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DAA Introduction

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Unit - 1 Unit - 1 Design and Analysis of Algorithm Design and Analysis of Algorithm INTRODUCTION INTRODUCTION Dr. K. RANJITH SINGH Dr. K. RANJITH SINGH Assistant Professor Assistant Professor Department of Computer Science Department of Computer Science Karpagam Academy of Higher Education Karpagam Academy of Higher Education (Deemed to be University) (Deemed to be University) Coimbatore Coimbatore
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-2 What is an algorithm? What is an algorithm? An An algorithm algorithm is a sequence of unambiguous instructions is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required for solving a problem, i.e., for obtaining a required output for any output for any legitimate legitimate input in a finite amount of input in a finite amount of time. time. “computer” problem algorithm input output
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-3 Algorithm Algorithm  An An algorithm algorithm is a sequence of unambiguous is a sequence of unambiguous instructions for solving a problem, i.e., for instructions for solving a problem, i.e., for obtaining a required output for any legitimate obtaining a required output for any legitimate input in a finite amount of time. input in a finite amount of time. • Can be represented various forms • Unambiguity/clearness • Effectiveness • Finiteness/termination • Correctness
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-4 Historical Perspective Historical Perspective  Euclid’s algorithm for finding the greatest common divisor Euclid’s algorithm for finding the greatest common divisor  Muhammad ibn Musa al-Khwarizmi – 9 Muhammad ibn Musa al-Khwarizmi – 9th th century century mathematician mathematician www.lib.virginia.edu/science/parshall/khwariz.html www.lib.virginia.edu/science/parshall/khwariz.html
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-5 Notion of algorithm and problem Notion of algorithm and problem “computer” algorithmic solution (different from a conventional solution) problem algorithm input (or instance) output
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-6 Example of computational problem: sorting Example of computational problem: sorting  Statement of problem: Statement of problem: • Input: Input: A sequence of A sequence of n n numbers <a numbers <a1 1, , a a2 2, …, a , …, an n> > • Output: Output: A reordering of the input sequence <a A reordering of the input sequence <a´ ´ 1 1, , a a´ ´ 2 2, …, a , …, a´ ´ n n> so that > so that a a´ ´ i i ≤ ≤ a a´ ´ j j whenever whenever i i < < j j  Instance: The sequence <5, 3, 2, 8, 3> Instance: The sequence <5, 3, 2, 8, 3>  Algorithms: Algorithms: • Selection sort Selection sort • Insertion sort Insertion sort • Merge sort Merge sort • (many others) (many others)
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-7 Selection Sort Selection Sort  Input: array Input: array a[1],…,a[n] a[1],…,a[n]  Output: array Output: array a a sorted in non-decreasing order sorted in non-decreasing order  Algorithm: Algorithm: for i=1 to n swap a[i] with smallest of a[i],…,a[n] • Is this unambiguous? Effective? • See also pseudocode, section 3.1
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-8 Some Well-known Computational Problems Some Well-known Computational Problems  Sorting Sorting  Searching Searching  Shortest paths in a graph Shortest paths in a graph  Minimum spanning tree Minimum spanning tree  Primality testing Primality testing  Traveling salesman problem Traveling salesman problem  Knapsack problem Knapsack problem  Chess Chess  Towers of Hanoi Towers of Hanoi  Program termination Program termination Some of these problems don’t have efficient algorithms, or algorithms at all!
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-9 Basic Issues Related to Algorithms Basic Issues Related to Algorithms  How to design algorithms How to design algorithms  How to express algorithms How to express algorithms  Proving correctness Proving correctness  Efficiency (or complexity) analysis Efficiency (or complexity) analysis • Theoretical analysis Theoretical analysis • Empirical analysis Empirical analysis  Optimality Optimality
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-10 Algorithm design strategies Algorithm design strategies  Brute force Brute force  Divide and conquer Divide and conquer  Decrease and conquer Decrease and conquer  Transform and conquer Transform and conquer  Greedy approach Greedy approach  Dynamic programming Dynamic programming  Backtracking and branch-and-bound Backtracking and branch-and-bound  Space and time tradeoffs Space and time tradeoffs
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-11 Analysis of Algorithms Analysis of Algorithms  How good is the algorithm? How good is the algorithm? • Correctness Correctness • Time efficiency Time efficiency • Space efficiency Space efficiency  Does there exist a better algorithm? Does there exist a better algorithm? • Lower bounds Lower bounds • Optimality Optimality
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-12 What is an algorithm? What is an algorithm?  Recipe, process, method, technique, procedure, routine,… Recipe, process, method, technique, procedure, routine,… with the following requirements: with the following requirements: 1. 1. Finiteness Finiteness  terminates after a finite number of steps terminates after a finite number of steps 2. 2. Definiteness Definiteness  rigorously and unambiguously specified rigorously and unambiguously specified 3. 3. Clearly specified input Clearly specified input  valid inputs are clearly specified valid inputs are clearly specified 4. 4. Clearly specified/expected output Clearly specified/expected output  can be proved to produce the correct output given a valid input can be proved to produce the correct output given a valid input 5. 5. Effectiveness Effectiveness  steps are sufficiently simple and basic steps are sufficiently simple and basic
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-13 Why study algorithms? Why study algorithms?  Theoretical importance Theoretical importance • the core of computer science the core of computer science  Practical importance Practical importance • A practitioner’s toolkit of known algorithms A practitioner’s toolkit of known algorithms • Framework for designing and analyzing algorithms for new Framework for designing and analyzing algorithms for new problems problems Example: Google’s PageRank Technology
