Computer Algorithms: Determine if a Number is Prime

algorithms, javascript, PHP, , , , , , , , , , ,

Introduction

Each natural number that is divisible only by 1 and itself is prime. Prime numbers appear to be more interesting to humans than other numbers. Why is that and why prime numbers are more important than the numbers that are divisible by 2, for instance? Perhaps the answer is that prime numbers are largely used in cryptography, although they were interesting for the ancient Egyptians and Greeks (Euclid has proved that the prime numbers are infinite circa 300 BC). The problem is that there is not a formula that can tell us which is the next prime number, although there are algorithms that check whether a given natural number is prime. It’s very important these algorithms to be very effective, especially for big numbers.

Overview

As I said each natural number that is divisible only by 1 and itself is prime. That means that 2 is the first prime number and 1 is not considered prime. It’s easy to say that 2, 3, 5 and 7 are prime numbers, but what about 983? Well, yes 983 is prime, but how do we check that? If we want to know whether n is prime the very basic approach is to check every single number between 2 and n. It’s kind of a brute force.

Implementation

The basic implementation in PHP for the very basic (brute force) approach is as follows.

Unfortunately this is one very ineffective algorithm. We don’t have to check every single number between 1 and n, it’s enough to check only the numbers between 1 and n/2-1. If we find such a divisor that will be enough to say that n isn’t prime.

Although that code above optimizes a lot our first prime checker, it’s clear that for large numbers it won’t be very effective. Indeed checking against the interval [2, n/2 -1] isn’t the optimal solution. A better approach is to check against [2, sqrt(n)]. This is correct, because if n isn’t prime it can be represented as p*q = n. Of course if p > sqrt(n), which we assume can’t be true, that will mean that q < sqrt(n).

Beside that these implementations shows how we can find prime number, they are a very good example of how an algorithm can be optimized a lot with some small changes.

Sieve of Eratosthenes

Although the sieve of Eratosthenes isn’t the exact same approach (to check whether a number is prime) it can give us a list of prime numbers quite easily. To remove numbers that aren’t prime, we start with 2 and we remove every single item from the list that is divisible by two. Then we check for the rest items of the list, as shown on the picture below.

http://www.stoimen.com/blog/wp-content/uploads/2012/05/SieveofEratosthenes.png

The PHP implementation of the Eratosthenes sieve isn’t difficult.

Application

As I said prime numbers are widely used in cryptography, so they are always of a greater interest in computer science. In fact every number can be represented by the product of two prime numbers and that fact is used in cryptography as well. That’s because if we know that number, which is usually very very big, it is still very difficult to find out what are its prime multipliers. Unfortunately the algorithms in this article are very basic and can be handy only if we work with small numbers or if our machines are tremendously powerful. Fortunately in practice there are more complex algorithms for finding prime numbers. Such are the sieves of Euler, Atkin and Sundaram.

Computer Algorithms: Lossy Image Compression with Run-Length Encoding

algorithms, , , , , , , , , , , , , , , ,

Introduction

Run-length encoding is a data compression algorithm that helps us encode large runs of repeating items by only sending one item from the run and a counter showing how many times this item is repeated. Unfortunately this technique is useless when trying to compress natural language texts, because they don’t have long runs of repeating elements. In the other hand RLE is useful when it comes to image compression, because images happen to have long runs pixels with identical color.

As you can see on the following picture we can compress consecutive pixels by only replacing each run with one pixel from it and a counter showing how many items it contains.

Lossless RLE for Images
Although lossless RLE can be quite effective for image compression, it is still not the best approach!

In this case we can save only counters for pixels that are repeated more than once. Such the input stream “aaaabbaba” will be compressed as “[4]a[2]baba”.

Actually there are several ways run-length encoding can be used for image compression. A possible way of compressing a picture can be either row by row or column by column, as it is shown on the picture below.

