 14 divided by 4 is 3 with remainder 2, so 14 mod 4 = 2. For example, 14 divided by 4 is 3 with remainder 2, so 14 mod 4 = 2. Java uses "%" to denote the modulo operator. By using the modulo operator, we can view Q as a circular array and implement each queue method in a constant amount of time (that is, O(1) time). We describe how to use this approach to implement a queue in Code Fragment 5.14. Java uses "%" to denote the modulo operator. By using the modulo operator, we can view Q as a circular array and implement each queue method in a constant amount of time (that is, O(1) time). We describe how to use this approach to implement a queue in Code Fragment 5.14.

46. Code Fragment 5.14: Implementation of a queue using a circular array. The implementation uses the modulo operator to "wrap" indices around the end of the array and it also includes two instance variables, f and r, which index the front of the queue and first empty cell after the rear of the queue respectively.

47. Using the Modulo Operator to Implement a Circular Array The implementation above contains an important detail, which might be missed at first. The implementation above contains an important detail, which might be missed at first. Consider the situation that occurs if we enqueue N objects into Q without dequeuing any of them. We would have f = r, which is the same condition that occurs when the queue is empty. Consider the situation that occurs if we enqueue N objects into Q without dequeuing any of them. We would have f = r, which is the same condition that occurs when the queue is empty. Hence, we would not be able to tell the difference between a full queue and an empty one in this case. Hence, we would not be able to tell the difference between a full queue and an empty one in this case. Fortunately, this is not a big problem, and a number of ways for dealing with it exist. Fortunately, this is not a big problem, and a number of ways for dealing with it exist. The solution we describe here is to insist that Q can never hold more than N − 1 objects. The solution we describe here is to insist that Q can never hold more than N − 1 objects. This simple rule for handling a full queue takes care of the final problem with our implementation, and leads to the pseudo-coded descriptions of the queue methods given in Code Fragment 5.14. This simple rule for handling a full queue takes care of the final problem with our implementation, and leads to the pseudo-coded descriptions of the queue methods given in Code Fragment 5.14. Note our introduction of an implementation-specific exception, called FullQueueException, to signal that no more elements can be inserted in the queue. Note our introduction of an implementation-specific exception, called FullQueueException, to signal that no more elements can be inserted in the queue. Also note the way we compute the size of the queue by means of the expression (N − f + r) mod N, which gives the correct result both in the "normal" configuration (when f ≤ r) and in the "wrapped around" configuration (when r < f). Also note the way we compute the size of the queue by means of the expression (N − f + r) mod N, which gives the correct result both in the "normal" configuration (when f ≤ r) and in the "wrapped around" configuration (when r < f). The Java implementation of a queue by means of an array is similar to that of a stack, and is left as an exercise (P-5.4). The Java implementation of a queue by means of an array is similar to that of a stack, and is left as an exercise (P-5.4). Table 5.2 shows the running times of methods in a realization of a queue by an array. As with our array-based stack implementation, each of the queue methods in the array realization executes a constant number of statements involving arithmetic operations, comparisons, and assignments. Thus, each method in this implementation runs in O(1) time. Table 5.2 shows the running times of methods in a realization of a queue by an array. As with our array-based stack implementation, each of the queue methods in the array realization executes a constant number of statements involving arithmetic operations, comparisons, and assignments. Thus, each method in this implementation runs in O(1) time.

48. Table 5.2: Performance of a queue realized by an array The space usage is O(N), where N is the size of the array, determined at the time the queue is created. Note that the space usage is independent from the number n < N of elements that are actually in the queue. The space usage is O(N), where N is the size of the array, determined at the time the queue is created. Note that the space usage is independent from the number n < N of elements that are actually in the queue. MethodTime SizeisEmptyfrontenqueuedequeue O(1) Problem is the capacity of the queue to be some fixed value. In a real application, we may actually need more or less queue capacity than this, but if we have a good capacity estimate, then the array-based implementation is quite efficient.

