Introduction to Algorithms Rabie A Ramadan rabierabieramadan org

Introduction to Algorithms Rabie A. Ramadan rabie@rabieramadan. org http: //www. rabieramadan. org Some of the sides are exported from different sources to clarify the topic

About my self l Rabie A. Ramadan l My website and publications • http: //www. rabieramadan. org 2

Class Rules l Attendance is a vey important l Assignments must be delivered on time • All assignments are individual assignments unless it is clearly stated that you can work on groups. • Assignments or part of them that are copied (including the programming assignments) will be punished by -2 assignments. 3

Class Rules l • • Projects There will be a term project The details of the project will be announced on the class website http: //rabieramadan. org/classes/2012/algorithms/ • Only 4 or less students per group Please no exception at all. 4

Class Rules l You can bring anything to drink but NO FOOD PLEASE l When you come in , DO NOT knock on the door as well as when you leave l I do not take attendance every class but if you miss one , it might greatly affect your grade

Text Book • Introduction to the Design and Analysis of Algorithms, 2 nd edition, 2007 by A. Levitin. • Other sources • • • Books Research papers … 6

Introduction 7

What is an algorithm? l An algorithm is a sequence of unambiguous instructions for solving a problem, i. e. , for obtaining a required output for any legitimate input in a finite amount of time. problem algorithm input “computer” output 8

What is an algorithm? l Procedure for getting answers to a specific problem l Problem solving strategy even if computers are not involved l Scope is limited I can not have a procedure for you to have very happy life or becoming rich and famous 9

Notion/Concept of algorithm and problem l l l Nonambiguity for each step Range of inputs is specified carefully The same algorithm can be represented in several ways Several algorithms for the same problem may exist Several algorithms with different ideas can solve the same problem with different speed. problem algorithm input “computer” output 10

Example of computational problem: sorting l Statement of problem: • Input: A sequence of n numbers <a 1, a 2, …, an> • Output: A reordering of the input sequence <a´ 1, a´ 2, …, a´n> so that a´i ≤ a´j whenever i < j l Instance: The sequence <5, 3, 2, 8, 3> l Algorithms: • • Selection sort Insertion sort Merge sort (many others) 1 -11

Some Well-known Computational Problems l l l l l Sorting Searching Shortest paths in a graph Minimum spanning tree Primality testing Traveling salesman problem Knapsack problem Chess Towers of Hanoi Some of problems don’t have efficient algorithms, or algorithms at all! 1 -12

Basic Issues Related to Algorithms l How to design algorithms l How to express algorithms l Proving correctness l Efficiency (or complexity) analysis l • Theoretical analysis • Empirical/ experimental analysis Optimality 1 -13

Algorithm design strategies b b Brute force • Try all possible combinations Divide and conquer breaking down a problem into two or more subproblems of the same (or related) type, until these become simple enough to be solved directly. b Decrease and conquer • Change an instance into one smaller instance • • b b b Transform and conquer • a simpler instance of the same problem, or • a different representation of the same problem, or • an instance of a different problem Greedy approach • The problem could be solved in iterations Dynamic programming • An instance is solved using the solutions for smaller • of the problem. Solve the smaller instance. Convert the solution of the smaller instance into a solution for the larger instance. • b instances. The solution for a smaller instance might be needed multiple times. The solutions to smaller instances are stored in a table, so that each smaller instance is solved only once. Additional space is used to save time. • Backtracking and branch-and-bound 1 -14

Analysis of Algorithms l How good is the algorithm? l Does there exist a better algorithm? • Correctness • Time efficiency • Space efficiency • Lower bounds • Optimality 1 -15

What is an algorithm? l 1. 2. 3. 4. 5. Recipe, process, method, technique, procedure, routine, … with the following requirements: Finiteness b terminates after a finite number of steps Definiteness b rigorously and unambiguously specified Clearly specified input b valid inputs are clearly specified Clearly specified/expected output b can be proved to produce the correct output given a valid input Effectiveness b steps are sufficiently simple and basic 1 -16