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-14 Euclid’s Algorithm Euclid’s Algorithm Problem: Find gcd( Problem: Find gcd(m,n m,n), the greatest common divisor of two ), the greatest common divisor of two nonnegative, not both zero integers nonnegative, not both zero integers m m and and n n Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Euclid’s algorithm is based on repeated application of equality Euclid’s algorithm is based on repeated application of equality gcd( gcd(m,n m,n) = gcd( ) = gcd(n, m n, m mod mod n n) ) until the second number becomes 0, which makes the problem until the second number becomes 0, which makes the problem trivial. trivial. Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12 Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-15 Two descriptions of Euclid’s algorithm Two descriptions of Euclid’s algorithm Step 1 If Step 1 If n n = 0, return = 0, return m m and stop; otherwise go to Step 2 and stop; otherwise go to Step 2 Step 2 Step 2 Divide Divide m m by by n n and assign the value of the remainder to and assign the value of the remainder to r r Step 3 Assign the value of Step 3 Assign the value of n n to to m m and the value of and the value of r r to to n. n. Go to Go to Step 1. Step 1. while while n n ≠ 0 ≠ 0 do do r ← m r ← m mod mod n n m← n m← n n ← r n ← r return return m m
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-16 Other methods for computing gcd( Other methods for computing gcd(m,n m,n) ) Consecutive integer checking algorithm Consecutive integer checking algorithm Step 1 Assign the value of min{ Step 1 Assign the value of min{m,n m,n} to } to t t Step 2 Step 2 Divide Divide m m by by t. t. If the remainder is 0, go to Step 3; If the remainder is 0, go to Step 3; otherwise, go to Step 4 otherwise, go to Step 4 Step 3 Step 3 Divide Divide n n by by t. t. If the remainder is 0, return If the remainder is 0, return t t and stop; and stop; otherwise, go to Step 4 otherwise, go to Step 4 Step 4 Decrease Step 4 Decrease t t by 1 and go to Step 2 by 1 and go to Step 2 Is this slower than Euclid’s algorithm? How much slower? O(n), if n <= m , vs O(log n)
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-17 Other methods for gcd( Other methods for gcd(m,n m,n) [cont.] ) [cont.] Middle-school procedure Middle-school procedure Step 1 Find the prime factorization of Step 1 Find the prime factorization of m m Step 2 Find the prime factorization of Step 2 Find the prime factorization of n n Step 3 Find all the common prime factors Step 3 Find all the common prime factors Step 4 Compute the product of all the common prime factors Step 4 Compute the product of all the common prime factors and return it as gcd and return it as gcd(m,n (m,n) ) Is this an algorithm? Is this an algorithm? How efficient is it? How efficient is it? Time complexity: O(sqrt(n))
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-18 Sieve of Eratosthenes Sieve of Eratosthenes Input: Input: Integer Integer n n ≥ ≥ 2 2 Output: List of primes less than or equal to Output: List of primes less than or equal to n n for for p p ← 2 ← 2 to to n n do do A A[ [p p] ← ] ← p p for for p p ← 2 ← 2 to to n n do do if if A A[ [p p] ]   0 // 0 //p p hasn’t been previously eliminated from the list hasn’t been previously eliminated from the list j j ← ← p p* * p p while while j j ≤ ≤ n n do do A A[ [j j] ] ← 0 ← 0 //mark element as eliminated //mark element as eliminated j j ← ← j j + p + p Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time complexity: O(n)
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-19 Two main issues related to algorithms Two main issues related to algorithms  How to design algorithms How to design algorithms  How to analyze algorithm efficiency How to analyze algorithm efficiency
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-20 Algorithm design techniques/strategies Algorithm design techniques/strategies  Brute force Brute force  Divide and conquer Divide and conquer  Decrease and conquer Decrease and conquer  Transform and conquer Transform and conquer  Space and time tradeoffs Space and time tradeoffs  Greedy approach Greedy approach  Dynamic programming Dynamic programming  Iterative improvement Iterative improvement  Backtracking Backtracking  Branch and bound Branch and bound
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-21 Analysis of algorithms Analysis of algorithms  How good is the algorithm? How good is the algorithm? • time efficiency time efficiency • space efficiency space efficiency • correctness ignored in this course correctness ignored in this course  Does there exist a better algorithm? Does there exist a better algorithm? • lower bounds lower bounds • optimality optimality
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-22 Important problem types Important problem types  sorting sorting  searching searching  string processing string processing  graph problems graph problems  combinatorial problems combinatorial problems  geometric problems geometric problems  numerical problems numerical problems
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-23 Sorting (I) Sorting (I)  Rearrange the items of a given list in ascending order. Rearrange the items of a given list in ascending order. • Input: A sequence of n numbers <a Input: A sequence of n numbers <a1 1, , a a2 2, …, a , …, an n> > • Output: A reordering <a Output: A reordering <a´ ´ 1 1, , a a´ ´ 2 2, …, a , …, a´ ´ n n> of the input sequence such that a > of the input sequence such that a ´ ´ 1 1≤ ≤ a a´ ´ 2 2 ≤ ≤ … … ≤ ≤ a a´ ´ n. n.  Why sorting? Why sorting? • Help searching Help searching • Algorithms often use sorting as a key subroutine. Algorithms often use sorting as a key subroutine.  Sorting key Sorting key • A specially chosen piece of information used to guide sorting. E.g., sort A specially chosen piece of information used to guide sorting. E.g., sort student records by names. student records by names.
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-24 Sorting (II) Sorting (II)  Examples of sorting algorithms Examples of sorting algorithms • Selection sort Selection sort • Bubble sort Bubble sort • Insertion sort Insertion sort • Merge sort Merge sort • Heap sort … Heap sort …  Evaluate sorting algorithm complexity: the number of key comparisons. Evaluate sorting algorithm complexity: the number of key comparisons.  Two properties Two properties • Stability Stability: A sorting algorithm is called stable if it preserves the relative order of : A sorting algorithm is called stable if it preserves the relative order of any two equal elements in its input. any two equal elements in its input. • In place In place : A sorting algorithm is in place if it does not require extra memory, : A sorting algorithm is in place if it does not require extra memory, except, possibly for a few memory units. except, possibly for a few memory units.
