CS101 - Introduction to Computing - Lecture Handout 16

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Algorithms

Focus of the last lecture was on Word Processing

First among the four lectures that we plan to have on productivity software, a subcategory of application software
That first lecture was on WP
We learnt about what we mean by WP and also desktop publishing
We also discussed the usage of various functions provided by common WP’s

The Objective of Today’s Lecture

To become familiar with the concept of algorithms:
What they are?
What is their use?
What do they consist of?
What are the techniques used for representing them?

Solving Problems (1)

When faced with a problem:
We first clearly define the problem
Think of possible solutions
Select the one that we think is the best under the prevailing circumstances
And then apply that solution
If the solution woks as desired, fine; else we go back to step 2

Solving Problems (2)

It is quite common to first solve a problem for a particular case
Then for another
And, possibly another
And watch for patterns and trends that emerge
And to use the knowledge form those patterns and trends in coming up with a general solution

Solving Problems (3)

It helps if we have experienced that problem or similar ones before
Generally, there are many ways of solving a given problem; the best problem-solvers come-up with the most appropriate solution more often than not!
The process that can be used to solve a problem is termed as the “algorithm”

Algorithm:

Sequence of steps that can be taken to solve a given problem

Algorithm

Examples

Addition
Conversion from decimal to binary
The process of boiling an egg
The process of mailing a letter
Sorting
Searching

Let us write down the algorithm for a problem that is familiar to us
Converting a decimal number into binary

Convert 75 to Binary

Algorithm for Decimal-to-Binary Conversion

Write the decimal number
Divide by 2; write quotient and remainder
Repeat step 2 on the quotient; keep on repeating until the quotient becomes zero
Write all remainder digits in the reverse order (last remainder first) to form the final result

Points to Note:

The process consists of repeated application of simple steps
All steps are unambiguous (clearly defined)
We are capable of doing all those steps
Only a limited no. of steps needs to be taken
Once all those steps are taken according to the prescribed sequence, the required result will be found
Moreover, the process will stop at that point

Algorithm (Better Definition)

Definition:
Sequence of steps that can be taken to solve a problem
Better Definition:
A precise sequence of a limited number of unambiguous, executable steps that terminates in the form of a solution

Three Requirements:

Sequence is:
Precise
Consists of a limited number of steps
Each step is:
Unambiguous
Executable
The sequence of steps terminates in the form of a solution

Why Algorithms are Useful?

Once we find an algorithm for solving a problem, we do not need to re-discover it the next time we are faced with that problem
Once an algorithm is known, the task of solving the problem reduces to following (almost blindly and without thinking) the instructions precisely
All the knowledge required for solving the problem is present in the algorithm

Why Write an Algorithm Down?

For your own use in the future, so that you don’t have spend the time for rethinking it
Written form is easier to modify and improve
Makes it easy when explaining the process to others

Analysis of Algorithms

Analysis in the context of algorithms is concerned with predicting the resources that re requires:
Computational time
Memory
Bandwidth
Logic functions
However, Time – generally measured in terms of the number of steps required to execute an algorithm - is the resource of most interest
By analyzing several candidate algorithms, the most efficient one(s) can be identified

Selecting Among Algorithms

When choosing among competing, successful solutions to a problem, choose the one which is the least complex
This principle is called the “Ockham’s Razor,” after William of Ockham - famous 13-th century English philosopher

Early History:

Search for a Generic Algorithm

The study of algorithms began with mathematicians and was a significant area of work in the early years
The goal of those early studies was to find a single, general algorithm that could solve all problems of a single type

Origin of the Term “Algorithm”

The name derives from the title of a Latin book: Algoritmi de numero Indorum
That book was a translation of an Arabic book: Al-Khwarizmi Concerning the Hindu Art of Reckoning
That book was written by the famous 9-th century Muslim mathematician, Muhammad ibn Musa al-Khwarizmi

Al-Khwarzmi

Al-Khwarizmi lived in Baghdad, where he worked at the Dar al-Hikma
Dar al-Hikma acquired and translated books on science and philosophy, particularly those in Greek, as well as publishing original research
The word Algebra has its origins in the title of another Latin book which was a translation of yet another book written by Al-Khwarzmi:
Kitab al-Mukhtasar fi Hisab al-Jabr wa'l-Muqabala