Algorithms Examples 1 -17

Addition and Subtraction 1 -18

2 - Digit Addition l Add the following (try it from left to right): right) l Simply , think of 32 add ( 30+2) ; then add it to 47 ; then add 2 to the results 1 -19

2 - Digit Addition (cont. ) l Add the following (try it from left to right): 1 -20

2 - Digit Addition (cont. ) l Try this on your own: 1 -21

3 - Digit Addition Add the following : Simply : Add the hundreds digit of the second number to the first number Add the tens digit to the result Add the unit digit to the result 1 -22

3 - Digit Addition Try this by yourself : 1 -23

3 - Digit Addition l Try this one : l Try it this way (subtraction instead of addition) 1 -24

Part of your first assignment l Find and write the best algorithm to sum up the numbers from 1 to 100 ? What is the complexity of your algorithm ? l Write an algorithm to add 3 -digit numbers (take the sign in your account- Allow the addition of positive or negative numbers)? Compare your algorithm to the computer does in similar cases? 1 -25

2 -Digit Subtraction 1 -26

3 -Digit Subtraction 1 -27

More about numbers Euclid’s Algorithm 1 -28

Euclid’s Algorithm Problem: Find gcd(m, n), the greatest common divisor of two nonnegative, not both zero integers m and n Examples: gcd(60, 24) = 12, gcd(60, 0) = 60, gcd(0, 0) = ? Euclid’s algorithm is based on repeated application of equality gcd(m, n) = gcd(n, m mod n) until the second number becomes 0, which makes the problem trivial. Example: gcd(60, 24) = gcd(24, 12) = gcd(12, 0) = 12 1 -29

Descriptions of Euclid’s algorithm l l l Step 1 If n = 0, return m and stop; otherwise go to Step 2 Divide m by n and assign the value of the remainder to r Step 3 Assign the value of n to m and the value of r to n. Go to Step 1. while n ≠ 0 do r ← m mod n m← n n←r return m 30

Other methods for gcd(m , n) [cont. ] l l Middle-school procedure Step 1 Find the prime factorization of m Step 2 Find the prime factorization of n Step 3 Find all the common prime factors Step 4 Compute the product of all the common prime factors and return it as gcd(m, n) E. g. find gcd(60, 24) Step 1: 60 = 2. 2. 3. 5 Step 2: 25 = 2. 2. 2. 3 Step 3: 2. 2. 3 Step 4: 2. 2. 3 = 12 Is this an algorithm? not qualified how do you get the prime factorization? How do you get common elements in two stored lists? How efficient is it? Time complexity: O(sqrt(n)) 31

More about numbers Sieve of Eratosthenes 1 -32

Sieve of Eratosthenes l To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: • • • Create a list of consecutive integers from two to n: (2, 3, 4, . . . , n). Initially, let p equal 2, the first prime number. Strike from the list all multiples of p less than or equal to n. (2 p, 3 p, 4 p, etc. ) Find the first number remaining on the list greater than p (this number is the next prime); replace p with this number. Repeat steps 3 and 4 until p 2 is greater than n. All the remaining numbers on the list are prime. 33

Sieve of Eratosthenes Input: Integer n ≥ 2 Output: List of primes less than or equal to n for p ← 2 to n do A[p] ← p for p ← 2 to n do if A[p] 0 //p hasn’t been previously eliminated from the list j ← p* p while j ≤ n do A[j] ← 0 //mark element as eliminated j←j+p Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 -34

Steps of Designing and Analyzing an Algorithm l l l Understand the Problem Decide on the machine to be used (sequential vs. parallel) Design an algorithm: select a specific techniques such as divide and conquer. Also, decide on how do you represent the algorithm for instance pseudocode , flowchart , etc. . Prove correctness may be by example or mathematically Analyze the algorithm for instance in space and time efficiency 35

Important Problem Types l Sorting l Searching l String processing l Graph problems l Combinatorial problems l Geometric problems l Numerical problems 1 -36

