HOW TO INVENT AN ALGORITHM

Transcription

1 HOW TO INVENT AN ALGORITHM Here are all the ideas from the previous topics combined into a strategy for inventing algorithms. How to Invent an Algorithm Step What? How? 1 Understand the problem by solving it 2 Determine the program s requirements 3 Use top-down design to develop a pseudo-code algorithm that solves the problem Use pencil and paper to solve the problem by hand. Use George Pólya s problem solving techniques. Write a list of any questions &or ambiguities you encounter and seek answers. Identify the program s input and output. Determine what computations transform the input into the output. Begin with a high-level algorithm Top-Down Algorithm Development Pseudo-code algorithm Is the algorithm correct, complete and detailed enough? NO YES Code it into a computer program Select some part of the algorithm and refine it Use pseudo-code; do not get sidetracked by trying to write the algorithm in a computer language. To refine some part of the algorithm use what-to-howconversion or decomposition. 4 Check that your algorithm is correct and complete Desk check the algorithm after each refinement step. See further explanations below How to Invent an Algorithm Page 1

2 More on Step 1 Understanding the Problem When given an assignment to write a computer program, always start by solving the problem with pencil and paper. Handwriting helps to stimulate your brain s learning activity 1 making this step absolutely essential. More on Step 2 Determining the Program s Requirements Characterizing the program in terms of input, processing and output is a fundamental second step. Trying to write a computer program without having done it is like trying to prepare a meal without having decided what you want to eat. All of the data needed for the program will be revealed when you solve the problem in Step 1. This data can be divided into three categories: Categories of Program Data What Input data Internal data Output data Description Must be supplied by a human being or a device that is external to the program Can be coded as constants directly within the program Must be displayed or transmitted to an external human being or device Generally, output data is easy to identify because it is the purpose of the program. Also, data that never changes (e.g. the value of π = ) should become internal data. Consider a program that calculates height by timing a falling object. The formula for computing this height is: height (meters) = ½gt 2 where t is the time (in seconds) and g is the force of gravity (9.81 meters per sec 2 ). For example, if t = 2.5 then: meters This shows that, if a dropped ball takes 2½ seconds to reach the ground, it was dropped from a height of meters. 1 Gwendolyn Bounds, How Handwriting Trains the Brain, The Wall Street Journal Online, October 5, How to Invent an Algorithm Page 2

3 Solving this problem involves three values (1) time t, (2) force of gravity 9.81 and (3) height which are readily divided into input, internal and output data. INPUT PROCESSING OUTPUT INTERNAL DATA g = 9.81 time t (sec) height (meters) = ½gt 2 height (m) Sometimes it can be a struggle deciding whether a datum should be input or internal data. The decision involves a balance between making the program easier to use and making it useful in a variety of situations. If you were to code all the data as constants within the program then it would be very easy to use the user would simply have to run it to get the answer. But it would give an answer for only one specific situation. Program is easy to use Program is useful in many situations All data is internal More data is made to be input You want to increase the usefulness of your program by requiring that more of the data be input by the user without requiring so much input that the program becomes onerous to run. Don t make these decisions in a vacuum. Consider how the program is to be used and be sure to ask your users what they want. $1,000 invested at 2% interest per year without compounding earns $1, = $20 after 1 year. Compounding means that interest earned is added to the investment so that it also begins to earn interest. Compounded monthly, for instance, means that interest is added to the investment at the end of every month. How to Invent an Algorithm Page 3

4 You can find the formula for calculating compound interest by Googling compound interest formula. r nt FV PV ( 1 ) PV = the present value of the investment t = the number of years the amount is invested, called the term r = the annual interest rate as a decimal (not a percentage) n = the number of times interest is compounded per year FV = the value of the investment after t years $1,000 invested at 2% interest per year compounded monthly will be worth, after 3 years: n FV ,000(1. 02 ) $1, Solving the problem involves the five values listed above. FV is clearly the output datum. PV, t and r ought to be input values to allow the user to do the calculation for various investment amounts, terms and interest rates. n is not so straightforward. Some users, if asked to input it, may not understand what it is. Furthermore, financial institutions generally stick with one compounding period it s not something they often change. So if you re creating this program for an institution or its customers it would seem better to make this internal data. INPUT present value PV ($) term t (years) annual interest rate r PROCESSING INTERNAL DATA n = 12 FV PV ( 1 ) r n nt OUTPUT future value FV ($) How to Invent an Algorithm Page 4

5 Step 3 Use Top-Down Design Like any top-down design process, top-down algorithm development repeatedly refines the pseudo-code algorithm until it is detailed enough to code into a programming language such as Java. There are two refinements that programmers typically make to an algorithm. One is decomposition, in which a design element is analyzed, broken up into its component parts and described in more detail. To illustrate decomposition, consider the top-level algorithm for finding the average of a list of students test scores. find the average score of a list of students test scores The average is calculated by dividing the sum of the scores by the number of scores. For example, the average of 30, 50, 60, 90 and 100 is This calculation can be 5 decomposed into three parts: (1) sum the scores, (2) count them and (3) divide the sum by the count. Rewriting the refined algorithm yields: find the sum of a list of students test scores count the number of students test scores divide the sum by the count A second algorithm refinement is the what-to-how conversion. High-level algorithms typically state what is desired; low-level algorithms explain how to accomplish it. Transitioning from what to how is an important part of algorithm development. An illustration of converting a description from what to how. What? calculate my car s fuel consumption in miles-per-gallon How? divide the number of miles driven by the number of gallons used How to Invent an Algorithm Page 5

6 Step 4 Desk Check the Algorithm Seasoned programmers desk check their algorithm after each refinement step. Desk check means to pretend you re a computer and follow your algorithm step-by-step, using pencil and paper to keep track of your computations. Look for errors and omissions and correct them. Let s revisit the problem of determining how much $1,000 invested at 2% interest per year compounded monthly will be worth after 3 years. This time, rather than use a fancy formula, we calculate the answer by fast-forwarding through the 3 years, keeping track of the investment at the end of each month. Like so: Month Investment Multiplied By Interest New Balance 1 $1, /12 $1.67 $1, , / , , / , , / , , / , etc. The monthly interest rate is 1 12 the annual rate. The algorithm treats each loop cycle as the passing of a month and quits after 36 months (i.e. 3 years). set the investment to 1,000 set the month count to 0 repeat calculate interest = investment *.02/12 add interest onto the investment count the month until month count = 36 output the investment How to Invent an Algorithm Page 6

7 To desk check an algorithm, create a table that you can use to track the values of the variables as you carry out the steps in the algorithm. Across the top of the table, list the names of the variables from left to right. Down the page show the values of the variables as you follow the actions specified by the algorithm. How to Invent an Algorithm Page 7