Row by row or column by column compression
Row by row or column by column compression.
Continue reading Computer Algorithms: Lossy Image Compression with Run-Length Encoding →

PHP Strings Don’t Need Quotes

PHP, , , , , , , , , , , , ,

I bet you didn’t know that PHP strings don’t need quotes! Indeed PHP developers work with strings with either single or double quotes, but actually in some cases you don’t need them.

PHP by Book

Here’s how PHP developer declare a string, which is something very common in any programming language.

$my_var = 'hello world';
// or
$my_var = "hello world";

PHP Tricks

What if you do the following:

echo hello;

That appears to be correct … Well, it’s not absolutely correct. You’ll be “noticed”.

// Notice: Use of undefined constant hello
echo hello;

However if you disable error reporting, the code will be completely fine.

error_reporting(0);
 
// no problem now
echo hello;

Variations

What follows from the thing above is that you can use strings without quotes:

// hello
echo hello;
 
// hello world (concatenated)
echo hello . ' world';
 
// helloworld
echo hello . world;

However you can’t have spaces and most of the “special” symbols.

// syntax error
echo hello world;
 
// syntax error
echo hello!;

Final Words

Although you can do this in PHP, that is completely wrong. The code becomes more difficult to read and understand. In the second place you can miss a $ sign in front of a variable declaration and thus the PHP interpreter will assume this is a string. So disable error reporting isn’t so great sometimes.

Computer Algorithms: How to Determine the Day of the Week

algorithms, , , , , , , , , , , , , , , , ,

Introduction

Do you know what day of the week was the day you were born? Monday or maybe Saturday? Well, perhaps you know that. Everybody know the day he’s born on, but do you know what day was the 31st January 1883? No? Well, there must be some method to determine any day in any century.

We know that 2012 started at Sunday. After we know that it’s easy to determine what day is the 2nd of January. It should be Monday. But things get a little more complex if we try to guess some date distant from January the 1st. Indeed 1st of Jan was on Sunday, but what day is 9th of May the same year. This is far more difficult to say. Of course we can go with a brute force approach and count from 1/Jan till 9/May, but that is quite slow and error prone.

Following Days
If 1st of January is Sunday the most logical thing to happen is 2nd of January to be Monday

So what we’ll do if we have to code a program that answers this question. The most easier way is to use a library. Almost every major library has built-in functions that can answer what day is on a given date. Such are date() in PHP or getDate() in JavaScript. But the question remains. How these library functions know the answer and how can we code such library function if our library doesn’t support such functionality?

There must be some algorithm to help us. Continue reading Computer Algorithms: How to Determine the Day of the Week →

Computer Algorithms: Boyer-Moore String Searching

algorithms, , , , , , , , , , , , , , , , , ,

Introduction

Have you ever asked yourself which is the algorithm used to find a word after clicking Ctrl+F and typing something? Well I guess you know the answer from the title, but in this article you’ll find out how exactly this is done.

As we saw from the Morris-Pratt string searching we don’t need to compare the text and the pattern character by character. Some comparisons can be skipped in order to improve the performance of the string searching. Indeed the brute force string searching and the Rabin-Karp algorithm are quite slow only because they compare the pattern and the text character by character.

In the other hand the Morris-Pratt algorithm is a very good improvement of the brute force string searching, but the question remains. Is there any algorithm that is faster than Morris-Pratt – is there any way to skip more comparisons and to move the pattern faster.

It’s clear that if we have to find whether a single character is contained into a text we need at least “n” steps, where n is the length of the text. Once we have to find whether a pattern with the length of “m” is contained into a text with length of “n” the case is getting a little more complex.

However the answer is that there is such algorithm that is faster and more suitable than Morris-Pratt. This is the Boyer-Moore string searching.

Overview

Boyer-Moore is an algorithm that improves the performance of pattern searching into a text by considering some observations. It is defined in 1977 by Robert S. Boyer and J Strother Moore and it consist of some specific features.

First of all this algorithm starts comparing the pattern from the leftmost part of text and moves it to the right, as on the picture below.

Boyer-Moore Shifting Direction
In Boyer-Moore the pattern is shifted from left to right!
Continue reading Computer Algorithms: Boyer-Moore String Searching →