Chapter 5 Stacks and Queues Objectives Stacks and

Chapter 5 Stacks and Queues Objectives – Stacks and implementations – Queues and implementations – Double-ended queues and implementation Stacks and Queues CSC 311: Data Structures 1

Abstract Data Types (ADTs) An abstract data type (ADT) is an abstraction of a data structure An ADT specifies: – Data stored – Operations on the data – Error conditions associated with operations Stacks and Queues CSC 311: Data Structures 2

ADT Example: ADT modeling a simple stock trading system – The data stored are buy/sell orders – The operations supported are order buy(stock, shares, price) order sell(stock, shares, price) void cancel(order) – Error conditions: Buy/sell a nonexistent stock Cancel a nonexistent order Stacks and Queues CSC 311: Data Structures 3

The Stack ADT stores arbitrary objects Insertions and deletions follow the last -in first-out scheme Think of a springloaded plate dispenser Main stack operations: – push(object): inserts an element – object pop(): removes and returns the last inserted element Stacks and Queues Auxiliary stack operations: – object top(): returns the last inserted element without removing it – integer size(): returns the number of elements stored – boolean is. Empty(): indicates whether no elements are stored CSC 311: Data Structures 4

Stack Interface in Java interface corresponding to our Stack ADT Requires the definition of class Empty. Stack. Except ion Different from the built-in Java class java. util. Stacks and Queues public interface Stack { public int size(); public boolean is. Empty(); public Object top() throws Empty. Stack. Exception; public void push(Object o); public Object pop() throws Empty. Stack. Exception; } CSC 311: Data Structures 5

Exceptions Attempting the execution of an operation of ADT may sometimes cause an error condition, called an exception Exceptions are said to be “thrown” by an operation that cannot be executed Stacks and Queues In the Stack ADT, operations pop and top cannot be performed if the stack is empty Attempting the execution of pop or top on an empty stack throws an Empty. Stack. Exception CSC 311: Data Structures 6

Applications of Stacks Direct applications – Page-visited history in a Web browser – Undo sequence in a text editor – Chain of method calls in the Java Virtual Machine Indirect applications – Auxiliary data structure for algorithms – Component of other data structures Stacks and Queues CSC 311: Data Structures 7

Method Stack in the JVM The Java Virtual Machine (JVM) keeps track of the chain main() { of active methods with a stack int i = 5; When a method is called, the foo(i); JVM pushes on the stack a } frame containing – Local variables and return value foo(int j) { – Program counter, keeping track int k; of the statement being executed k = j+1; When a method ends, its bar(k); frame is popped from the } stack and control is passed to the method on top of the stack Allows for recursion Stacks and Queues bar(int m) { … } CSC 311: Data Structures bar PC = 1 m=6 foo PC = 3 j=5 k=6 main PC = 2 i=5 8

Array-based Stack A simple way of implementing the Stack ADT uses an array We add elements from left to right A variable keeps track of the index of the top element Algorithm size() return t + 1 Algorithm pop() if is. Empty() then throw Empty. Stack. Exception else t t 1 return S[t + 1] … S 0 1 2 Stacks and Queues t CSC 311: Data Structures 9

Array-based Stack (cont. ) The array storing the stack elements may become full A push operation will then throw a Full. Stack. Exception – Limitation of the array-based implementation – Not intrinsic to the Stack ADT Algorithm push(o) if t = S. length 1 then throw Full. Stack. Exception else t t+1 S[t] o … S 0 1 2 Stacks and Queues t CSC 311: Data Structures 10

Performance and Limitations Performance – Let n be the number of elements in the stack – The space used is O(n) – Each operation runs in time O(1) Limitations – The maximum size of the stack must be defined a priori and cannot be changed – Trying to push a new element into a full stack causes an implementation-specific exception Stacks and Queues CSC 311: Data Structures 11

Array-based Stack in Java public class Array. Stack implements Stack { // holds the stack elements private Object S[ ]; // index to top element private int top = -1; // constructor public Array. Stack(int capacity) { S = new Object[capacity]); } Stacks and Queues public Object pop() throws Empty. Stack. Exception { if is. Empty() throw new Empty. Stack. Exception (“Empty stack: cannot pop”); Object temp = S[top]; // facilitates garbage collection S[top] = null; top = top – 1; return temp; CSC 311: Data Structures 12

Parentheses Matching Each “(”, “{”, or “[” must be paired with a matching “)”, “}”, or “[” – correct: ( )(( )){([( )])} – correct: ((( )){([( )])} – incorrect: )(( )){([( )])} – incorrect: ({[ ])} – incorrect: ( Stacks and Queues CSC 311: Data Structures 13

