Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology Taiwan CSIE, NTUT, TAIWAN

Instructor Chuan-Ming Liu (劉傳銘) Office: 1530 Technology Building Computer Science and Information Engineering National Taipei University of Technology TAIWAN Phone:  (02) 2771-2171 ext. 4251 Email: cmliu@csie.ntut.edu.tw Office Hours: Mon: 11:10-12:00, 13:10-14:00 and Thu:10:10 - 12:00, OR by appointment. CSIE, NTUT, TAIWAN

Teaching Assisant Bill In-Chi Su (蘇英啟) Office: 1226 Technology Building Office Hours: Tue:10:00~12:00 and Wed: 10:00 ~ 12:00, OR by appointment Email: billandcs@gmail.com Phone: 02-2771-2171 ext. 4262 CSIE, NTUT, TAIWAN

Text Books Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Frees, Fundamentals of Data Structures in C, 2nd edition, Silicon Press, 2008. Supplementary Texts Michael T. Goodrich and Roberto Tamassia, Data Structures and Algorithms in JAVA, 4th edition, John Wiley & Sons, 2006. ISBN: 0-471-73884-0. Sartaj Sahni, Data Structures, Algorithms, and Applications in JAVA, 2nd edition, Silicon Press, 2005. ISBN: 0-929306-33-3. Frank M. Carranno and Walter Savitch, Data Structures and Abstractions with Java, Prentice Hall, 2003. ISBN: 0-13-017489-0. CSIE, NTUT, TAIWAN

Course Outline Introduction and Recursion Analysis Tools Arrays Stacks and Queues Linked Lists Sorting Hashing Trees Priority Queues Search Trees Graphs CSIE, NTUT, TAIWAN

Course Work Assignments (50%): 6-8 homework sets Midterm (20%) Final exam (30%) CSIE, NTUT, TAIWAN

Course Policy (1) No late homework is acceptable. For a regrade please contact me for the question within 10 days from the date when the quiz or exam was officially returned. No regrading after this period. Cheating directly affects the reputation of the Department and the University and lowers the morale of other students. Cheating in homework and exam will not be tolerated.  An automatic grade of 0 will be assigned to any student caught cheating. Presenting another person's work as your own constitutes cheating. Everything you turn in must be your own doing. CSIE, NTUT, TAIWAN

Course Policy (2) The following activities are specifically forbidden on all graded course work: Theft or possession of another student's solution or partial solution in any form (electronic, handwritten, or printed). Giving a solution or partial solution to another student, even with the explicit understanding that it will not be copied. Working together to develop a single solution and then turning in copies of that solution (or modifications) under multiple names. CSIE, NTUT, TAIWAN

First Thing to Do Please visit the course web site http://www.cc.ntut.edu.tw/~cmliu/DS/NTUT_DS_S09u-GIT/ Send an email to me using the email address:cmliu@csie.ntut.edu.tw. I will make a mailing list for this course. All the announcements will be broadcast via this mailing list. CSIE, NTUT, TAIWAN

Introduction Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology Taiwan CSIE, NTUT, TAIWAN

Outline Data Structures and Algorithms Pseudo-code Recursion CSIE, NTUT, TAIWAN

What is Data Structures A data structure * in computer science is a way of storing data in a computer so that it can be used efficiently. An organization of mathematical and logical concepts of data Implementation using a programming language A proper data structure can make the algorithm or solution more efficient in terms of time and space * Wikipedia: http://en.wikipedia.org/wiki/Data_structure CSIE, NTUT, TAIWAN

Why We Learn Data Structures Knowing data structures well can make our programs or algorithms more efficient In this course, we will learn Some basic data structures How to tell if the data structures are good or bad The ability to create some new and advanced data structures CSIE, NTUT, TAIWAN

What is an Algorithm (1) An algorithm is a finite set of instructions that, if followed, accomplishes a particular task. All the algorithms must satisfy the following criteria: Input Output Precision (Definiteness) Effectiveness Finiteness CSIE, NTUT, TAIWAN

What is an Algorithm (2) Definiteness: each instruction is clear and unambiguous Effectiveness: each instruction is executable; in other words, feasibility Finiteness: the algorithm terminates after a finite number of steps. CSIE, NTUT, TAIWAN

What is an Algorithm (3) Definiteness Output Input Effectiveness Computational Procedures CSIE, NTUT, TAIWAN

Procedures vs. Algorithms Termination or not One example for procedure is OS Program, a way to express an algorithm CSIE, NTUT, TAIWAN

Expressing Algorithms Ways to express an algorithm Graphic (flow chart) Programming languages (C/C++) Pseudo-code representation CSIE, NTUT, TAIWAN

Outline Data Structures and Algorithms Pseudo-code Recursion CSIE, NTUT, TAIWAN

Example – Selection Sort Suppose we must devise a program that sorts a collection of n1 elements. Idea: Among the unsorted elements, select the smallest one and place it next in the sorted list. for (int i=1; i<=n; i++) { examine a[i] to a[n] and suppose the smallest element is at a[j]; interchange a[i] and a[j]; } CSIE, NTUT, TAIWAN