Al-Khwarizmi’s Golden Principle

All complex problems can be and must be solved using the following simple steps:
Break down the problem into small, simple sub-problems
Arrange the sub-problems in such an order that each of them can be solved without effecting any other
Solve them separately, in the correct order

Combine the solutions of the sub-problems to form the solution of the original problem That was some info on history.
Now, let us to take a look at several types of algorithms & algorithmic strategies

Greedy Algorithm

An algorithm that always takes the best immediate, or local solution while finding an answer
Greedy algorithms may find the overall or globally optimal solution for some optimization problems, but may find less-than-optimal solutions for some instances of other problems
KEY ADVANTAGE: Greedy algorithms are usually faster, since they don't consider the details of possible alternatives

Greedy Algorithm: Counter Example

During one of the international cricket tournaments, one of the teams intentionally lost a match, so that they could qualify for the next round
If they had won that particular match, some other team would have qualified
This is an example of a non-greedy algorithm

Greedy Algorithm: Example

A skier skiing downhill on a mountain wants to get to the bottom as quickly as possible What sort of an algorithm should the skier be using?
The greedy-algorithm approach will be to always have the skies pointed towards the largest downhill slope (dy/dx), at all times
What is the problem with that approach?
In what situations that will be the best algorithm?
In which situations would it perform poorly?

Deterministic Algorithm (1)

An algorithm whose behavior can be completely predicted from the inputs
That is, each time a certain set of input is presented, the algorithm gives the same results as any other time the set of input is presented.

Randomized Algorithm (1)

Any algorithm whose behavior is not only determined by the input, but also values produced by a random number generator
These algorithms are often simpler and more efficient than deterministic algorithms for the same problem
Simpler algorithms have the advantages of being easier to analyze and implement.

Randomized Algorithm (2)

These algorithm work for all practical purposes but have a theoretical chance of being wrong:
Either in the form of incorrect results
Or in the form of impractically long running time
Example: Monte Carlo algorithms.

Deterministic Algorithm (2)

There can be degrees of deterministic behavior: an algorithm that also uses a random number generator might not be considered deterministic
However, if the "random numbers" come from a pseudo-random number generator, the behavior may be deterministic
Most computing environments offer a “pseudo random number generators,” therefore, most randomized algorithms, in practice, behave deterministically!

Heuristic

A procedure that usually, but not always, works or that gives nearly the right answer Some problems, such as the traveling salesman problem, take far too long to compute an exact, optimal solution. A few good heuristics have been devised that are fast and find a near-optimal solution more often than not
Is a heuristic, an algorithm? Yes? No? Why?

The Traveling Salesman Problem

A Few Questions

Is that the best possible sequence?
How do you know?
How do I determine the best sequence?

The Brute Force Strategy (1)

A strategy in which all possible combinations are examined and the best among them is selected
What is the problem with this approach?
A: Doesn’t scale well with the size of the problem
How many possible city sequences for n=6? For n=60? For n=600?

The Brute Force Strategy (2)

However, with the relentless increase in computing power, certain problems that – only a few years ago - were impossible to solve with brute force, are now solvable with this technique

A Selection of Algorithmic Application Areas

Search
Sort
Cryptography
Parallel
Numeric
Graphical
Quantum computing
Combinatory
We’ll now talk about the various ways of representing algorithms.
But, before we do that please allow me to say a few words about …

Syntax & Semantics

Now onto Algorithm Representation

We have said enough about algorithms – their definition, their types, etc.
But, how do we actually represent them?
Generally, SW developers represent them in one of three forms:
Pseudo code
Flowcharts
Actual code
Pseudo Code
Language that is typically used for writing algorithms
Similar to a programming language, but not as rigid
The method of expression most suitable for a given situation is used:
At times, plain English
At others, a programming language like syntax

Flowchart

A graphical representation of a process (e.g. an algorithm), in which graphic objects are used to indicate the steps & decisions that are taken as the process moves along from start to finish
Individual steps are represented by boxes and other shapes on the flowchart, with arrows between those shapes indicating the order in which the steps are taken

Flowchart

In Today’s Lecture, We …

Became familiar with the concept of algorithms:
What they are?
What is their use?
What do they consist of?
What are the techniques used for representing them?

Next Lecture: Algorithms II

We will continue our discussion on algorithms during the next lecture
In particular, we will discuss the pseudo code and flowcharts for particular problems
We will also discuss the pros and cons of these two algorithm representation techniques i.e. pseudo code and flow charts