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Designing Algorithms Csci 107 Lecture 3

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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]

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Computing the sum Problem: 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

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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

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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

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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

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

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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, 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

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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: responsegallons 12000 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 13000 1000

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Visualizing Fig 2.5 response end distance Yesgallons start mpg 0 Set response … … 11 Stop Computer Input (keyboard) Output (screen) 25 12000 13000

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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.

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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 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)

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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

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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.

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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

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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 a 3 a 2 a 1 a 0 b 3 b 2 b 1 b 0 43 2 7 67 3 4 5 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

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E.g., Visualizing Fig 1.2 m a 3 a 2 a 1 a 0 b 3 b 2 b 1 b 0 4 3 2 7 6 7 3 4 5 i0carry0 c 4 c 3 c 2 c 1 c 0 0 Get values for … … 8 Print value of … Computer Input (keyboard) Output (screen) 4 3276 7345

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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:

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The (sequential) search algorithm (Fig 2.9) Variables: target, n, list 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

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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

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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

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