Stacks1 CS2468 Data Structures and Data Management Lecturer: Lusheng Wang Office: B6422 Phone: Welcome to ask questions at ANY time. Java Source code: The course Website: Text Book: Michael T. Goodrich and Roberto Tamassia, Data Structurea and Algorithms in Java, John Wiley & Sons, Inc.

Stacks2 What We Cover Analysis of Algorithms: worst case time and space complexity Data Structures: stack, queue, linked list, tree, priority queue, heap, and hash; Searching algorithms: binary and AVL search trees; Sorting algorithms: merge sort, quick sort, bucket sort and radix sort; (Reduce some contents) Graph: data structure, depth first search and breadth first search. (add some interesting contents). LCS and shortest path.

Stacks3 Why This Course? You will be able to evaluate the quality of a program (Analysis of Algorithms: Running time and memory space ) You will be able to write fast programs You will be able to solve new problems You will be able to give non-trivial methods to solve problems. ( Your algorithm (program) will be faster than others.)

Stacks4 Course Evaluations Course work: 30% Final Exam: 70% Course Work: Three assignments (do them in groups of two people) +a mid term exam.

Stacks5 Pre-requisites: CS2360CS2360 Java Programming /or CS2362 Computer Programming for Engineers & Scientists /or CS2363 Computer Programming /or CS2372 Fundamentals of Programming /or equivalent If you are not familiar with Java, it is difficult to take this course (but still possible). CS2362 CS2363 CS2372 Spend the rest of 8-10 days to study Java. Test: Design a class with two integers C1 and C2 and a method f() to calculate the average value of c1 and c2. If you cannot do that, you should worry…

Data Structures: A systematic way of organizing and accessing data. -- No single data structure works well for ALL purposes. Algorithm Input Output An algorithm is a step-by-step procedure for solving a problem in a finite amount of time.

Stacks7 Algorithm Descriptions Nature languages: Chinese, English, etc. Pseudo-code: codes very close to computer languages, e.g., C programming language. Programs: C programs, C++ programs, Java programs. Goal: Allow a well-trained programmer to be able to implement. Allow an expert to be able to analyze the running time.

Stacks8 An Example of an Algorithm Algorithm sorting(X, n) Input array X of n integers Output array X sorted in a non-decreasing order for i  0 to n  1 do for j  i+1 to n-1 do if (X[i]>X[j]) then { s=X[i]; X[i]=X[j]; X[j]=s; } return X // after i-th round, the i-th smallest number is at i-th position.

Part-B1 Stacks

Stacks10 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 Example: ADT modeling a students record The data stored are  Student name, id No., as1, as2,as3, exam The operations supported are  int averageAs(as1,as2,as3)  Int finalMark(as1, as2,as3, exam) ) Error conditions:  Calculate the final mark for absent student

Stacks11 The Stack ADT (§4.2) The Stack ADT stores arbitrary objects Insertions and deletions follow the last-in first-out scheme Think of a spring-loaded plate dispenser Main stack operations: push(object): inserts an element object pop(): removes and returns the last inserted element Auxiliary stack operations: object top(): returns the last inserted element without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored

Stacks12 Applications of Stacks Direct applications 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

Stacks13 Parentheses Matching An expression, i.e.,[(2+3)*x+5]*2. Each “(”, “{”, or “[” must be paired with a matching “)”, “}”, or “[” correct: ( )(( )){([( )])} correct: ((( )(( )){([( )])})) incorrect: )(( )){([( )])} incorrect: ({[ ])} incorrect: (

Stacks14 Parentheses Matching Algorithm Algorithm ParenMatch( 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. isEmpty () 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. isEmpty () then return true { every symbol matched } else return false { some symbols were never matched }

Stacks15 Parentheses Matching Example 1 Input: () (() [()]) i X[i] OperationStackOutput 0(Push (( 1)Pop ( Test if ( and X[i] match? YES 2(Push (( 3( (( 4)Pop ( Test if ( and X[i] match? YES ( 5[Push [([

Stacks16 Parentheses Matching Example 1 Input: () (() [()]) i X[i] OperationStackOutput 6(Push (([( 7)Pop ( Test if ( and X[i] match? YES ([ 8]Pop [ Test if [ and X[i] match? YES ( 9)Pop ( Test if ( and X[i] match? YES Test if stack is Empty? YES TRUE

Stacks17 Parentheses Matching Example 2 Input: ( () [] ]() i X[i] OperationStackOutput 0(Push (( 1( (( 2)Pop ( Test if ( and X[i] match? YES ( 3[Push [([ 4]Pop [ Test if [ and X[i] match? YES ( 5]Pop ( Test if ( and X[i] match ? NOFASLE

Stacks18 Java Program VS C Program Refer to the java program QuadraticSort.java and QuadraticSort.C. Objects: items we deal with Class: type of objects Objects store data and provides methods. Structure of data (components of data)  Forexamlpe: A stack holding: (([. Methods (functions) to do things  For example: push(), pop(), Top(), etc.

