Chapter 1 Introduction

1. Example 1 Describe an algorithm for finding the maximum (largest) value in a finite sequence of integers. Instead of using a particular computer language, we use a form of pseudocode. Chapter 1 Introduction Definition of Algorithm An algorithm is a finite sequence of precise instructions for performing a computation or for solving a problem. • Solution • Set the temporary maximum equal to the first integer in the sequence. • Compare the next integer in the sequence to the temporary maximum, and set the larger one to be temporary maximum. • Repeat the previous step if there are more integers in the sequence. • Stop when there are no integers left in the sequence. The temporary maximum at this point is the maximum in the sequence.

2. Understand the problem Design an algorithm and proper data structures Analyze the algorithm Code the algorithm Algorithmic Problem Solving • Ascertaining the Capabilities of a Computational Device • Choosing between Exact and Approximate Problem Solving • Deciding on Appropriate Data Structures • Algorithm Design • Algorithm Analysis • Coding

3. Important Problem Types • Sorting • Searching • String processing (e.g. string matching) • Graph problems (e.g. graph coloring problem) • Combinatorial problems (e.g. maximizes a cost) • Geometric problems (e.g. convex hull problem) • Numerical problems (e.g. solving equations )

4. 10 6 9 4 7 17 20 13 1 4 9 Fundamental Data Structures 1. Array a[1] a[2] a[3] a[n] 2. Link List Head of list 3. Binary Tree

5. A[0] 20 A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9] A[1] A[2] 19 17 A[3] A[4] A[5] A[6] 13 16 2 4 A[7] A[8] A[9] 20 19 17 13 4 16 2 10 8 3 10 8 3 Node i’s parent and children Parent: Left child: 2i +1 Right child: 2i+2 4. Heap Definition 1A max-heap (or min-heap) A is an array object that can be reviewed as a complete binary tree (each node has two children, except the right part of the last level) , where at each node, there is a key which is larger (or smaller) or equal to the keys of its children. The height of a max-heap with n elements is O(log n) • Operations in a max-Heap (or min-heap): • Extract the item with the maximum (or minimum ) key. • Insert a new item into the heap

6. 15 6 18 3 7 17 20 13 1 4 9 5. Binary Search Tree • Definition 1 Binary Search Tree is a Binary Tree satisfying the following condition: • Each vertex contains an item called as key which belongs to a total ordering set and two links to its left child and right child, respectively. • In each node, its key is larger than the keys of all vertices in its left subtree and smaller than the keys of all the vertices in its right subtree. Operations a binary search tree support: search, insert, and delete an element with a given key. How to select a data structure for a given problem?

7. rear front Fundamental Abstract Data Structures 1. Stack • Operations a stack supports: last-in-first-out (pop, push) • Implementation: using an array or a list (need to check underflows and overflows) 2. Queue • Operations a queue supports: first-in-first-out (enqueue, dequeue) • Implementation: using an array or a list (need to check underflows and overflows) • 3. Priority Queue (each elements of a priority queue has a key) • Operations a priority queue supports: inserting an element, returning an element with maximum/minimum key. • Implementation: heap

8. 4. Dynamic Dictionary • A Dynamic Dictionary is a data structure of item • with keys that support the following basic operations: • Insert a new item • Remove an item with a given key • (3) Search an item with a given key

9. Graph • Operations a graph support: finding neighbors • Implementation: using list or matrix a c b c a d e f c f b e c a e d f c e f Adjacent lists Some Advanced Data Structure Adjacent matrix

10. Assignment 1 • (1) Read the sections of the text book about array, linked list, heap, binary search tree, stack, queue, priority queue, and dynamic dictionary. • (i) Implement stack S and queue Q using both of array and linked list. • (ii) Write a main function to do the following operations for S: pop, push 10 times, pop, pop, repeat push and pop 7 times. Do the following operations for Q: dequeue, enqueue 10 times, dequeue, depueue, repeat enqueue and dequeue 7 times. When you use array, declare the size of the array to be 10. Printout all the elements in S and in Q. • Select one of (3) and (4) • (3) Implement a min-heap of integer. The program must contain the following operations: • Heapify the min-heap • Build the min-heap from a given array. • Extract an item with the minimum key. • Insert a new item into the min-heap. • Print the heap. • Also, write a Main method to test your program as follows: • Create an array A of integers. • Create an empty min-heap. • Assign10integers one by one into the array A in the following order: 12, 34, 76, 21, 4, 3, 33, 88, 90, 15. • Build the min-heap from array A, and print the heap. • Extract the smallest integer, and print the heap. • Insert a new integer into the heap, and print the heap.

