Introduction to Algorithms

What is Computer Science? Computer Science is the study of computers (??) This leaves aside the theoretical work in CS, which does not make use of real computers, but of formal models of computers A lot of work in CS is done with pen and paper! Actually, the early work in CS took place before the development of the first computer Computer Science is no more about computers than astronomy is about telescopes, biology is about microscopes, or chemistry is about test tubes. Science is not about tools. It is about how we use them, and what we find out we can do.

What is Computer Science? Computer Science is the study of how to write computer programs (programming) (??) Programming is a big part of CS....but it is not the most important part. Computer Science is the study of the uses and applications of computers and software (??) Learning to use software packages is no more a part of CS than driver’s education is part of automotive engineering. CS is responsible for building and designing software.

What is computer science? The study of algorithms: – their formal properties correctness, limits efficiency/cost – their hardware realizations computer design – their linguistic realizations programming languages – their applications network design, ocean modeling, bioinformatics,...

What is an algorithm? … a well-defined procedure that allows an agent to solve a problem. Algorithms must: 1. Be well-ordered and unambigous 2. Be executable (understandable) 3. Solve the problem, and 4. Terminate. Note: often the agent is a computer or a robot…

Example Algorithms Cooking a dish Making a peanut-butter jelly sandwich Shampooing hair Programming a VCR Making a pie

Examples Is this an algorithm? Step 1: Wet hair Step 2: Lather Step 3: Rinse Step 4: Repeat

Example Problem: Adding two n-digit numbers How would you write an algorithm to solve this problem? Assume the basic operation is adding one-digit numbers.

Types of Operations Basic operations – Wet hair – Rinse – Turn on VCR Conditional operations – If batter is too dry add water Repeat/looping operations – Repeat step 1 and 2 three times – Repeat steps 2,3,4,…10 until batter becomes soft.

Expressing algorithms Is natural language good? – For daily life, yes…but for CS it lacks structure and would be hard to follow – Too rich, ambiguous, depends on context How about a programming language? – Good, but not when we try to solve a problem..we want to think at an abstract level – It shifts the emphasis from how to solve the problem to tedious details of syntax and grammar.

Pseudocode Pseudocode = English but looks like programming Good compromise – Simple, readable, no rules, don’t worry about punctuation. Lets you think at an abstract level about the problem. – Contains only instructions that have a well-defined structure and resemble programming languages

Pseudocode elements Basic operations – Read the input from user – Print the output to the user – Carry out basic arithmetical computations Conditional operations – Execute an operation if a condition is true Repeat operations – Execute a block of operation multiple times until a certain condition is met

Basic operations – Read the input from user Get x Get a, b, c – Print the output to the user Print x Print “Your mileage is ” x – Cary out basic arithmetical computations Set x to 10 Set y to x*x/3

Conditional statements Specify a statement that may or may not be done: if then else Example if the value of carry is 0 then set the value of a to 0 else set the vale of a to a+1

Loop statements specify a group of statements that may be done several times (repeated): repeat until How does this work? – Condition is evaluated – If it is true than the loop terminates and the next instruction to be executed will be the instruction immediately following the loop – If it is false, then the algorithm executes the in order, one by one

Example Variables: count, square Step 1: set count to 1 Step 2: repeat step 3 to step 5 until count is > 10 Step 3: set square to count *count Step 4: print value of square and value of count Step 5: add 1 to count Step 6: end What does this algorithm do? Note: indentation – Not necessary, but makes reading/understanding algorithms easier

Pseudocode examples Equivalent: –Set the value of a to 1 –Set a to 1 –a=1 Equivalent –Add 1 to count –Set count to count + 1 –Increment the value of count by 1 –count = count + 1 Writing in pseudocode gives you the freedom to choose any of these!

Pseudocode Examples Incorrect –Set 1 to a –Add a + b (what do you do with the result?) –Set a+b +3 Note: not the same –set a to b –set b to a Example: what is the output of the following algorithms?set a to 2set b to 4 set a to bset b to aprint a, b

Problem Adding two n-digit numbers How would you write an algorithm to solve this problem? Assume the basic operation is adding one-digit numbers.

