Lecture 2: Introduction to Algorithms

Lecture 2: Introduction to Algorithms • Review • Linear Search • Binary Search • Selection Sort • Reading

Review Algorithm: Textbook Definition: A well-ordered collection of unambiguous and effectively computable operations that when executed produces a result and halts in a finite amount of time.

Review • Computer Science • Textbook Definition: The study of algorithms including • Their formal and mathematical properties • Their hardware realizations • Their linguistic realizations • Their applications

Pseudocode • A programming language is a notation for specifying instructions that a computer can execute • Every (programming) language has a syntax and a semantics • Syntax specifies how something is said (grammar) • Semantics specifies what it means • Pseudocode is an informal programming language with English-like constructs modeled to look like statements in a Java-like language • Anyone able to program in any computer language should understand how to read pseudocode instructions. • When in doubt just write it out in English.

Your first Algorithm Linear Search: (also called sequential search) Algorithm to determine whether a given name x appears on a list. Input: A list of n ≥1 names A[1], A[2], ... , A[n], and a given name x. Output. The message "Sorry, x is not on the list" if x is not on the list. Otherwise, the message: "x occurs at position i on the list."

Linear Search Here are the instructions for linear search written in pseudocode: found = "no"; i=1; while (found == "no" and i <= n) { if (A[i] == x) { found = "yes"; location = i; } i++; } if (found == "no") { print ("Sorry, " + x + " is not on the list"); } else { print (x + " occurs at position " + location + " on the list"); }

Introduction to Algorithms • Definition of an algorithm: A well-ordered collection of unambiguous and effectively computable operations that, when executed, produces a result and halts in a finite amount of time. • There are three components to an algorithm: • A set of inputs, where each input is a finite sequence of items. • A set of outputs, where each output is a finite sequence of items. • A method consisting of a finite sequence of instructions, each one of which can be mechanically executed in a fixed length of time with a fixed amount of resources. The method must produce an output for each input in the set of possible inputs in a finite amount of time.

Introduction to Algorithms • Input size: associated with each input is a measure of its size. How we measure the size can vary from problem to problem. Examples: • Number of digits as in the number of digits in a number (Often we use binary digits). • Number of characters in a string. • Number of items as in the number of items to be searched or sorted.

Binary Search • Searching a sorted list • How can we exploit the structure that a sorted list provides? • Start in the middle and determine in which half of the list our target lives • Repeat this procedure eliminating half of the remaining list elements on each iteration • What is the maximum number of comparisons required here?

Binary Search Input: A list of n ≥1 numbers A[1], A[2], ... , A[n], and a target number x. Output. The message "Sorry, x is not on the list" if x is not on the list. Otherwise, the message: "x occurs at position i on the list."

Pseudocode for Binary Search found = "no"; begin = 1; end = n; while (found == "no" and begin  end) { m = begin + floor((end-begin)/2) if(A[m] == x) { found = "yes"; location = m; } if(A[m]>x) end = m-1; if(A[m]<x) begin = m+1; } if (found == "no") { print ("Sorry, " + x + " is not on the list"); } else { print (x + " occurs at position " + location + " on the list"); }

Selection Sort Algorithm to sort the items on a list into nondecreasing order Input. A list of n ≥1 elements A[1], A[2], ... , A[n]. Output. The list sorted in nondecreasing order. for (i = 1; i < length(A); i++) { locmin=i; for (j=i+1;j <= length(A); j++) { if (A[locmin] > A[j]) locmin=j; } exchange(A[locmin], A[i]); }

Reading • Chapters 1-3 in Schneider and Gersting

Lecture 2: Introduction to Algorithms