Stacks19 Stack Interface in Java Java interface corresponding to our Stack ADT Requires the definition of class EmptyStackException public interface Stack { public int size(); public boolean isEmpty(); public Object top() throws EmptyStackException; public void push(Object o); public Object pop() throws EmptyStackException; }

Stacks20 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 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 EmptyStackException

Stacks21 Array-based Stack (Implementation) A simple way of implementing the Stack ADT uses an array We add elements from left to right A variable t keeps track of the index of the top element S 012 t … Algorithm size() return t + 1 Algorithm pop() if isEmpty() then throw EmptyStackException else t  t  1 return S[t + 1]

Stacks22 Array-based Stack (cont.) The array storing the stack elements may become full A push operation will then throw a FullStackException Limitation of the array- based implementation Not intrinsic to the Stack ADT S 012 t … Algorithm push(o) if t = S.length  1 then throw FullStackException else t  t + 1 S[t]  o

Stacks23 Array-based Stack (Cont.) A Stack might be empty top returns the element at the top of the Stack, but does not remove the top element. When the Stack is empty, an error occurs. S 012 t … Algorithm isEmpty() if t<0 then return true else return false Algorithm top() if isEmpty() then throw EmptyStackException return S[t ]

Stacks24 Performance and Limitations for array-based Stack 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

Stacks25 Array-based Stack in Java public class ArrayStack implements Stack{ // holds the stack elements private Object S[ ]; // index to top element private int top = -1; // constructor public ArrayStack(int capacity) { S = new Object[capacity]); }

Stacks26 Array-based Stack in Java public Object pop() throws EmptyStackException{ if isEmpty() throw new EmptyStackException (“Empty stack: cannot pop”); Object temp = S[top]; // facilitates garbage collection S[top] = null; top = top – 1; return temp; } public int size() { return (top + 1);

Stacks27 Array-based Stack in Java public boolean isEmpty() { return (top < 0); } public void push(Object obj) throws FullStackException { if (size() == capacity) throw new FullStackException("Stack overflow."); S[++top] = obj; } public Object top() throws EmptyStackException { if (isEmpty()) throw new EmptyStackException("Stack is empty."); return S[top]; }

Stacks28 Computing Spans (not in book) We show how to use a stack as an auxiliary data structure in an algorithm Given 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[j+1] Spans have applications to financial analysis E.g., stock at 52-week high X S

Stacks29 Quadratic Algorithm Algorithm spans1(X, n) Input array X of n integers Output array S of spans of X # S  new array of n integers n for i  0 to n  1 don s  1n while s  i  X[i  s]  X[i-s+1] 1  2  …  (n  1) s  s  1 1  2  …  (n  1) S[i]  s n return S 1 X[]= 1,2,3, …, n-1, n. Algorithm spans1 runs in O(n 2 ) time

Stacks30 Computing Spans with a Stack We push the n elements in the stack We sacn the elements in the stack in the reverse order. We find the last element x[j] with x[j] x[i] and set s[i]=i-j+1. If no such a j exists, s[i]=i+1. Compute the s[i] for the remaining i’s as follows: for (i=n-2; i>=0; i--) if (s[i+1]>1 & s[i]==0) then s[i]=s[i+1]-1

Stacks31 Example: i= 0, 1, 2, 3, 4, 5, 6, 7 X[i]= 6, 3, 4, 1, 2, 3, 5, 4 S[i]= 1, 1, 2, 1, 2, 3, 4, Stack Stack Stack Stack Stack

Stacks32 Linear Algorithm Algorithm spans2(X, n) # S  new array of n integers n A  new empty stack 1 for i  0 to n  1 doA.push(i) n i=n-1 ; j=n-1; 1 while (i<=0) do while (  A.isEmpty()  X[A.top()]  X[j] ) do n j  A.pop() n if A.isEmpty() then n S[i]  i  1 1 else S[i]  i  j+1 n i=j-1; n return S 1 Each index of the array Is pushed into the stack exactly one Is popped from the stack at most once The statements in the two while loops are executed at most n times Algorithm spans2 runs in O(n) time for (i=n-2; i>=0; i--) if (s[i+1]>1 & s[i]==0) then s[i]=s[i+1]-1

Stacks33 Summary of Stack Understand the definition of Stack (basic) Applications 1. Parentheses (moderate) 2. Computing Spans (not required)  Implementation of Stack is not required.  You should be able to use stack to write programs to solve problems

Stacks34 Remarks Emphasize the concept of ADT. More examples about ADT Delete the Span application examples. Add Queue part (perhaps to week 3) Teach slowly.