Pseudo-code for Selection Sort SelectionSort(A) /* Sort the array A[1:n] into nondecreasing order. */ for i 1 to length[A] do j  i for k  (i+1) to length[A] do if A[k]<A[j] then j  k; t  A[i] A[i]  A[j] A[j]  t CSIE, NTUT, TAIWAN

Maximum of Three Numbers This algorithm finds the largest of the numbers a, b, and c. Input Parameters: a, b, c Output Parameter: x max(a,b,c,x) { x = a if (b > x) // if b is larger than x, update x x = b if (c > x) // if c is larger than x, update x x = c } CSIE, NTUT, TAIWAN

Pseudo-Code Conventions Indentation as block structure Loop and conditional constructs similar to those in PASCAL, such as while, for, repeat( do – while) , if-then-else // as the comment in a line Using = for the assignment operator Variables local to the given procedure CSIE, NTUT, TAIWAN

Pseudo-Code Conventions Relational operators: ==, != , , . Logical operators: &&, ||, !. Array element accessed by A[i] and A[1..j] as the subarray of A CSIE, NTUT, TAIWAN

Outline Data Structures and Algorithms Pseudo-code Recursion CSIE, NTUT, TAIWAN

Recursion Recursion is the concept of defining a method that makes a call to itself A method calling itself is making a recursive call A method M is recursive if it calls itself (direct recursion) or another method that ultimately leads to a call back to M (indirect recursion) CSIE, NTUT, TAIWAN

Repetition Repetition can be achieved by Factorial function Loops (iterative) : for loops and while loops Recursion (recursive) : a function calls itself Factorial function General definition: n! = 1· 2· 3· ··· · (n-1)· n Recursive definition CSIE, NTUT, TAIWAN

Method for Factorial Function recursive factorial function public static int recursiveFactorial(int n) { if (n == 0) return 1; else return n * recursiveFactorial(n- 1); } CSIE, NTUT, TAIWAN

Content of a Recursive Method Base case(s) Values of the input variables for which we perform no recursive calls are called base cases (there should be at least one base case). Every possible chain of recursive calls must eventually reach a base case. Recursive calls Calls to the current method. Each recursive call should be defined so that it makes progress towards a base case. CSIE, NTUT, TAIWAN

Visualizing Recursion Example recursion trace: Recursion trace A box for each recursive call An arrow from each caller to callee An arrow from each callee to caller showing return value return 4*6 recursiveFactorial(4) return 3*2 recursiveFactorial(3) return 2*1 recursiveFactorial(2) return 1*1 recursiveFactorial(1) return 1 recursiveFactorial(0) CSIE, NTUT, TAIWAN

About Recursion Advantages Examples Avoiding complex case analysis Avoiding nested loops Leading to a readable algorithm description Efficiency Examples File-system directories Syntax in modern programming languages CSIE, NTUT, TAIWAN

Linear Recursion The simplest form of recursion A method M is defined as linear recursion if it makes at most one recursive call Example: Summing the Elements of an Array Given: An integer array A of size m and an integer n, where m ≧n≧1. Problem: the sum of the first n integers in A. CSIE, NTUT, TAIWAN

Summing the Elements of an Array Solutions: using (for) loop using recursion Algorithm LinearSum(A, n): Input: integer array A, an integer n ≧ 1, such that A has at least n elements Output: The sum of the first n integers in A if n = 1 then return A[0] else return LinearSum(A, n - 1) + A[n - 1] Note: the index starts from 0. CSIE, NTUT, TAIWAN

Recursive Method An important property of a recursive method – the method terminates An algorithm using linear recursion has the following form: Test for base cases Recur CSIE, NTUT, TAIWAN

Analyzing Recursive Algorithms Recursion trace Box for each instance of the method Label the box with parameters Arrows for calls and returns call return 15 + A [ 4 ] = 15 + 5 = 20 LinearSum ( A , 5 ) call return 13 + A [ 3 ] = 13 + 2 = 15 LinearSum ( A , 4 ) call return 7 + A [ 2 ] = 7 + 6 = 13 LinearSum ( A , 3 ) call return 4 + A [ 1 ] = 4 + 3 = 7 LinearSum ( A , 2 ) call return A [ ] = 4 LinearSum ( A , 1 ) CSIE, NTUT, TAIWAN

Example Reversing an Array by Recursion Solutions Given: An array A of size n Problem: Reverse the elements of A (the first element becomes the last one, …) Solutions Nested loop ? Recursion CSIE, NTUT, TAIWAN

Reversing an Array Algorithm ReverseArray(A, i, j): Input: An array A and nonnegative integer indices i and j Output: The reversal of the elements in A starting at index i and ending at j if i < j then Swap A[i] and A[ j] ReverseArray(A, i + 1, j - 1) return CSIE, NTUT, TAIWAN

Facilitating Recursion In creating recursive methods, it is important to define the methods in ways that facilitate recursion. This sometimes requires we define additional parameters that are passed to the method. For example, we defined the array reversal method as ReverseArray(A, i, j), not ReverseArray(A). CSIE, NTUT, TAIWAN