List variables How to represent (in pseudocode) inputs of arbitrary size? Suppose that we need to read 100 numbers from the user, or 1000, or.. – we could give each variable a different name…tedious!! Use a list variable: –Variable: list a of size n – This means that a is a list of n elements: a 1, a 2, a 3,…,a n – To read the list from the user use Get n and a 1, a 2, a 3,…,a n – To print the list use Print a 1, a 2, a 3,…,a n – We can treat each element in the list as a variable Set a 3 to 5 Set a 4 to a 3 +2

Algorithm for adding two m-digit numbers (Fig 1.2) Given: m ≥ 1 and two positive numbers a and b, each containing m digits, compute the sum c = a + b. Variables: m, list a 0..a m-1, list b 0 …. b m-1, list c 0 …c m-1 c m, carry, i 0Get values for m, a m-1 … a 0 and b m-1 … b 0 1Set the value of carry to 0. 2Set the value of i to 0. 3Repeat steps 4-6 until i > m-1 4 Set the value of c i to a i + b i + carry 5 if c i ≥ 10 then subtract 10 from c i and set the value of carry to 1 else set the value of carry to 0 6Add 1 to i 7Set the value of c m to carry 8Print value of c = c m c m-1 c m-2 … c 0

So, how does this work? For example, the input is m = 4, a = 3276, and b = After step 0, the variables m, a, and b have those values: m a 3 a 2 a 1 a 0 b 3 b 2 b 1 b After steps 1 and 2, the variables i and carry are initialized. i0carry0 Next, steps 4-6 are repeated until the value of i > 3. Each repetition computes a single digit of c. c 4 c 3 c 2 c 1 c 0

A model for visualizing an algorithm’s behavior Algorithm Computer Input (keyboard) Output (screen) Variables

E.g., Visualizing Fig m a 3 a 2 a 1 a 0 b 3 b 2 b 1 b i0carry0 c 4 c 3 c 2 c 1 c 0 0 Get values for … … 8 Print value of … Computer Input (keyboard) Output (screen)

Algorithm for computing MPG Write a pseudocode algorithm to compute the distance traveled and the average miles per gallon on a trip when given as input the number of gallons used and the starting and ending mileage readings on the odometer. Variables: response, gallons, start, end, distance, mpg 0 Set response to “Yes” 1 Repeat steps 2-8 until response = “No” 2 Get gallons, start, end 3 Set distance to end - start 4 Set mpg to distance ÷ gallons 5 Print mpg 6 if mpg > 25.0 then print “You are getting good gas mileage” else print “You are NOT getting good gas mileage” 7 Print “Do you want to do this again, Yes or No?” 8 Get response 9Stop

So, how does this work? For example, suppose we use 25 gallons, beginning at and ending at on the odometer. Then, after step 2, some variables have the following values: responsegallons start Yes25 After step 4, the variables distance and mpg are computed. mpg40 Steps 5-9 displays these results on the output screen: 40 You are getting good gas mileage Do you want to do this again, Yes or No? end distance

Visualizing Fig response end distance Yesgallons start mpg 0 Set response … … 11 Stop Computer Input (keyboard) Output (screen)

Designing Algorithms: A Methodology 1.Read the problem, identifying the input and the output. 2.What variables are needed? 3.What computations are required to achieve the output? 4.Usually, the first steps in your algorithm bring input values to the variables. 5.Usually, the last steps display the output 6.So, the middle steps will do the computation. 7.If the process is to be repeated, add loops.

How was the MPG program designed? Problem statement: Write a pseudocode algorithm to compute the distance traveled and the average miles per gallon on a trip when given as input the number of gallons used and the starting and ending mileage readings on the odometer. Input: number of gallons, starting mileage, ending mileage Output: distance traveled, average miles per gallon Variables: gallons, start, end, distance, mpg Calculate: distance = end - start mpg = distance / gallons Put the steps in order: input, calculate, output (steps 2-8) Determine if a loop is needed (steps 0, 1, 9, 10)

A Search Algorithm Problem statement: Write a pseudocode algorithm to find the location of a target value in a list of values. Input: a list of values and the target value Output: the location of the target value, or else a message that the value does not appear in the list. Variables:

The (sequential) search algorithm Variables: target, n, list of n values Get the value of target, n, and the list of n values Set index to 1 Set found to false Repeat until found = true or index > n If the value of list index = target then Output the index Set found to true else Increment the index by 1 If not found then Output a message that target was not found Stop