Introduction to Algorithms

Introduction to Algorithms
> Click or hit Control-Enter to run Example.main above

Algorithm: a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer.

As computer scientists, we implement algorithms by having computers:

Perform simple calculations
Store the results
Make simple decisions
Do things over and over again as fast as possible

Data structure: a collection of data values, the relationships among them, and the functions or operations that can be applied to the data.

As Java programmers we implement more complicated data structures using a mix of:

Primitive types and objects to store and organize data values
Existing data structures like arrays
References to reflect relationships among objects

Algorithms and data structures are highly complementary:

We will implement algorithms that utilize specific features of data structures
We will implement data structures to support specific algorithms
We will use our existing imperative and object-oriented ideas along the way
And we’ll introduce a few more important ideas along the way

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

What’s a simpler approach?

Brute force solution: a very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem’s statement.

Computers today are very, very fast
So try the simple thing first
If it’s too slow, try something a bit more sophisticated

> Click or hit Control-Enter to run Example.main above

That’s a good thing!

Even if you don’t at the beginning, you will eventually start to care about how fast your code runs. For any number of the following reasons:

You start to have larger problems to solve.
You’re embarrassed that your algorithm makes your incredibly fast computer seem slow
You have to start paying for machines
Your customer tells you that your program is too slow
You’re in a job interview

How long will our brute force GCD algorithm take?

To compute the GCD of 4 and 6
To compute the GCD of 185 and 2045
To compute the GCD of M and N

Algorithm analysis: the determination of the computational complexity of algorithms, that is the amount of time, storage and/or other resources necessary to execute them.

We’re usually want to analyze an algorithm in the general case, rather than for a specific set of inputs.

How does the algorithm perform on arbitrarily difficult or large inputs?
What are the best, average, and worst-case running times?
How is the algorithm’s performance related to its inputs?

Big-O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

Put another way: we want to estimate what happens as the problem gets really, really hard.

(At this point non-critical components of the algorithm’s performance also cease to matter.)

A dumb algorithm can move a problem up in the runtime categorization: for example, from O(n) to O(n^2). (Our sort test is dumb. The problem is O(n).)

A smart algorithm can move a problem down in the runtime categorization: for example, from O(n^2) to O(n log n). (Euclid’s Method GCD is smart. The problem is O(log(N)).)

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Algorithms

Data Structures

Algorithms and Data Structures

Example: Greatest Common Denominator

But If We’re In A Hurry…

Brute Force Solution

You Don’t Need the Fastest Algorithm to Change the World

But Speed Eventually Matters

So How Long Will It Take?

Algorithm Analysis

At The Limit

Big-O Notation

Dumb Algorithm, Clever Algorithm

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