49. Implementing a Queue with a Generic Linked List We can efficiently implement the queue ADT using a generic singly linked list. We can efficiently implement the queue ADT using a generic singly linked list. For efficiency reasons, For efficiency reasons, the front of the queue to be at the head of the list, the front of the queue to be at the head of the list, and the rear of the queue to be at the tail of the list. and the rear of the queue to be at the tail of the list. In this way, we remove from the head and insert at the tail. (Why would it be bad to insert at the head and remove at the tail?) Note that we need to maintain references to both the head and tail nodes of the list. Rather than go into every detail of this implementation, we simply give a Java implementation for the fundamental queue methods in Code Fragment 5.15. In this way, we remove from the head and insert at the tail. (Why would it be bad to insert at the head and remove at the tail?) Note that we need to maintain references to both the head and tail nodes of the list. Rather than go into every detail of this implementation, we simply give a Java implementation for the fundamental queue methods in Code Fragment 5.15.

50. Code Fragment 5.15: Methods enqueue and dequeue in the implementation of the queue ADT by means of a singly linked list, using nodes from class Node of Code Fragment 5.6. Each of the methods of the singly linked list implementation of the queue ADT runs in O(1) time. We also avoid the need to specify a maximum size for the queue, but this benefit comes at the expense of increasing the amount of space used per element. Still, the methods in the singly linked list queue implementation are more complicated than we might like, for we must take extra care in how we deal with special cases where the queue is empty before an enqueue or where the queue becomes empty after a dequeue.

51. Round Robin Schedulers A popular use of the queue data structure is to implement a round robin scheduler, where A popular use of the queue data structure is to implement a round robin scheduler, where we iterate through a collection of elements in a circular fashion and "service" each element by performing a given action on it. we iterate through a collection of elements in a circular fashion and "service" each element by performing a given action on it. Such a schedule is used, Such a schedule is used, for example, to fairly allocate a resource that must be shared by a collection of clients. for example, to fairly allocate a resource that must be shared by a collection of clients. For instance, we can use a round robin scheduler to allocate a slice of CPU time to various applications running concurrently on a computer. For instance, we can use a round robin scheduler to allocate a slice of CPU time to various applications running concurrently on a computer. We can implement a round robin scheduler using a queue, Q, by repeatedly performing the following steps (see Figure 5.5): We can implement a round robin scheduler using a queue, Q, by repeatedly performing the following steps (see Figure 5.5): 1.e ← Q.dequeue() 2.Service element e 3.Q.enqueue(e)

52. Figure 5.5: The three iterative steps for using a queue to implement a round robin scheduler.

53. The Josephus Problem In the children's game "hot potato," a group of n children sit in a circle passing an object, called the "potato," around the circle. The potato begins with a starting child in the circle, and the children continue passing the potato until a leader rings a bell, at which point the child holding the potato must leave the game after handing the potato to the next child in the circle. After the selected child leaves, the other children close up the circle. This process is then continued until there is only one child remaining, who is declared the winner. If the leader always uses the strategy of ringing the bell after the potato has been passed k times, for some fixed value k, then determining the winner for a given list of children is known as the Josephus problem. In the children's game "hot potato," a group of n children sit in a circle passing an object, called the "potato," around the circle. The potato begins with a starting child in the circle, and the children continue passing the potato until a leader rings a bell, at which point the child holding the potato must leave the game after handing the potato to the next child in the circle. After the selected child leaves, the other children close up the circle. This process is then continued until there is only one child remaining, who is declared the winner. If the leader always uses the strategy of ringing the bell after the potato has been passed k times, for some fixed value k, then determining the winner for a given list of children is known as the Josephus problem.

54. Solving the Josephus Problem Using a Queue We can solve the Josephus problem for a collection of n elements using a queue, by associating the potato with the element at the front of the queue and storing elements in the queue according to their order around the circle. Thus, passing the potato is equivalent to dequeuing an element and immediately enqueuing it again. After this process has been performed k times, we remove the front element by dequeuing it from the queue and discarding it. We show a complete Java program for solving the Josephus problem using this approach in Code Fragment 5.16, which describes a solution that runs in O(nk) time. (We can solve this problem faster using techniques beyond the scope of this book.) We can solve the Josephus problem for a collection of n elements using a queue, by associating the potato with the element at the front of the queue and storing elements in the queue according to their order around the circle. Thus, passing the potato is equivalent to dequeuing an element and immediately enqueuing it again. After this process has been performed k times, we remove the front element by dequeuing it from the queue and discarding it. We show a complete Java program for solving the Josephus problem using this approach in Code Fragment 5.16, which describes a solution that runs in O(nk) time. (We can solve this problem faster using techniques beyond the scope of this book.)

55. Code Fragment 5.16: A complete Java program for solving the Josephus problem using a queue. Class NodeQueue is shown in Code Fragment 5.15.