Parentheses Matching Algorithm Paren. Match(X, n): Input: An array X of n tokens, each of which is either a grouping symbol, a variable, an arithmetic operator, or a number Output: true if and only if all the grouping symbols in X match Let S be an empty stack for i=0 to n-1 do if X[i] is an opening grouping symbol then S. push(X[i]) else if X[i] is a closing grouping symbol then if S. is. Empty() then return false {nothing to match with} if S. pop() does not match the type of X[i] then return false {wrong type} if S. is. Empty() then return true {every symbol matched} else return false {some symbols were never matched} Stacks and Queues CSC 311: Data Structures 14

HTML Tag Matching For fully-correct HTML, each <name> should pair with a matching </name> <body> <center> <h 1> The Little Boat </h 1> </center> <p> The storm tossed the little boat like a cheap sneaker in an old washing machine. The three drunken fishermen were used to such treatment, of course, but not the tree salesman, who even as a stowaway now felt that he had overpaid for the voyage. </p> <ol> <li> Will the salesman die? </li> <li> What color is the boat? </li> <li> And what about Naomi? </li> </ol> </body> Stacks and Queues The Little Boat The storm tossed the little boat like a cheap sneaker in an old washing machine. The three drunken fishermen were used to such treatment, of course, but not the tree salesman, who even as a stowaway now felt that he had overpaid for the voyage. 1. Will the salesman die? 2. What color is the boat? 3. And what about Naomi? CSC 311: Data Structures 15

Tag Matching Algorithm Is similar to parentheses matching: import java. util. String. Tokenizer ; import datastructures. Stack; import datastructures. Node. Stack; import java. io. *; /** Simpli. ed test of matching tags in an HTML document. */ public class HTML { /** Nested class to store simple HTML tags */ public static class Tag { String name; // The name of this tag boolean opening; // Is true i. this is an opening tag public Tag() { // Default constructor name = ""; opening = false; } public Tag(String nm, boolean op) { // Preferred constructor name = nm; opening = op; } /** Is this an opening tag? */ public boolean is. Opening() { return opening; } /** Return the name of this tag */ public String get. Name() {return name; } } /** Test if every opening tag has a matching closing tag. */ public boolean is. HTMLMatched(Tag[ ] tag) { Stacks and Queues } Stack S = new Node. Stack(); // Stack for matching tags for (int i=0; (i<tag. length) && (tag[i]] != null); i++) { if (tag[i]. is. Opening()) S. push(tag[i]. get. Name()); // opening tag; push its name on the stack else { if (S. is. Empty()) // nothing to match return false; if (!((String) S. pop()). equals(tag[i]. get. Name())) // wrong match return false; } } if (S. is. Empty()) return true; // we matched everything return false; // we have some tags that never were matched CSC 311: Data Structures 16

Tag Matching Algorithm, cont. public final static int CAPACITY = 1000; // Tag array size upper bound /* Parse an HTML document into an array of html tags */ public Tag[ ] parse. HTML(Buffered. Reader r) throws IOException { String line; // a line of text boolean in. Tag = false ; // true iff we are in a tag Tag[ ] tag = new Tag[CAPACITY]; // our tag array (initially all null) int count = 0 ; // tag counter while ((line = r. read. Line()) != null) { // Create a string tokenizer for HTML tags (use < and > as delimiters) String. Tokenizer st = new String. Tokenizer(line , "<> t", true); while (st. has. More. Tokens()) { } String token = (String) st. next. Token(); if (token. equals("<")) // opening a new HTML tag in. Tag = true; else if (token. equals(">")) // ending an HTML tag in. Tag = false; else if (in. Tag) { // we have a opening or closing HTML tag if ( (token. length() == 0) | | (token. char. At(0) != ’/’) ) tag[count++] = new Tag(token, true); // opening tag else // ending tag[count++] = new Tag(token. substring(1), false); // skip the } // Note: we ignore anything not in an HTML tag } return tag; // our array of tags } } /** Tester method */ public static void main(String[ ] args) throws IOException { Buffered. Reader stdr; // Standard Input Reader stdr; stdr = new Buffered. Reader(new Input. Stream. Reader(System. in)); HTML tag. Checker = new HTML(); if (tag. Checker. is. HTMLMatched(tag. Checker. parse. HTML(stdr))) System. out. println("The input file is a matched HTML document. "); else System. out. println("The input file is not a matched HTML document. "); } Stacks and Queues CSC 311: Data Structures 17

Computing Spans We show to use a stack as an auxiliary data structure in an algorithm Given an an array X, the span S[i] of X[i] is the maximum number of consecutive elements X[j] immediately preceding X[i] and such that X[j] X [i ] Spans have applications to financial analysis – E. g. , stock at 52 -week high Stacks and Queues X S CSC 311: Data Structures 6 1 3 1 4 2 5 3 2 1 18

