AlgorithmsRecipes for disaster?

Topics Covered Programming vs. problem solving What is an algorithm? Algorithm development

common constructs (loops, conditionals, etc)

Representing algorithms flowcharts pseudocode

Programming vs. Problem Solving Two very distinct tasks! Problem solving:

developing a solution to a problem result is an algorithm or a plan for the solution

Programming: converting the solution into a computer readable form (C,

assembly, VB, Delphi, etc..) Programmers are cheap – problem solvers are

not! eg, architect, engineer vs. bricklayers, carpenters, etc..

What is an Algorithm? Formal definition:

An algorithm is a step-by-step procedure for accomplishing a stated task

A recipe for solving a problem History:

derived from 9th century mathematician Abu Abdullah Muhammad ibn Musa al-Khwarizmi

initially referred only to solving arithmetic problems (multiplication, division, etc) known as algorism

definition expanded to include all mathematical procedures formalised by Alan Turing in 1936 (Turing Machine) the foundation of computer science

Algorithm Examples Humans solve problems intuitively without

usually formalising the steps involved Consider:

passing this course making a cup of tea multiplying two numbers determining if a number is prime finding the largest number in a list

Properties of Algorithms Finiteness: must eventually terminate Definiteness: each step is precisely defined, there

is no ambiguity Input: information which is required in order to

start the algorithm Output: result of the algorithm, with some relation

to the input Effectiveness: should be able to be emulated

manually (paper and pencil)

Types of Algorithms Iterative or Recursive:

recursive algorithms call themselves, iterative algorithms do not Serial or Parallel:

one instruction at a time or multiple instructions Deterministic or Stochastic:

identical execution for identical input or some randomness involved

Exact or Approximate: is the solution exact or merely good enough

Complexity: how long the algorithm takes to run (for a given size input)

Algorithmic Constructs There are a number of standard ‘tools’ used when

developing algorithms variables statements loops conditional statements modules

These are the same actions we use for every activity you just don’t realise it!

Variables Variables are areas of storage used to keep track of data

in an algorithm, the size or type of the data in a variable is not necessarily defined, but should be consistent

can hold numbers, letters, words, etc.. Each variable requires a distinct name

should be related to what the variable is holding, eg. age, iq, student_number, etc

The equivalent of writing something down for later when manually running through an algorithm, it is helpful to use

labelled boxes for variables Input/Output values are also considered variables Variables are the only data that is available to the algorithm!

Statements Statements are the building blocks of algorithms

each statement is an instruction to perform an action set or change a variable’s value perform calculations write data to the screen, read from keyboard call another module

Statements should be either: simple, atomic actions that are easily understood module calls which are defined elsewhere

ALL relevant information needs to be included in the statement!

Read a, b, c from keyboardSET r1 = (-b + b2-4ac) / 2Print r1 to screen

Conditional Statements Most non-trivial algorithms are not simply a

sequence of statements require some decisions at some point!

Conditional statements are used to branch the execution of an algorithm logical expressions are used to make such decisions

IF ( something is true ) THEN do something

ELSE do something else

Loops and Repetition Most algorithms require some actions to be

repeated numerous times, either: a fixed number of times, OR until a condition is met (or no longer met)

Such constructs are known as loops, and are a powerful tool in algorithm design

Representing Algorithms Algorithms can be represented in numerous ways

plain English descriptions graphically (flowcharts, Nasi-Schneiderman diagrams, etc) structured English (pseudocode)

Each approach has advantages and disadvantages ease of understanding ambiguity speed of creation ease of conversion to actual source code

Plain English Descriptions As name implies, a plain language description of

the algorithm Usually in the form of an English paragraph Advantages:

easy for layperson to understand quick to create

Disadvantages: often ambiguous does not convert easily to code no well defined structure, so large variations between

authors

Graphical Representations Visual representation of algorithm

each structure has a set of symbols to represent it Flowcharts are one of the earliest and most popular forms

boxes = actions diamonds = decisions arrows = program flow

Advantages: some people find visual approach easy to understand unambiguous

Disadvantages: can be large and difficult to create direct conversion to code is possible, but not always simple

