CS2468 Data Structures and Data Management

1. CS2468 Data Structures and Data Management • Lecturer: Lusheng Wang • Office: B6422 • Phone: 2788 9820 • E-mail lwang@cs.cityu.edu.hk Welcome to ask questions at ANY time. • Java Source code: http://net3.datastructures.net/download.html • The course Website: http://www.cs.cityu.edu.hk/~lwang/cs2468.html Text Book: Michael T. Goodrich and Roberto Tamassia, Data Structurea and Algorithms in Java, John Wiley & Sons, Inc. Stacks

2. 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. Stacks

3. 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.) Stacks

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

5. Pre-requisites: • CS2360 Java Programming /orCS2362 Computer Programming for Engineers & Scientists /orCS2363 Computer Programming /orCS2372 Fundamentals of Programming /or equivalentIf you are not familiar with Java, it is difficult to take this course (but still possible). • 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… Stacks

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

7. 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. Stacks

8. An Example of an Algorithm Algorithmsorting(X, n) Inputarray X of n integers Outputarray X sorted in a non-decreasing order fori0ton 1do for ji+1ton-1do if(X[i]>X[j])then { s=X[i]; X[i]=X[j]; X[j]=s; } returnX • // after i-th round, the i-th smallest number is at i-th position. Stacks

9. Part-B1Stacks

10. 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 Abstract Data Types (ADTs) Stacks

11. 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 The Stack ADT (§4.2) Stacks

12. 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 Stacks

13. 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: ( Stacks

14. 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} Stacks

15. Parentheses Matching Example 1 • Input: () (() [()]) Stacks

16. Parentheses Matching Example 1 • Input: () (() [()]) Stacks

17. Parentheses Matching Example 2 • Input: ( () [] ]() Stacks

18. 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. Java Program VS C Program Stacks

19. Stack Interface in Java public interfaceStack{ public int size(); public boolean isEmpty(); public Object top()throwsEmptyStackException; public voidpush(Object o); public Object pop()throwsEmptyStackException;} • Java interface corresponding to our Stack ADT • Requires the definition of class EmptyStackException Stacks

20. 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 Exceptions Stacks

21. 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 Array-based Stack (Implementation) Algorithmsize() returnt +1 Algorithmpop() ifisEmpty()then throw EmptyStackException else tt 1 returnS[t +1] … S 0 1 2 t Stacks

22. … S 0 1 2 t 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 Algorithmpush(o) ift=S.length 1then throw FullStackException else tt +1 S[t] o Stacks

23. 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. Array-based Stack (Cont.) AlgorithmisEmpty() ift<0thenreturntrue elsereturnfalse Algorithmtop() ifisEmpty()then throw EmptyStackException returnS[t ] … S 0 1 2 t Stacks

24. 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 Stacks

25. Array-based Stack in Java public classArrayStack implements Stack{ // holds the stack elementsprivate Object S[ ]; // index to top elementprivate int top = -1; // constructorpublicArrayStack(int capacity){S = new Object[capacity]);} Stacks

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

27. 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]; } Stacks

28. 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 Stacks

29. Quadratic Algorithm Algorithmspans1(X, n) Inputarray X of n integers Outputarray S of spans of X # S new array of n integers n fori0ton 1do n s 1n while s i X[i - s]X[i-s+1]1 + 2 + …+ (n 1) ss+ 11 + 2 + …+ (n 1) S[i]sn returnS 1 X[]= 1,2,3, …, n-1, n. • Algorithm spans1 runs in O(n2) time Stacks

30. 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 x[j-1]>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 Stacks

31. 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, 1. 4 5 5 3 3 2 2 1 1 4 4 4 3 3 3 6 6 6 6 Stack Stack Stack Stack Stack Stacks

32. Linear Algorithm Algorithmspans2(X, n)# S new array of n integers n A new empty stack 1 fori0ton 1do A.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()thenn S[i] i +1 1 else S[i]i - j +1 n i=j-1; n returnS 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 Stacks

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

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