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-25 Selection Sort Selection Sort Algorithm Algorithm SelectionSort(A[0..n-1]) SelectionSort(A[0..n-1]) //The algorithm sorts a given array by selection sort //The algorithm sorts a given array by selection sort //Input: An array A[0..n-1] of orderable elements //Input: An array A[0..n-1] of orderable elements //Output: Array A[0..n-1] sorted in ascending order //Output: Array A[0..n-1] sorted in ascending order for i for i   0 to n – 2 do 0 to n – 2 do min min   i i for j for j   i + 1 to n – 1 do i + 1 to n – 1 do if A[j] < A[min] if A[j] < A[min] min min   j j swap A[i] and A[min] swap A[i] and A[min]
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-26 Searching Searching  Find a given value, called a Find a given value, called a search key search key, in a given set. , in a given set.  Examples of searching algorithms Examples of searching algorithms • Sequential search Sequential search • Binary 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)
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-27 String Processing String Processing  A string is a sequence of characters from an alphabet. A string is a sequence of characters from an alphabet.  Text strings: letters, numbers, and special characters. Text strings: letters, numbers, and special characters.  String matching: searching for a given word/pattern in a String matching: searching for a given word/pattern in a text. 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
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-28 Graph Problems Graph Problems  Informal definition Informal definition • A graph is a collection of points called A graph is a collection of points called vertices vertices, some of , some of which are connected by line segments called which are connected by line segments called edges edges. .  Modeling real-life problems Modeling real-life problems • Modeling WWW Modeling WWW • Communication networks Communication networks • Project scheduling … Project scheduling …  Examples of graph algorithms Examples of graph algorithms • Graph traversal algorithms Graph traversal algorithms • Shortest-path algorithms Shortest-path algorithms • Topological sorting Topological sorting
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-29 Fundamental data structures Fundamental data structures  list list • array array • linked list linked list • string string  stack stack  queue queue  priority queue/heap priority queue/heap  graph graph  tree and binary tree tree and binary tree  set and dictionary set and dictionary
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-30 Linear Data Structures Linear Data Structures  Arrays Arrays • A sequence of n items of the same A sequence of n items of the same data type that are stored data type that are stored contiguously in computer memory contiguously in computer memory and made accessible by specifying a and made accessible by specifying a value of the array’s index. value of the array’s index.  Linked List Linked List • A sequence of zero or more nodes A sequence of zero or more nodes each containing two kinds of each containing two kinds of information: some data and one or information: some data and one or more links called pointers to other more links called pointers to other nodes of the linked list. nodes of the linked list. • Singly linked list (next pointer) Singly linked list (next pointer) • Doubly linked list (next + previous Doubly linked list (next + previous pointers) pointers)  Arrays  fixed length (need preliminary reservation of memory)  contiguous memory locations  direct access  Insert/delete  Linked Lists  dynamic length  arbitrary memory locations  access by following links  Insert/delete … a1 an a2 .
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-31 Stacks and Queues Stacks and Queues  Stacks Stacks • A stack of plates A stack of plates – insertion/deletion can be done only at the top. insertion/deletion can be done only at the top. – LIFO LIFO • Two operations (push and pop) Two operations (push and pop)  Queues Queues • A queue of customers waiting for services A queue of customers waiting for services – Insertion/enqueue from the rear and deletion/dequeue from Insertion/enqueue from the rear and deletion/dequeue from the front. the front. – FIFO FIFO • Two operations (enqueue and dequeue) Two operations (enqueue and dequeue)
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-32 Priority Queue and Heap Priority Queue and Heap  Priority queues (implemented using heaps)  A data structure for maintaining a set of elements, each associated with a key/priority, with the following operations  Finding the element with the highest priority  Deleting the element with the highest priority  Inserting a new element  Scheduling jobs on a shared computer 9 6 8 5 2 3 9 6 5 8 2 3
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-33 Graphs Graphs  Formal definition Formal definition • A graph A graph G = <V, E> G = <V, E> is defined by a pair of two sets: a is defined by a pair of two sets: a finite set V of items called finite set V of items called vertices vertices and a set E of vertex and a set E of vertex pairs called pairs called edges edges. .  Undirected Undirected and and directed directed graphs ( graphs (digraphs digraphs). ).  What’s the maximum number of edges in an undirected What’s the maximum number of edges in an undirected graph with |V| vertices? graph with |V| vertices?  Complete, dense, Complete, dense, and and sparse sparse graphs graphs • A graph with every pair of its vertices connected by an A graph with every pair of its vertices connected by an edge is called complete, K edge is called complete, K|V| |V| 1 2 3 4
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-34 Graph Representation Graph Representation  Adjacency matrix Adjacency matrix • n x n boolean matrix if |V| is n. n x n boolean matrix if |V| is n. • The element on the ith row and jth column is 1 if there’s an 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. edge from ith vertex to the jth vertex; otherwise 0. • The adjacency matrix of an undirected graph is symmetric. The adjacency matrix of an undirected graph is symmetric.  Adjacency linked lists Adjacency linked lists • A collection of linked lists, one for each vertex, that contain all A collection of linked lists, one for each vertex, that contain all the vertices adjacent to the list’s vertex. the vertices adjacent to the list’s vertex.  Which data structure would you use if the graph is a 100-node Which data structure would you use if the graph is a 100-node star shape? star shape? 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 2 3 4 4 4
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-35 Weighted Graphs Weighted Graphs  Weighted graphs Weighted graphs • Graphs or digraphs with numbers assigned to the edges. Graphs or digraphs with numbers assigned to the edges. 1 2 3 4 6 8 5 7 9
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-36 Graph Properties -- Paths and Connectivity Graph Properties -- Paths and Connectivity  Paths Paths • A path from vertex u to v of a graph G is defined as a sequence of 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 adjacent (connected by an edge) vertices that starts with u and ends with v. with v. • Simple paths Simple paths: All edges of a path are distinct. : All edges of a path are distinct. • Path lengths: the number of edges, or the number of vertices – 1. Path lengths: the number of edges, or the number of vertices – 1.  Connected graphs Connected graphs • A graph is said to be connected if for every pair of its vertices u and 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. v there is a path from u to v.  Connected component Connected component • The maximum connected subgraph of a given graph. The maximum connected subgraph of a given graph.
