Algorithms
- 1. Discrete Mathematics And it’s Applications Algorithms
- 2. Group # 01 BSIT-3rd –B-Morning Members Muhammad Waqar Munir Rehan Ali Farooq Shah Jahanzaib Shahid Muhammad Hammad
- 3. Algorithms Methodology: Construct a model that translates the problem into a mathematical context Build a method that will solve the general problem using the model Ideally, we need a procedure that follows a sequence of steps that leads to the desired answer, such a sequence is called an Algorithm. History: The term Algorithm is a corruption of the name Al-Khowarizmi (mathematician of the 9th century)
- 4. Algorithms What is an algorithms? An Algorithm is a finite set of precise instructions for performing a computation or for solving a problem.
- 5. • Finding the Maximum Element in a Finite Sequence procedure max(a1, a2, . . . , an: integers) max := a1 for i := 2 to n if max < ai then max := ai return max{max is the largest element}
- 6. Properties of algorithms: Inputfrom a specified set. Outputfrom a specified set (solution) Correctness of output for every possible input. Definiteness of evert step in the computation Effectiveness of each calculation step and Generality for a class of problems.
- 7. Searching Algorithms The problem of locating an element in an ordered list occurs in many contexts. For instance, a program that checks the spelling of words searches for them in a dictionary, which is just an ordered list of words. Problems of this kind are called searching problems.
- 8. THE LINEAR SEARCH The first algorithm that we will present is called the linear search, or sequential search, algorithm. • ALGORITHM For The Linear Search procedure linear search(x: integer, a1, a2, . . . , an: distinct integers) i := 1 while (i ≤ n and x = ai ) i := i + 1 if i ≤ n then location := i else location := 0 return location{location is the subscript of the term that equals x, or is 0 if x is not found}
- 9. THE BINARY SEARCH Some Instructions: • if the terms are numbers, they are listed from smallest to largest • if they are words, they are listed in lexicographic, or alphabetic, order can be used when the list has terms occurring in order of increasing size.
- 10. Binary Search May only b used on a sorted array. Eliminates one half of the elements after each comparison. Locate the middle of the array. Compare the value at that location with the search key. If they are equal-done. Otherwise, decide which half of the array contains the search key. Repeat the search on that half of the array and ignore the other half. The search continues until the key is matched or no elements remain to be sorted.
- 11. • ALGORITHM For The Binary Search procedure binary search (x: integer, a1, a2, . . . , an: increasing integers) i := 1{i is left endpoint of search interval} j := n {j is right endpoint of search interval} while i < j m := (i + j)/2 if x > am then i := m + 1 else j := m if x = ai then location := i else location := 0 return location{location is the subscript i of the term ai equal to x, or 0 if x is not found}
- 12. Sorting Goal “Ordering the elements” for example, sorting the list 7,2,1,4,5,9 produces the list 1,2,4,5,7,9. Similarly, sorting of list d, h, c, a, f produces a, c, d, f, h.
- 13. BUBBLE SORT The bubble sort is one of the simplest sorting algorithms, but not one of the most efficient. It puts a list into increasing order by successively comparing adjacent elements, interchanging them if they are in the wrong order. The Bubble Sort algorithm looks at the pairs of entries in the array, and swaps their order if needed.
- 15. • ALGORITHM For The Bubble Sort procedure bubble sort(a1, . . . , an : real numbers with n ≥ 2) for i := 1 to n − 1 for j := 1 to n − i if aj > aj+1 then interchange aj and aj+1 {a1, . . . , an is in increasing order}
- 16. INSERTION SORT • The insertion sort is a simple sorting algorithm, but it is usually not the most efficient. • To sort a list with n elements, • insertion sort begins with the second element.
- 17. • ALGORITHM For Insertion Sort procedure insertion sort(a1, a2, . . . , an: real numbers with n ≥ 2) for j := 2 to n i := 1 while aj > ai i := i + 1 m := aj for k := 0 to j − i − 1 aj−k := aj−k−1 ai := m {a1, . . . , an is in increasing order}
- 18. Greedy Algorithms • to find a solution to the given problem that either minimizes or maximizes the value of some parameter • one of the simplest approaches often leads to a solution of an optimization Problem. Algorithms that make what seems to be the “best” choice at each step are called greedy algorithms.
- 19. • ALGORITHM For Greedy Change- Making Algorithm procedure change(c1, c2, . . . , cr : values of denominations of coins, where c1 > c2 > ・ ・ ・ > cr ; n: a positive integer) for i := 1 to r di := 0 {di counts the coins of denomination ci used} while n ≥ ci di := di + 1 {add a coin of denomination ci} n := n − ci {di is the number of coins of denomination ci in the change for i = 1, 2, . . . , r}
- 20. TheHaltingProblem • these are unsolvable problems, called halting problem. • a problem that cannot be solved using any procedure. • We will describe Turing’s proof that the halting problem is unsolvable.