Designing Algorithms

Csci 107

Lecture 3

Designing algorithms

• Last time– Pseudocode– Algorithm: computing the sum 1+2+…+n– Gauss formula for 1+2+…+n

• Today – Variations of the sum algorithm– Computing MPG– List variables– Adding two m-digit numbers– [Search and variations]

Computing the sumProblem: Write an algorithm which reads a positive integer from

the user and computes the sum of all positive integers smaller than or equal to te number entered by the user. – Example: if the user enters 10, the algorithm should compute 1+2+3+…

+10

Algorithm:• Variables: n(the number entered by the user),

sum, i• Print “Please enter a positive integer”• Get n• Sum=0, i=0• Repeat until i>n

– Sum = sum + i– i = i+1

• Print “The sum is” sum

• An algorithm is not unique!!!• There are many ways to solve a problem• Moreover, given a certain way to solve a

problem, there are many ways to express that into pseudocode!!

• Etiquette:– Give variables meaningful names– Write explanations/comments of what your

code does

Another algorithm for the sum problem

Use while instead of repeat loop

Algorithm 2:• Variables: n (the number entered by the user), sum,

i• Print “Please enter a positive integer”• Get n• Sum=0, i=0• While i <= n

– Sum = sum + i– i = i+1

• Print “The sum is” sum

Variations..

Given a number n from the user, write an algorithm..

• To compute the sum of all even numbers <= n

• To compute the sum of all odd numbers <= n

• To compute the product of all numbers <= n (starting at 1)

• To compute the sum of all numbers strictly smaller than n

A model for visualizing an algorithm’s behavior

Algorithm

Computer

Input (keyboard)

Output (screen)

Variables

Algorithm for computing MPG (Fig 2.5)

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. Repeat this process until user wants to stop.

Variables: response, gallons, start, end, distance, mpg0 Set response to “Yes”1 Repeat steps 2-8 until response = “No”

2 Get gallons, start, end3 Set distance to end - start4 Set mpg to distance ÷ gallons5 Print mpg6 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

9 Stop

So, how does this work???

For example, suppose we use 25 gallons, beginning at 12000 and ending at 13000 on the odometer. Then, after step 2, some variables have the following values:

response gallons12000start

Yes 25

After step 4, the variables distance and mpg are computed.

mpg 40

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

13000

1000

Visualizing Fig 2.5

response

end

distance

Yes gallons

start

mpg

0 Set response ……11 Stop

Computer

Input (keyboard)

Output (screen)

251200013000

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 output6. So, the middle steps will do the computation.7. If the process is to be repeated, add loops.

How was the MPG program (Fig 2.5) 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 mileageOutput: distance traveled, average miles per gallonVariables: gallons, start, end, distance, mpgCalculate: distance = end - start

mpg = distance / gallonsPut the steps in order: input, calculate, output (steps 2-8)Determine if a loop is needed (steps 0, 1, 9, 10)

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: a1, a2, a3,…,an

– To read the list from the user use• Get n and a1, a2, a3,…,an

– To print the list use• Print a1, a2, a3,…,an

– We can treat each element in the list as a variable• Set a3 to 5• Set a4 to a3 +2

Problem

• Adding two m-digit numbers

7597831 + 1287525-------------------

8885356

Write an algorithm to solve this problem. Assume the basic operation is adding one-digit numbers.

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 a0 ..am-1, list b0 …. bm-1 , list c0 …cm-1 cm, carry, i

0 Get values for m, am-1 … a0 and bm-1 … b0

1 Set the value of carry to 0.2 Set the value of i to 0.3 Repeat steps 4-6 until i > m-1

4 Set the value of ci to ai + bi + carry

5 if ci ≥ 10 then

subtract 10 from ci and set the value of carry to 1

else set the value of carry to 06 Add 1 to i

7 Set the value of cm to carry

8 Print value of c = cm cm-1 cm-2 … c0

So, how does this work???

For example, the input is m = 4, a = 3276, and b = 7345.

After step 0, the variables m, a, and b have those values:

m

a3 a2 a1 a0 b3 b2 b1 b0

4 3 2 7 6 7 3 4 5

After steps 1 and 2, the variables i and carry are initialized.

i 0 carry 0

Next, steps 4-6 are repeated until the value of i > 3. Each repetition computes a single digit of c.

c4 c3 c2 c1 c0

E.g., Visualizing Fig 1.2

m

a3 a2 a1 a0

b3 b2 b1 b0

4

3 2 7 6

7 3 4 5

i 0 carry 0

c4 c3 c2 c1 c0

0 Get values for ……8 Print value of …

Computer

Input (keyboard)

Output (screen)

432767345

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 (Fig 2.9)

Variables: target, n, list list of n valuesGet the value of target, n, and the list of n valuesSet index to 1Set found to falseRepeat until found = true or index > n

If the value of listindex = target then

Output the indexSet found to true

elseIncrement the index by 1

If not found thenOutput a message that target was not found

Stop

Variations of sequential search..

• Modify the sequential search algorithm in order

– To find all occurrences of target in the list and print the positions where they occur

– To count the number of occurrences of target in the list

– To count how many elements in the list are larger than target

More algorithms• Write algorithms to find

– the largest number in a list of numbers (and the position where it occurs)

– the smallest number in a list of numbers (and the position where it occurs)

– the range of a list of numbers • Range= largest - smallest

– the average of a list of numbers

– the sum of a list of numbers