Structured English Known as pseudocode A cross between English and program language

rules, conventions and structures similar to programming languages

statements are written in plain English statements Advantages:

very easy to convert to program code unambiguous in most cases

Disadvantages: not as easy to read as plain language can be ambiguous if written poorly

Algorithm Representation Examples

Simple directions to a destination: Plain English:

Go straight on to the traffic lights, turn left and take the third turning on the right. If the red gate is open then go through it, otherwise go through the green gate.

Algorithm Representation Examples

Flowchart:

Algorithm Representation Examples Pseudocode:

Go straight on to the traffic lights

Turn left

Take the third turning on the rightIF the red gate is open THEN      go through the red gateELSE      go through the green gateEND IF

Examples Reconsider these problems:

passing this course making a cup of tea multiplying two numbers determining if a number is prime finding the largest number in a list

Now define a solution in each representation…

Breaking Down a Problem Problems are often too complicated to be completely defined

in a single algorithm In such cases, some steps are defined only in general terms

add milk to tea study determine factors of x

Such tasks are not simple cannot be done in one action – one line of code

This is OK, provided that they are expanded upon elsewhere This is known as a modular approach, and is how all large

software systems are developed each module is designed and built separately, and assumes the

other modules will do their job…

Modular Algorithm Examples Traditional program breakdown:

get input process input display output

For very simple problems all such steps can be done in one algorithm

For more complex tasks, each step becomes one or more modules eg, solving quadratic equation:

Read a, b, c simple, not a moduleCalculate r1, r2 NOT simple, requires a moduleDisplay r1, r2 simple, not a module

Defining Modules When modules have been identified, it is

necessary to define them What is required?

inputs outputs their own algorithm!

Note that the calling module doesn’t need to know the algorithm but does need to know the inputs and outputs!!

“Top-Down” Design Systematic method of algorithm

development start at a high, conceptual level specify operations in general terms identify sub-modules

inputs, outputs repeat for each sub-module

Can benefit from re-use of some common modules

Top Down Example Missile guidance system Basic algorithm:

inputs: target coordinates (t_location)

while ( haven’t hit target yet )get instrument readingscalculate position and directioncalculate required trajectorycalculate fin/motor adjustmentsimplement adjustments

end while

Clearly, all statements here require some refinement…

Top Down Example Get_instrument_readings

outputs: gps, speed, altitude, gyro

activate gpsget gps informationread speedread altitude from altimeterread spin from gyroRETURN ( gps, speed, altitude, spin )

Calculate_position_and_velocity input: gps, speed, altitude, spin output: position, velocity

Algorithm: who knows?!? (somebody else’s job!)

Top Down Example Calculate_trajectory

inputs: position, velocity, t_location outputs: req_trajectory

Calculate_fin_motor input: position, velocity, req_trajectory output: fin_adjust, motor_adjust

Implement_adjustments input: fin_adjust, motor_adjust

Top Down Example

MissileGuidance

Get_inst calc_pos calc_traj calc_fin_adjst imp_fin_adjst

… … …

When to Break Down Problem? In general, an algorithm (module) should be no more than 20 lines

long >20, probably should be modularised not a hard and fast rule, but a guideline – many ‘simple’ modules may

contain hundreds of lines Any moderately lengthy section of an algorithm which is used a

number of times should be made into a module Modules should be logical!

well-defined purpose (make sense independently) the module should be cohesive not just to break apart a long module!

Modules should be able to be tested independently! must have a good input/output relationship testing is done before integration occurs

Types of Modules Some languages define different module types Procedures:

perform a specific task do NOT provide an output can take inputs however

Functions: return at least one output in general, their purpose is to calculate those values, and

nothing else can of course take input as well

C makes no distinction between these types of modules – all are called functions