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.

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.

When you want to null the sliders on an page, using jQuery UI Slider, somehow the following code works in any browser except IE.

// wrong
$('.my-slider').slider('value', 0);

I say somehow, because according to the documentation this code is simply wrong. It can be used to get the value of the slider as it is a getter. You can null the value with the following snippet.

// correct
$('.my-slider').slider('option', 'value', 0);

IE Fix

Another way to null the value is with an anonymous object.

// correct
$('.my-slider').slider({ value: 0 });

This works on every browser including IE.

If you have read some preformance tests on JavaScript loops, you may have heard that “while” is faster than “for”. However the question is how faster is “while”? Here are some results, but first let’s take a look on the JavaScript code.

The for experiment

console.time('for');
for (var i = 0; i < 10000000; i++) {
	i / 2;
}
console.timeEnd('for');

The while experiment

console.time('while');
var i = 0;
while (i++ < 10000000) {
	i / 2;
}
console.timeEnd('while');

Note - these tests are performed and measured with Firebug on Firefox.

Results

It's a fact, that you'll get different results as many times as you run this snippet. It depends also on the enviroment and software/hardware specs. That is why I performed them 10 times and then I took the average value. Here are the values of my performance tests. Note that both for and while perform 10,000,000 iterations.

And the Winner Is

While is the winner with an average result of 83.5 milliseconds, while "for" result is 88 average milliseconds.

As the diagram bellow shows, the while loop is slightly faster. However we should be aware that these performance gains are significant for large number of iterations!

The expression “linear search in sorted lists” itself sounds strange. Why should we use this algorithm for sorted lists when there are lots of other algorithms that are far more effective? As I mentioned in

my previous post the sequential search is very ineffective in most of the cases and it is primary used for unordered lists. Indeed sometimes it is more useful first to sort the data and then use a faster algorithm like the binary search. On the other hand the analysis shows that for lists with less than ten items the linear search is much faster than the binary search. Although, for instance, binary search is more effective on sorted lists, sequential search can be a better solution in some specific cases with minor changes. The problem is that when developers hear the expression “sorted list” they directly choose an algorithm different from the linear search. Perhaps the problem lays in the way we understand what an ordered list is?

What is a sorted list?

We used to think that this list (1, 1, 2, 3, 5, 8, 13) is sorted. Actually we think so because it is … sorted, but the list (3, 13, 1, 3, 3.14, 1.5, -1) is also sorted, except that we don’t know how. Thus we can think that any array is sorted, although it is not always obvious how. There are basically two cases when sequential search can be very useful. First when the list is very short or when we know in advance that there are some values that are very frequently searched. Let’s say we have a very large list, with hundreds of thousands of items, but actually most of the searches in that list always find the same ten values. This additional information tells us that using a binary search will be quite ineffective in this case. A possible approach, of course, is to place those values at the front of the list and to perform a sequential search.

Self-Organization

Self-organization practically means that every time we search and find the desired value, we somehow change the list so the next search will be far more effective. There are basically two approaches to do that.

1. To move the item one position forward to the front of the list;
2. To move the item directly at the front of the list;

Of course it depends on your case which approach you’ll choose, but it is assumed that the second option, the one that we choose to move the item directly at the front of the list, is better. Indeed if we choose the first option and the list is (…, 24, 31) after constantly searching for those two values the array will be changing from (…, 24, 31) to (…, 31, 24) and once again to (…, 24, 31) and so on and so on. Thus a better solution is to move the desired item directly to the front of the list. Now if we look for the value of “5” in the list

(1, 2, 4, …, 5, …, 398) it will become (5, 1, 2, …, 398) after the value is found. By choosing this approach we can be sure that as the number of searches increases, the most frequently searched values are placed at the front of the list. Now the sequential search is quite a good solution! Here’s an example of sequential search from my previous article. The only change is that after we find the desired value we need to move it to the front of the list.