Sorting (I) l l l Rearrange the items of a given list in ascending order. • • Input: A sequence of n numbers <a 1, a 2, …, an> Output: A reordering <a´ 1, a´ 2, …, a´n> of the input sequence such that a´ 1≤ a´ 2 ≤ … ≤ a´n. Why sorting? • • Help searching Algorithms often use sorting as a key subroutine. Sorting key • A specially chosen piece of information used to guide sorting. E. g. , sort student records by names. 1 -37

Sorting (II) l Examples of sorting algorithms • Selection sort • Bubble sort • Insertion sort • Merge sort • …. . l Evaluate sorting algorithm complexity: the number of key comparisons. Two properties • Stability: A sorting algorithm is called stable if it preserves the relative order of any two equal elements in its input. • In place : A sorting algorithm is in place if it does not require extra memory, except, possibly for a few memory units. l 1 -38

Selection Sort Algorithm Selection. Sort(A[0. . n-1]) //The algorithm sorts a given array by selection sort //Input: An array A[0. . n-1] of orderable elements //Output: Array A[0. . n-1] sorted in ascending order for i 0 to n – 2 do min i for j i + 1 to n – 1 do if A[j] < A[min] min j swap A[i] and A[min] 1 -39

Searching l l Find a given value, called a search key, in a given set. Examples of searching algorithms • Sequential search • Binary search … Input: sorted array a_i < … < a_j and key x; m (i+j)/2; while i < j and x != a_m do if x < a_m then j m-1 else i m+1; if x = a_m then output a_m; Time: O(log n) 1 -40

String Processing l l l A string is a sequence of characters from an alphabet. Text strings: letters, numbers, and special characters. String matching: searching for a given word/pattern in a text. Examples: (i) searching for a word or phrase on WWW or in a Word document (ii) searching for a short read in the reference genomic sequence 1 -41

String Matching l l Given a text string T of length n and a pattern string P of length m, the exact string matching problem is to find all occurrences of P in T. Example: T=“AGCTTGA” P=“GCT” Applications: • Searching keywords in a file • Searching engines (like Google and Openfind) • Database searching (Gen. Bank) More string matching algorithms (with source codes): http: //www-igm. univ-mlv. fr/~lecroq/string/ 1 -42

String Matching l Brute Force algorithm • The brute force algorithm consists in checking, at all positions in the text between 0 and n-m, whether an occurrence of the pattern starts there or not. Then, after each attempt, it shifts the pattern by exactly one position to the right. Time: O(mn) where m=|P| and n=|T|. 1 -43

Main Features l l l No preprocessing phase; Constant extra space needed; Always shifts the window by exactly 1 position to the right; Comparisons can be done in any order; Searching phase in O(mn) time complexity; 1 -44

Your Second part of assignment • Report three different methods for string matching and write a program to test them? 1 -45

Graph Problems l l l Informal definition • A graph is a collection of points called vertices, some of which are connected by line segments called edges. Modeling Real-life Problems • Modeling WWW • Communication networks • Project scheduling … Examples of Graph Algorithms • Graph traversal algorithms • Shortest-path algorithms • ……. . 1 -46

Fundamental data structures l list • array • linked list • string l stack l queue l priority queue b graph b tree and binary tree 1 -47

Linear Data Structures n l Arrays • A sequence of n items of the same data type that are stored contiguously in computer memory and made accessible by specifying a value of the array’s index. l Linked List • A sequence of zero or more nodes each n containing two kinds of information: some data and one or more links called pointers to other nodes of the linked list. • Singly linked list (next pointer) • Doubly linked list (next + previous pointers) … a 1 a 2 Arrays n fixed length (need preliminary reservation of memory) n contiguous memory locations n direct access n Insert/delete Linked Lists n dynamic length n arbitrary memory locations n access by following links n Insert/delete an . 1 -48