Quadratic Algorithm spans 1(X, n) Input array X of n integers Output array S of spans of X S new array of n integers for i 0 to n 1 do s 1 while s i X[i s] X[i] s s+1 S [i ] s return S # n n n 1 + 2 + …+ (n 1) n 1 Algorithm spans 1 runs in O(n 2) time Stacks and Queues CSC 311: Data Structures 19

Computing Spans with a Stack We keep in a stack the indices of the elements visible when “looking back” We scan the array from left to right – Let i be the current index – We pop indices from the stack until we find index j such that X[i] X[j] – We set S[i] i j – We push x onto the stack Stacks and Queues CSC 311: Data Structures 20

Linear Algorithm Each index of the array – Is pushed into the stack exactly once – Is popped from the stack at most once The statements in the while-loop are executed at most n times Algorithm spans 2 runs in O(n) time Stacks and Queues Algorithm spans 2(X, n) # S new array of n integers n A new empty stack 1 for i 0 to n 1 do n while ( A. is. Empty() X[A. top()] X[i] ) do n A. pop() n if A. is. Empty() then n S [i ] i + 1 n else S[i] i A. top() n A. push(i) n return S 1 CSC 311: Data Structures 21

The Queue ADT stores arbitrary objects Insertions and deletions follow the first-in first-out scheme Insertions are at the rear of the queue and removals are at the front of the queue Main queue operations: – enqueue(object): inserts an element at the end of the queue – object dequeue(): removes and returns the element at the front of the queue Stacks and Queues Auxiliary queue operations: – object front(): returns the element at the front without removing it – integer size(): returns the number of elements stored – boolean is. Empty(): indicates whether no elements are stored Exceptions – Attempting the execution of dequeue or front on an empty queue throws an Empty. Queue. Exception CSC 311: Data Structures 22

Queue Example Operation enqueue(5) enqueue(3) dequeue() enqueue(7) dequeue() front() dequeue() is. Empty() enqueue(9) enqueue(7) size() enqueue(3) enqueue(5) dequeue() Stacks and Queues Output Q – (5) – (5, 3) 5 (3) – (3, 7) 3 (7) 7 () “error” () true () – (9, 7) 2 (9, 7) – (9, 7, 3, 5) 9 (7, 3, 5) CSC 311: Data Structures 23

Applications of Queues Direct applications – Waiting lists, bureaucracy – Access to shared resources (e. g. , printer) – Multiprogramming Indirect applications – Auxiliary data structure for algorithms – Component of other data structures Stacks and Queues CSC 311: Data Structures 24

Array-based Queue Use an array of size N in a circular fashion Two variables keep track of the front and rear f index of the front element r index immediately past the rear element Array location r is kept empty normal configuration Q 0 1 2 f r wrapped-around configuration Q 0 1 2 Stacks and Queues r f CSC 311: Data Structures 25

Queue Operations We use the modulo operator (remainder of division) Algorithm size() return (N f + r) mod N Algorithm is. Empty() return (f = r) Q 0 1 2 f 0 1 2 r r Q Stacks and Queues f CSC 311: Data Structures 26

Queue Operations (cont. ) Operation enqueue throws an exception if the array is full This exception is implementationdependent Algorithm enqueue(o) if size() = N 1 then throw Full. Queue. Exception else Q[r] o r (r + 1) mod N Q 0 1 2 f 0 1 2 r r Q Stacks and Queues f CSC 311: Data Structures 27

Queue Operations (cont. ) Operation dequeue Algorithm dequeue() if is. Empty() then throws an exception if the throw Empty. Queue. Exception queue is empty else This exception is o Q[f] specified in the f (f + 1) mod N queue ADT return o Q 0 1 2 f 0 1 2 r r Q Stacks and Queues f CSC 311: Data Structures 28

Queue Interface in Java interface corresponding to our Queue ADT Requires the definition of class Empty. Queue. Excepti on No corresponding built-in Java class Stacks and Queues public interface Queue { public int size(); public boolean is. Empty(); public Object front() throws Empty. Queue. Exception; public void enqueue(Object o); public Object dequeue() throws Empty. Queue. Exception; CSC 311: Data Structures 29

Application: Round Robin Schedulers We can implement a round robin scheduler using a queue, Q, by repeatedly performing the following steps: 1. 2. 3. e = Q. dequeue() Service element e Q. enqueue(e) The Queue 1. Deque the next element 2. Service the next element 3. Enqueue the serviced element Shared Service Stacks and Queues CSC 311: Data Structures 30