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-37 Graph Properties -- Acyclicity Graph Properties -- Acyclicity  Cycle Cycle • A simple path of a positive length that starts and A simple path of a positive length that starts and ends a the same vertex. ends a the same vertex.  Acyclic graph Acyclic graph • A graph without cycles A graph without cycles • DAG DAG (Directed Acyclic Graph) (Directed Acyclic Graph) 1 2 3 4
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-38 Trees Trees  Trees Trees • A tree (or A tree (or free tree free tree) is a connected acyclic graph. ) is a connected acyclic graph. • Forest: a graph that has no cycles but is not necessarily connected. Forest: a graph that has no cycles but is not necessarily connected.  Properties of trees Properties of trees • For every two vertices in a tree there always exists exactly one For every two vertices in a tree there always exists exactly one simple path from one of these vertices to the other. simple path from one of these vertices to the other. Why? Why? – Rooted trees Rooted trees: : The above property makes it possible to select an The above property makes it possible to select an arbitrary vertex in a free tree and consider it as the root of the arbitrary vertex in a free tree and consider it as the root of the so called rooted tree. so called rooted tree. – Levels in a rooted tree. Levels in a rooted tree.  |E| = |V| - 1 1 3 2 4 5 1 3 2 4 5 rooted
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-39 Rooted Trees (I) Rooted Trees (I)  Ancestors Ancestors • For any vertex For any vertex v v in a tree in a tree T T, all the vertices on the simple path , all the vertices on the simple path from the root to that vertex are called ancestors. from the root to that vertex are called ancestors.  Descendants Descendants • All the vertices for which a vertex All the vertices for which a vertex v v is an ancestor are said to is an ancestor are said to be descendants of be descendants of v v. .  Parent, child Parent, child and and siblings siblings • If If (u, v) (u, v) is the last edge of the simple path from the root to is the last edge of the simple path from the root to vertex vertex v v, , u u is said to be the parent of is said to be the parent of v v and and v v is called a child is called a child of of u u. . • Vertices that have the same parent are called siblings. Vertices that have the same parent are called siblings.  Leaves Leaves • A vertex without children is called a leaf. A vertex without children is called a leaf.  Subtree Subtree • A vertex A vertex v v with all its descendants is called the subtree of with all its descendants is called the subtree of T T rooted at rooted at v v. .
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-40 Rooted Trees (II) Rooted Trees (II)  Depth Depth of a vertex of a vertex • The length of the simple path from the root to the vertex. The length of the simple path from the root to the vertex.  Height Height of a tree of a tree • The length of the longest simple path from the root to a leaf. The length of the longest simple path from the root to a leaf. 1 3 2 4 5 h = 2
Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-41 Ordered Trees Ordered Trees  Ordered trees Ordered trees • An ordered tree is a rooted tree in which all the children of each An ordered tree is a rooted tree in which all the children of each vertex are ordered. vertex are ordered.  Binary trees Binary trees • A binary tree is an ordered tree in which every vertex has no more A binary tree is an ordered tree in which every vertex has no more than two children and each children is designated s either a left child than two children and each children is designated s either a left child or a right child of its parent. or a right child of its parent.  Binary search trees Binary search trees • Each vertex is assigned a number. Each vertex is assigned a number. • A number assigned to each parental vertex is larger than all the 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 numbers in its left subtree and smaller than all the numbers in its right subtree. right subtree.   log log2 2n n    h h   n – 1 n – 1, where h is the height of a binary tree and n the size. , where h is the height of a binary tree and n the size. 9 6 8 5 2 3 6 3 9 2 5 8

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DAA Introduction - in Chapter - I and lecture

  • 1.