Stacks and Queues l Stacks • A stack of plates • insertion/deletion can be done only at the top. • LIFO • Two operations (push and pop) l Queues • A queue of customers waiting for services • Insertion/enqueue • • from the rear and deletion/dequeue from the front. FIFO Two operations (enqueue and dequeue) 1 -49

Priority Queue and Heap n Priority queues (implemented using heaps) A data structure for maintaining a set of elements, each associated with a key/priority, with the following operations: n n n Finding the element with the highest priority n Deleting the element with the highest priority n Inserting a new element Scheduling jobs on a shared computer 1 -50

Graphs l Formal definition l A graph G = <V, E> is defined by a pair of two sets: a finite set V of items called vertices and a set E of vertex pairs called edges. Undirected and directed graphs (digraphs). l • Complete, dense, and sparse graphs • A graph with every pair of its vertices connected by an edge is called complete, K|V| • In Dense graph, the number of edges is close to the maximal number of edges. Complete 1 2 3 4 Dense 1 -51

Graph Representation l Adjacency matrix • • • n x n boolean matrix if |V| is n. The element on the ith row and jth column is 1 if there’s an edge from ith vertex to the jth vertex; otherwise 0. The adjacency matrix of an undirected graph is symmetric. l Adjacency linked lists l A collection of linked lists, one for each vertex, that contain all the vertices adjacent to the list’s vertex. Which data structure would you use if the graph is a 100 -node star shape? • 0 1 1 1 0 0 0 1 0 0 2 4 4 3 4 1 -52

Weighted Graphs • Graphs or digraphs with numbers assigned to the edges. 1 6 5 2 9 3 4 8 7 1 -53

Graph Properties -- Paths and Connectivity l Paths • A path from vertex u to v of a graph G is defined as a sequence of adjacent (connected by an edge) vertices that starts with u and ends with v. • Simple paths: a path in a graph which does not have repeating vertices. . • Path lengths: the number of edges, or the number of vertices – 1. l Connected graphs • A graph is said to be connected if for every pair of its vertices u and v there is a path from u to v. Connected component • The maximum connected subgraph of a given graph. l 1 -54

Graph Properties – A cyclicity l Cycle • A simple path of a positive length that starts and ends a the same vertex. l Acyclic graph • A graph without cycles • DAG (Directed Acyclic Graph) 1 2 3 4 1 -55

Trees l l Trees rooted • A tree (or free tree) is a connected 3 acyclic graph. • Forest: a graph that has no cycles but 4 1 5 is not necessarily connected. Properties of trees 2 • For every two vertices in a tree there always exists exactly one simple path from one of these vertices to the other. • Rooted trees: The above property makes it possible to select an arbitrary vertex in a free tree and consider it as the root of the so called rooted tree. • Levels in a rooted tree. 1 3 2 4 Forest 1 -56 5

Rooted Trees (I) l l l Ancestors/predecessors • For any vertex v in a tree T, all the vertices on the simple path from the root to that vertex are called ancestors. Descendants/children • All the vertices for which a vertex v is an ancestor are said to be 4 descendants of v. Parent, child and siblings • If (u, v) is the last edge of the simple path from the root to vertex v, u is said to be the parent of v and v is called a child of u. • Vertices that have the same parent are called siblings. Leaves • A vertex without children is called a leaf. Subtree • A vertex v with all its descendants is called the subtree of T rooted at v. rooted 3 1 2 1 -57 5

Rooted Trees (II) l Depth of a vertex • The length of the simple path from the root to the vertex. l Height of a tree • The length of the longest simple path from the root to a leaf. h = 2 3 4 1 5 2 1 -58

Ordered Trees l l l Ordered trees • An ordered tree is a rooted tree in which all the children of each vertex are ordered. Binary trees • A binary tree is an ordered tree in which every vertex has no more than two children and each children is designated either a left child or a right child of its parent. Binary search trees • Each vertex is assigned a number. • A number assigned to each parental vertex is larger than all the numbers in its left subtree and smaller than all the numbers in its right subtree. 9 6 5 8 2 3 6 3 2 9 5 1 -59 8