11. (4) Write the program for BST of student records (student’s id is used as key). The program must contain the following 5 operations: • Search a node (given a student id), and print the record when it is found. • Insert a node (given a new student record) • Delete a node (given a student id) • Print out of all student records in the BST in in-order. • Also, write a Main method to test your program as follows: • Create an empty BST. • Insert 10 student records one by one into the BST in the following order: record1: 1884, name1, score1 record2: -174, name2, score2 record3: 1464, name3, score3 record4: 527, name4, score4 record5: -1158, name5, score5 record6: 1860, name6, score6 record7: -1783, name7, score7 record8: -2014, name8, score8 record9: 1979, name9, score9 record10: 1892, name10, score10. • and print the BST in in-order to confirm 10 records are inserted correctly. • Search a student record (giving a student id), and print the student record if the id is found. • Delete a student record (giving a student id), and print the BST in in-order to confirm that the recordis deleted.

12. Chapter 2Fundamentals of the Analysis of Algorithm • Implementation and Empirical Analysis • Challenge in empirical analysis: • Develop a correct and complete implementation. • Determine the nature of the input data and other factors influencing on the experiment. Typically, There are three choices: actual data, random data, or perverse data. • Compare implementations independent to programmers, machines, compilers, or other related systems. • Consider performance characteristics of algorithms, especially for those whose running time is big. Can we analyze algorithms that haven’t run yet?

13. Example 1 Describe an algorithm for finding an element x in a list of distinct elements

15. Linear Search and Binary Search written in C++ Linear search int lsearch(int a[], int v, int l, int r) { for (int i=l; i<=n; i++) if (v==a[i]) return i; return -1; } Binary search int bsearch(int a[], int v, int l, int n) { while (r>=l) {int m=(l+r)/2; if (v==a[m]) return m; if (v<a[m]) r=m-1; else l=m+1; } return -1; }

16. Complexity of Algorithms Assume that both algorithms A and B solve the problem P. Which one is better? • Time complexity: the time required to solve a problem of a specified size. • Space complexity: the computer memory required to solve a problem of a specified size. The time complexity is expressed in terms of the number of operations used by the algorithm. • Worst case analysis: the largest number of operations needed to solve the given problem using this algorithm. • Average case analysis: the average number of operations used to solve the problem over all inputs.

17. Example 2 Analyze the time complexities of linear search algorithm and binary search algorithm 1 2(n+1) n 2 3n+5

18. 1 1 1 1 2 2 Number of operations = 4 log n+4 2

19. Running time for a problem with size Orders of Growth Using silicon computer, no matter how fast CPU will be you can never solve the problem whose running time is exponential !!!

20. Definition 2.1 A function t(n) is said to be O(g(n)) if there exist some constant Asymptotic Notations: O-notation n

21. Example 5 List the following function in O-notation in increasing order: Example 3 Prove 2n+1=O(n)

22. Example 7 Analyze the time complexities of linear search algorithm and binary search algorithm asymptotically. Totally(addition): O(n) Asymptotic Analysis of algorithms (using O-notation)

23. Totally(comparison): O(log n)

24. Example 8 Analyze the time complexities of following algorithm asymptotically. Repeat n times Repeat n times Totally(addition): Matrix addition algorithm Procedure MatricAddition(A[0..n-1,0..n-1],B[0..n-1,0..n-1]) for i=0ton-1do forj=0ton-1do C[i,j] = A[i,j] + B[i,j]; returnC;

25. F(n) can be defined recursively as follows: recurrence Recursive Algorithms Example 9 Computing the factorial function F(n)=n!. Factorial Algorithm Procedure factorial(n) if n = 0 return 1 else return factorial(n-1) * n; Algorithm factorial calls itself in its body! Time complexity(multiplication):T(0) = 0 T(n) = T(n-1) +1 when n>0

26. Basic Recurrences Example 10 Solving the following recurrence T(0) = 1 T(n) = T(n-1) + 1 n>0 T(n) = T(n-1) + 1 = T(n-2) + 1 + 1 = T(n-2) + 2 = T(n-3) + 1 + 2 = T(n-3) + 3 … = T(n-i) + i … = T(n-n) + n = n

27. Example 12 Solve the recurrence