    Unit - 1 Unit- 1 Design and Analysis of Algorithm Design and Analysis of Algorithm INTRODUCTION INTRODUCTION Dr. K. RANJITH SINGH Dr. K. RANJITH SINGH Assistant Professor Assistant Professor Department of Computer Science Department of Computer Science Karpagam Academy of Higher Education Karpagam Academy of Higher Education (Deemed to be University) (Deemed to be University) Coimbatore Coimbatore
  • 2.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-2 What is an algorithm? What is an algorithm? An An algorithm algorithm is a sequence of unambiguous instructions is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required for solving a problem, i.e., for obtaining a required output for any output for any legitimate legitimate input in a finite amount of input in a finite amount of time. time. “computer” problem algorithm input output
  • 3.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-3 Algorithm Algorithm  An An algorithm algorithm is a sequence of unambiguous is a sequence of unambiguous instructions for solving a problem, i.e., for instructions for solving a problem, i.e., for obtaining a required output for any legitimate obtaining a required output for any legitimate input in a finite amount of time. input in a finite amount of time. • Can be represented various forms • Unambiguity/clearness • Effectiveness • Finiteness/termination • Correctness
  • 4.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-4 Historical Perspective Historical Perspective  Euclid’s algorithm for finding the greatest common divisor Euclid’s algorithm for finding the greatest common divisor  Muhammad ibn Musa al-Khwarizmi – 9 Muhammad ibn Musa al-Khwarizmi – 9th th century century mathematician mathematician www.lib.virginia.edu/science/parshall/khwariz.html www.lib.virginia.edu/science/parshall/khwariz.html
  • 5.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-5 Notion of algorithm and problem Notion of algorithm and problem “computer” algorithmic solution (different from a conventional solution) problem algorithm input (or instance) output
  • 6.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-6 Example of computational problem: sorting Example of computational problem: sorting  Statement of problem: Statement of problem: • Input: Input: A sequence of A sequence of n n numbers <a numbers <a1 1, , a a2 2, …, a , …, an n> > • Output: Output: A reordering of the input sequence <a A reordering of the input sequence <a´ ´ 1 1, , a a´ ´ 2 2, …, a , …, a´ ´ n n> so that > so that a a´ ´ i i ≤ ≤ a a´ ´ j j whenever whenever i i < < j j  Instance: The sequence <5, 3, 2, 8, 3> Instance: The sequence <5, 3, 2, 8, 3>  Algorithms: Algorithms: • Selection sort Selection sort • Insertion sort Insertion sort • Merge sort Merge sort • (many others) (many others)
  • 7.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-7 Selection Sort Selection Sort  Input: array Input: array a[1],…,a[n] a[1],…,a[n]  Output: array Output: array a a sorted in non-decreasing order sorted in non-decreasing order  Algorithm: Algorithm: for i=1 to n swap a[i] with smallest of a[i],…,a[n] • Is this unambiguous? Effective? • See also pseudocode, section 3.1
  • 8.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-8 Some Well-known Computational Problems Some Well-known Computational Problems  Sorting Sorting  Searching Searching  Shortest paths in a graph Shortest paths in a graph  Minimum spanning tree Minimum spanning tree  Primality testing Primality testing  Traveling salesman problem Traveling salesman problem  Knapsack problem Knapsack problem  Chess Chess  Towers of Hanoi Towers of Hanoi  Program termination Program termination Some of these problems don’t have efficient algorithms, or algorithms at all!
  • 9.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-9 Basic Issues Related to Algorithms Basic Issues Related to Algorithms  How to design algorithms How to design algorithms  How to express algorithms How to express algorithms  Proving correctness Proving correctness  Efficiency (or complexity) analysis Efficiency (or complexity) analysis • Theoretical analysis Theoretical analysis • Empirical analysis Empirical analysis  Optimality Optimality
  • 10.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-10 Algorithm design strategies Algorithm design strategies  Brute force Brute force  Divide and conquer Divide and conquer  Decrease and conquer Decrease and conquer  Transform and conquer Transform and conquer  Greedy approach Greedy approach  Dynamic programming Dynamic programming  Backtracking and branch-and-bound Backtracking and branch-and-bound  Space and time tradeoffs Space and time tradeoffs
  • 11.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-11 Analysis of Algorithms Analysis of Algorithms  How good is the algorithm? How good is the algorithm? • Correctness Correctness • Time efficiency Time efficiency • Space efficiency Space efficiency  Does there exist a better algorithm? Does there exist a better algorithm? • Lower bounds Lower bounds • Optimality Optimality
  • 12.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-12 What is an algorithm? What is an algorithm?  Recipe, process, method, technique, procedure, routine,… Recipe, process, method, technique, procedure, routine,… with the following requirements: with the following requirements: 1. 1. Finiteness Finiteness  terminates after a finite number of steps terminates after a finite number of steps 2. 2. Definiteness Definiteness  rigorously and unambiguously specified rigorously and unambiguously specified 3. 3. Clearly specified input Clearly specified input  valid inputs are clearly specified valid inputs are clearly specified 4. 4. Clearly specified/expected output Clearly specified/expected output  can be proved to produce the correct output given a valid input can be proved to produce the correct output given a valid input 5. 5. Effectiveness Effectiveness  steps are sufficiently simple and basic steps are sufficiently simple and basic
  • 13.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-13 Why study algorithms? Why study algorithms?  Theoretical importance Theoretical importance • the core of computer science the core of computer science  Practical importance Practical importance • A practitioner’s toolkit of known algorithms A practitioner’s toolkit of known algorithms • Framework for designing and analyzing algorithms for new Framework for designing and analyzing algorithms for new problems problems Example: Google’s PageRank Technology