Example – Computing Powers The power function, p(x, n)=xn, can be defined recursively: Following the definition leads to an O(n) time recursive algorithm (for we make n recursive calls). We can do better than this, however. CSIE, NTUT, TAIWAN

Recursive Squaring We can derive a more efficient linearly recursive algorithm by using repeated squaring: For example, 24 = 2(4/2)2 = (24/2)2 = (22)2 = 42 = 16 25 = 21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 32 26 = 2(6/2)2 = (26/2)2 = (23)2 = 82 = 64 27 = 21+(6/2)2 = 2(26/2)2 = 2(23)2 = 2(82) = 128 CSIE, NTUT, TAIWAN

A Recursive Squaring Method Algorithm Power(x, n): Input: A number x and integer n ≧ 0 Output: The value xn if n = 0 then return 1 if n is odd then y = Power(x, (n - 1)/ 2) return x · y ·y else /* n is even */ y = Power(x, n/ 2) return y · y CSIE, NTUT, TAIWAN

Analyzing the Recursive Squaring Method Algorithm Power(x, n): Input: A number x and integer n ≧0 Output: The value xn if n = 0 then return 1 if n is odd then y = Power(x, (n - 1)/ 2) return x · y · y else y = Power(x, n/ 2) return y · y Each time we make a recursive call we halve the value of n; hence, we make log n recursive calls. That is, this method runs in O(log n) time. It is important that we used a variable twice here rather than calling the method twice. CSIE, NTUT, TAIWAN

Tail Recursion Tail recursion occurs when a linearly recursive method makes its recursive call as its last step. Such methods can be easily converted to non-recursive methods (which saves on some resources). The array reversal method is an example. CSIE, NTUT, TAIWAN

Example – Using Iteration Algorithm IterativeReverseArray(A, i, j ): Input: An array A and nonnegative integer indices i and j Output: The reversal of the elements in A starting at index i and ending at j while i < j do Swap A[i ] and A[ j ] i = i + 1 j = j - 1 return CSIE, NTUT, TAIWAN

Binary Recursion Binary recursion occurs whenever there are two recursive calls for each non-base case. Example: BinaySum CSIE, NTUT, TAIWAN

Example – Summing n Elements in an Array Recall that this problem has been solved using linear recursion Using binary recursion instead of linear recursion Algorithm BinarySum(A, i, n): Input: An array A and integers i and n Output: The sum of the n integers in A starting at index i if n = 1 then return A[i ] return BinarySum(A,i,n/2)+BinarySum(A,i+n/2,n/2) CSIE, NTUT, TAIWAN

Recursion Trace Note the floor and ceiling used in the method 3 , 1 2 4 8 7 6 5 CSIE, NTUT, TAIWAN

Fibonacci Numbers Fibonacci numbers are defined recursively: F0 = 0 Fi = Fi-1 + Fi-2 for i > 1. Example: 0, 1, 1, 2, 3, 5, 8, … CSIE, NTUT, TAIWAN

Fibonacci Numbers – Binary Recursion Algorithm BinaryFib(k): Input: Nonnegative integer k Output: The kth Fibonacci number Fk if k ≦ 1 then return k else return BinaryFib(k-1) + BinaryFib(k-2) CSIE, NTUT, TAIWAN

Analyzing the Binary Recursion Algorithm BinaryFib makes a number of calls that are exponential in k By observation, there are many redundant computations: F0 = 0; F1 = 1; F2 = F1 + F0; F3 = F2 + F1 =(F1 + F0)+ F1; … The above two results show the inefficiency of the method using binary recursion CSIE, NTUT, TAIWAN

A Better Fibonacci Algorithm Using linear recursion instead – avoid the redundant computation: Algorithm LinearFibonacci(k): Input: A nonnegative integer k Output: Pair of Fibonacci numbers (Fk, Fk-1) if k = 1 then return (k, 0) else (i, j) = LinearFibonacci(k - 1) return (i +j, i) Runs in O(k) time. CSIE, NTUT, TAIWAN

Multiple Recursion Motivating example: summation puzzles pot + pan = bib dog + cat = pig boy + girl = baby Multiple recursion: makes potentially many recursive calls (not just one or two). CSIE, NTUT, TAIWAN

Algorithm for Multiple Recursion Algorithm PuzzleSolve(k,S,U): Input: An integer k, sequence S, and set U (the universe of elements to test) Output: An enumeration of all k-length extensions to S using elements in U without repetitions for all e in U do Remove e from U {e is now being used} Add e to the end of S if k = 1 then Test whether S is a configuration that solves the puzzle if S solves the puzzle then return “Solution found: ” S else PuzzleSolve(k - 1, S,U) Add e back to U {e is now unused} Remove e from the end of S CSIE, NTUT, TAIWAN

Visualizing PuzzleSolve() 3 , () ,{ a b c } ) Initial call 2 1 ab ac cb ca bc ba abc acb bac bca cab cba CSIE, NTUT, TAIWAN