  • 14.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-14 Euclid’s Algorithm Euclid’s Algorithm Problem: Find gcd( Problem: Find gcd(m,n m,n), the greatest common divisor of two ), the greatest common divisor of two nonnegative, not both zero integers nonnegative, not both zero integers m m and and n n Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Euclid’s algorithm is based on repeated application of equality Euclid’s algorithm is based on repeated application of equality gcd( gcd(m,n m,n) = gcd( ) = gcd(n, m n, m mod mod n n) ) until the second number becomes 0, which makes the problem until the second number becomes 0, which makes the problem trivial. trivial. Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12 Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12
  • 15.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-15 Two descriptions of Euclid’s algorithm Two descriptions of Euclid’s algorithm Step 1 If Step 1 If n n = 0, return = 0, return m m and stop; otherwise go to Step 2 and stop; otherwise go to Step 2 Step 2 Step 2 Divide Divide m m by by n n and assign the value of the remainder to and assign the value of the remainder to r r Step 3 Assign the value of Step 3 Assign the value of n n to to m m and the value of and the value of r r to to n. n. Go to Go to Step 1. Step 1. while while n n ≠ 0 ≠ 0 do do r ← m r ← m mod mod n n m← n m← n n ← r n ← r return return m m
  • 16.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-16 Other methods for computing gcd( Other methods for computing gcd(m,n m,n) ) Consecutive integer checking algorithm Consecutive integer checking algorithm Step 1 Assign the value of min{ Step 1 Assign the value of min{m,n m,n} to } to t t Step 2 Step 2 Divide Divide m m by by t. t. If the remainder is 0, go to Step 3; If the remainder is 0, go to Step 3; otherwise, go to Step 4 otherwise, go to Step 4 Step 3 Step 3 Divide Divide n n by by t. t. If the remainder is 0, return If the remainder is 0, return t t and stop; and stop; otherwise, go to Step 4 otherwise, go to Step 4 Step 4 Decrease Step 4 Decrease t t by 1 and go to Step 2 by 1 and go to Step 2 Is this slower than Euclid’s algorithm? How much slower? O(n), if n <= m , vs O(log n)
  • 17.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-17 Other methods for gcd( Other methods for gcd(m,n m,n) [cont.] ) [cont.] Middle-school procedure Middle-school procedure Step 1 Find the prime factorization of Step 1 Find the prime factorization of m m Step 2 Find the prime factorization of Step 2 Find the prime factorization of n n Step 3 Find all the common prime factors Step 3 Find all the common prime factors Step 4 Compute the product of all the common prime factors Step 4 Compute the product of all the common prime factors and return it as gcd and return it as gcd(m,n (m,n) ) Is this an algorithm? Is this an algorithm? How efficient is it? How efficient is it? Time complexity: O(sqrt(n))
  • 18.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-18 Sieve of Eratosthenes Sieve of Eratosthenes Input: Input: Integer Integer n n ≥ ≥ 2 2 Output: List of primes less than or equal to Output: List of primes less than or equal to n n for for p p ← 2 ← 2 to to n n do do A A[ [p p] ← ] ← p p for for p p ← 2 ← 2 to to n n do do if if A A[ [p p] ]   0 // 0 //p p hasn’t been previously eliminated from the list hasn’t been previously eliminated from the list j j ← ← p p* * p p while while j j ≤ ≤ n n do do A A[ [j j] ] ← 0 ← 0 //mark element as eliminated //mark element as eliminated j j ← ← j j + p + p Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time complexity: O(n)
  • 19.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-19 Two main issues related to algorithms Two main issues related to algorithms  How to design algorithms How to design algorithms  How to analyze algorithm efficiency How to analyze algorithm efficiency
  • 20.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-20 Algorithm design techniques/strategies Algorithm design techniques/strategies  Brute force Brute force  Divide and conquer Divide and conquer  Decrease and conquer Decrease and conquer  Transform and conquer Transform and conquer  Space and time tradeoffs Space and time tradeoffs  Greedy approach Greedy approach  Dynamic programming Dynamic programming  Iterative improvement Iterative improvement  Backtracking Backtracking  Branch and bound Branch and bound
  • 21.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-21 Analysis of algorithms Analysis of algorithms  How good is the algorithm? How good is the algorithm? • time efficiency time efficiency • space efficiency space efficiency • correctness ignored in this course correctness ignored in this course  Does there exist a better algorithm? Does there exist a better algorithm? • lower bounds lower bounds • optimality optimality
  • 22.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-22 Important problem types Important problem types  sorting sorting  searching searching  string processing string processing  graph problems graph problems  combinatorial problems combinatorial problems  geometric problems geometric problems  numerical problems numerical problems
  • 23.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-23 Sorting (I) Sorting (I)  Rearrange the items of a given list in ascending order. Rearrange the items of a given list in ascending order. • Input: A sequence of n numbers <a Input: A sequence of n numbers <a1 1, , a a2 2, …, a , …, an n> > • Output: A reordering <a Output: A reordering <a´ ´ 1 1, , a a´ ´ 2 2, …, a , …, a´ ´ n n> of the input sequence such that a > of the input sequence such that a ´ ´ 1 1≤ ≤ a a´ ´ 2 2 ≤ ≤ … … ≤ ≤ a a´ ´ n. n.  Why sorting? Why sorting? • Help searching Help searching • Algorithms often use sorting as a key subroutine. Algorithms often use sorting as a key subroutine.  Sorting key Sorting key • A specially chosen piece of information used to guide sorting. E.g., sort A specially chosen piece of information used to guide sorting. E.g., sort student records by names. student records by names.
  • 24.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-24 Sorting (II) Sorting (II)  Examples of sorting algorithms Examples of sorting algorithms • Selection sort Selection sort • Bubble sort Bubble sort • Insertion sort Insertion sort • Merge sort Merge sort • Heap sort … Heap sort …  Evaluate sorting algorithm complexity: the number of key comparisons. Evaluate sorting algorithm complexity: the number of key comparisons.  Two properties Two properties • Stability Stability: A sorting algorithm is called stable if it preserves the relative order of : A sorting algorithm is called stable if it preserves the relative order of any two equal elements in its input. any two equal elements in its input. • In place In place : A sorting algorithm is in place if it does not require extra memory, : A sorting algorithm is in place if it does not require extra memory, except, possibly for a few memory units. except, possibly for a few memory units.
  • 25.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-25 Selection Sort Selection Sort Algorithm Algorithm SelectionSort(A[0..n-1]) SelectionSort(A[0..n-1]) //The algorithm sorts a given array by selection sort //The algorithm sorts a given array by selection sort //Input: An array A[0..n-1] of orderable elements //Input: An array A[0..n-1] of orderable elements //Output: Array A[0..n-1] sorted in ascending order //Output: Array A[0..n-1] sorted in ascending order for i for i   0 to n – 2 do 0 to n – 2 do min min   i i for j for j   i + 1 to n – 1 do i + 1 to n – 1 do if A[j] < A[min] if A[j] < A[min] min min   j j swap A[i] and A[min] swap A[i] and A[min]
  • 26.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-26 Searching Searching  Find a given value, called a Find a given value, called a search key search key, in a given set. , in a given set.  Examples of searching algorithms Examples of searching algorithms • Sequential search Sequential search • Binary 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)
  • 27.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-27 String Processing String Processing  A string is a sequence of characters from an alphabet. A string is a sequence of characters from an alphabet.  Text strings: letters, numbers, and special characters. Text strings: letters, numbers, and special characters.  String matching: searching for a given word/pattern in a String matching: searching for a given word/pattern in a text. 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
  • 28.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-28 Graph Problems Graph Problems  Informal definition Informal definition • A graph is a collection of points called A graph is a collection of points called vertices vertices, some of , some of which are connected by line segments called which are connected by line segments called edges edges. .  Modeling real-life problems Modeling real-life problems • Modeling WWW Modeling WWW • Communication networks Communication networks • Project scheduling … Project scheduling …  Examples of graph algorithms Examples of graph algorithms • Graph traversal algorithms Graph traversal algorithms • Shortest-path algorithms Shortest-path algorithms • Topological sorting Topological sorting
  • 29.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-29 Fundamental data structures Fundamental data structures  list list • array array • linked list linked list • string string  stack stack  queue queue  priority queue/heap priority queue/heap  graph graph  tree and binary tree tree and binary tree  set and dictionary set and dictionary
  • 30.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-30 Linear Data Structures Linear Data Structures  Arrays Arrays • A sequence of n items of the same A sequence of n items of the same data type that are stored data type that are stored contiguously in computer memory contiguously in computer memory and made accessible by specifying a and made accessible by specifying a value of the array’s index. value of the array’s index.  Linked List Linked List • A sequence of zero or more nodes A sequence of zero or more nodes each containing two kinds of each containing two kinds of information: some data and one or information: some data and one or more links called pointers to other more links called pointers to other nodes of the linked list. nodes of the linked list. • Singly linked list (next pointer) Singly linked list (next pointer) • Doubly linked list (next + previous Doubly linked list (next + previous pointers) pointers)  Arrays  fixed length (need preliminary reservation of memory)  contiguous memory locations  direct access  Insert/delete  Linked Lists  dynamic length  arbitrary memory locations  access by following links  Insert/delete … a1 an a2 .
  • 31.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-31 Stacks and Queues Stacks and Queues  Stacks Stacks • A stack of plates A stack of plates – insertion/deletion can be done only at the top. insertion/deletion can be done only at the top. – LIFO LIFO • Two operations (push and pop) Two operations (push and pop)  Queues Queues • A queue of customers waiting for services A queue of customers waiting for services – Insertion/enqueue from the rear and deletion/dequeue from Insertion/enqueue from the rear and deletion/dequeue from the front. the front. – FIFO FIFO • Two operations (enqueue and dequeue) Two operations (enqueue and dequeue)
  • 32.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-32 Priority Queue and Heap Priority Queue and Heap  Priority queues (implemented using heaps)  A data structure for maintaining a set of elements, each associated with a key/priority, with the following operations  Finding the element with the highest priority  Deleting the element with the highest priority  Inserting a new element  Scheduling jobs on a shared computer 9 6 8 5 2 3 9 6 5 8 2 3
  • 33.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-33 Graphs Graphs  Formal definition Formal definition • A graph A graph G = <V, E> G = <V, E> is defined by a pair of two sets: a is defined by a pair of two sets: a finite set V of items called finite set V of items called vertices vertices and a set E of vertex and a set E of vertex pairs called pairs called edges edges. .  Undirected Undirected and and directed directed graphs ( graphs (digraphs digraphs). ).  What’s the maximum number of edges in an undirected What’s the maximum number of edges in an undirected graph with |V| vertices? graph with |V| vertices?  Complete, dense, Complete, dense, and and sparse sparse graphs graphs • A graph with every pair of its vertices connected by an A graph with every pair of its vertices connected by an edge is called complete, K edge is called complete, K|V| |V| 1 2 3 4
  • 34.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-34 Graph Representation Graph Representation  Adjacency matrix Adjacency matrix • n x n boolean matrix if |V| is n. n x n boolean matrix if |V| is n. • The element on the ith row and jth column is 1 if there’s an 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. edge from ith vertex to the jth vertex; otherwise 0. • The adjacency matrix of an undirected graph is symmetric. The adjacency matrix of an undirected graph is symmetric.  Adjacency linked lists Adjacency linked lists • A collection of linked lists, one for each vertex, that contain all A collection of linked lists, one for each vertex, that contain all the vertices adjacent to the list’s vertex. the vertices adjacent to the list’s vertex.  Which data structure would you use if the graph is a 100-node Which data structure would you use if the graph is a 100-node star shape? star shape? 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 2 3 4 4 4
  • 35.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-35 Weighted Graphs Weighted Graphs  Weighted graphs Weighted graphs • Graphs or digraphs with numbers assigned to the edges. Graphs or digraphs with numbers assigned to the edges. 1 2 3 4 6 8 5 7 9
  • 36.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-36 Graph Properties -- Paths and Connectivity Graph Properties -- Paths and Connectivity  Paths Paths • A path from vertex u to v of a graph G is defined as a sequence of 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 adjacent (connected by an edge) vertices that starts with u and ends with v. with v. • Simple paths Simple paths: All edges of a path are distinct. : All edges of a path are distinct. • Path lengths: the number of edges, or the number of vertices – 1. Path lengths: the number of edges, or the number of vertices – 1.  Connected graphs Connected graphs • A graph is said to be connected if for every pair of its vertices u and 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. v there is a path from u to v.  Connected component Connected component • The maximum connected subgraph of a given graph. The maximum connected subgraph of a given graph.
  • 37.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-37 Graph Properties -- Acyclicity Graph Properties -- Acyclicity  Cycle Cycle • A simple path of a positive length that starts and A simple path of a positive length that starts and ends a the same vertex. ends a the same vertex.  Acyclic graph Acyclic graph • A graph without cycles A graph without cycles • DAG DAG (Directed Acyclic Graph) (Directed Acyclic Graph) 1 2 3 4
  • 38.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-38 Trees Trees  Trees Trees • A tree (or A tree (or free tree free tree) is a connected acyclic graph. ) is a connected acyclic graph. • Forest: a graph that has no cycles but is not necessarily connected. Forest: a graph that has no cycles but is not necessarily connected.  Properties of trees Properties of trees • For every two vertices in a tree there always exists exactly one For every two vertices in a tree there always exists exactly one simple path from one of these vertices to the other. simple path from one of these vertices to the other. Why? Why? – Rooted trees Rooted trees: : The above property makes it possible to select an The above property makes it possible to select an arbitrary vertex in a free tree and consider it as the root of the arbitrary vertex in a free tree and consider it as the root of the so called rooted tree. so called rooted tree. – Levels in a rooted tree. Levels in a rooted tree.  |E| = |V| - 1 1 3 2 4 5 1 3 2 4 5 rooted
  • 39.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-39 Rooted Trees (I) Rooted Trees (I)  Ancestors Ancestors • For any vertex For any vertex v v in a tree in a tree T T, all the vertices on the simple path , all the vertices on the simple path from the root to that vertex are called ancestors. from the root to that vertex are called ancestors.  Descendants Descendants • All the vertices for which a vertex All the vertices for which a vertex v v is an ancestor are said to is an ancestor are said to be descendants of be descendants of v v. .  Parent, child Parent, child and and siblings siblings • If If (u, v) (u, v) is the last edge of the simple path from the root to is the last edge of the simple path from the root to vertex vertex v v, , u u is said to be the parent of is said to be the parent of v v and and v v is called a child is called a child of of u u. . • Vertices that have the same parent are called siblings. Vertices that have the same parent are called siblings.  Leaves Leaves • A vertex without children is called a leaf. A vertex without children is called a leaf.  Subtree Subtree • A vertex A vertex v v with all its descendants is called the subtree of with all its descendants is called the subtree of T T rooted at rooted at v v. .
  • 40.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-40 Rooted Trees (II) Rooted Trees (II)  Depth Depth of a vertex of a vertex • The length of the simple path from the root to the vertex. The length of the simple path from the root to the vertex.  Height Height of a tree of a tree • The length of the longest simple path from the root to a leaf. The length of the longest simple path from the root to a leaf. 1 3 2 4 5 h = 2
  • 41.
    Karpagam Academy ofHigher Education Design & Analysis of Algorithms, 1-41 Ordered Trees Ordered Trees  Ordered trees Ordered trees • An ordered tree is a rooted tree in which all the children of each An ordered tree is a rooted tree in which all the children of each vertex are ordered. vertex are ordered.  Binary trees Binary trees • A binary tree is an ordered tree in which every vertex has no more A binary tree is an ordered tree in which every vertex has no more than two children and each children is designated s either a left child than two children and each children is designated s either a left child or a right child of its parent. or a right child of its parent.  Binary search trees Binary search trees • Each vertex is assigned a number. Each vertex is assigned a number. • A number assigned to each parental vertex is larger than all the 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 numbers in its left subtree and smaller than all the numbers in its right subtree. right subtree.   log log2 2n n    h h   n – 1 n – 1, where h is the height of a binary tree and n the size. , where h is the height of a binary tree and n the size. 9 6 8 5 2 3 6 3 9 2 5 8

Editor's Notes

  • #4 Euclid’s algorithm is good for introducing the notion of an algorithm because it makes a clear separation from a program that implements the algorithm. It is also one that is familiar to most students. Al Khowarizmi (many spellings possible...) – “algorism” (originally) and then later “algorithm” come from his name.
  • #7 The algorithm is given *very* informally here. Show students the pseudocode in section 3.1. This is a good opportunity to discuss pseudocode conventions.
  • #8 1-4 have well known efficient (polynomial-time) solutions 5: primality testing has recently been found to have an efficient solution This is a great problem to discuss because it has recently been in the news (see mathworld news at: http://mathworld.wolfram.com/news/2002-08-07_primetest/ or original article: http://www.cse.iitk.ac.in/primality.pdf) 6(TSP)-9(chess) are all problems for which no efficient solution has been found it is possible to informally discuss the “try all possibilities” approach that is required to get exact solutions to such problems 10: Towers of Hanoi is a problem that has only exponential-time solutions (simply because the output required is so large) 11: Program termination is undecidable
  • #12 The formalization of the notion of an algorithm led to great breakthroughs in the foundations of mathematics in the 1930s.
  • #14 Euclid’s algorithm is good for introducing the notion of an algorithm because it makes a clear separation from a program that implements the algorithm. It is also one that is familiar to most students. Al Khowarizmi (many spellings possible...) – “algorism” (originally) and then later “algorithm” come from his name.
  • #36 Examples of a simple path and a not simple path. Connected graphs: starting from any vertex, we can always find a path to reach all the other vertices. (Ball-String example.) From NIST: Connected graphs: Definition: An undirected graph that has a path between every pair of vertices. Strongly connected graphs: Definition: A directed graph that has a path from each vertex to every other vertex. Connected component: … Strongly connected component: a strongly connected component of a digraph G is a maximal strongly connected subgraph of G.