Application

Using sequential search in sorted lists can be very useful and fast, the only thing is that we need to know in advance that there are some values that are frequently searched. A typical example of this case is the contact list on your phone. Perhaps you have lots of names in there, but most of the times you search in it is to find your best friends’ and family phone numbers. That is why most of the cell phone manufacturers add to their phones the ability to predefine shortcut keys for the most frequently dialed numbers. Here’s another use case. Let’s say that we have the same scenario as in my previous

post, where username/name pairs are stored into a CSV file. We can fetch those values in a PHP array.

Every time a user enters the site we search for his name by his username and a welcome message is displayed. We know that some users enter the site very frequently while others do that once per month so we cannot only perform a sequential search but also we can use self-organization for the array and change the CSV file at the end.

The result is:

Hello, Darth Vader 
Found after 5 iterations!
Hello, Darth Vader
Found after 1 iterations!

Now every time Darth Vader tries to sign in, you won’t bother him to wait a lot for sure. However I bet nobody uses CSV files to store such information, but this is only an example.

This is the easiest to implement and the most frequently used search algorithm in practice. Unfortunately the sequential search is also the most ineffective searching algorithm. However, it is so commonly used that it is appropriate to consider several ways to optimize it. In general the sequential search, also called linear search, is the method of consecutively check every value in a list until we find the desired one.

Basic Implementation

The most natural approach is to loop through the list until we find the desired value. Here’s an implementation on PHP using FOR loop, something that can be easily written into any other computer language.

This is really the most ineffective implementation. There are two big mistakes in this code. First of all we calculate the length of the list on every iteration of the array, and secondly after we find the desired element, we don’t break the loop, but continue to loop through the array.

Yes, if the element is repeated without the “break” we can find its last occurrence, but if not the loop will iterate over the end of the array with no practical value.

Optimization of the forward sequential search

… and javascript:

Even with this little optimization the algorithm remains ineffective. As we can see, on every iteration we have two conditional expressions. First we check whether we’ve reached the end of the list, and then we check whether the current element equals to the searched element. So the question is can we reduce the number of the conditional expressions?

Searching in reverse order

Yes, we can reduce the number of comparison instructions from the forward approach of the linear search algorithm by using reverse order searching. Although it seems to be pretty much the same by reversing the order of the search we can discard one of the conditional expressions.

Note that we need to adjust index because of $index—expression.

Indeed here we have only one conditional expression, but the problem is that this implementation is correct ONLY when the element exists in the list, which is not always true. If the element doesn’t appears into the list, then this code can lead to an infinite loop. OK, but how can we stop the loop even when the list doesn’t contain the desired value? The answer is, by adding the searched value to the list.

Sentinel

The above problem can be solved by inserting the desired item as a sentinel value. Thus we’re sure that the list contains the value, so the loop will stop for sure even if at the beginning the value didn’t appear to be part of the list.

This approach can be used to overcome the problem of the reverse linear search approach from the previous section.

Complexity

As I said at the beginning of this post this is one of the most ineffective searching algorithms. Of course the best case is when the searched value is at the very beginning of the list. Thus on the first comparison we can find it. On the other hand the worst case is when the element is located at the very end of the list. Assuming that we don’t know where the element is and the possibility to be anywhere in the list is absolutely equal, then the complexity of this algorithm is O(n).

Different cases

We must remember, however, that the algorithm’s complexity can vary depending on whether the element occurs once.

Is it so ineffective?

Sequential search can be very slow compared to binary search on an ordered list. But actually this is not quite true. Sequential search can be faster than binary search for small arrays, but it is assumed that for n < 8 the sequential search is faster.

Application

The linear search is really very simple to implement and most web developers go to the forward implementation, which is the most ineffective one. On the other hand this algorithm is quite useful when we search in an unordered list. Yes, searching in an ordered list is something that can dramatically change the search algorithm. Actually searching and sorting algorithms are often used together.

A typical case is pulling something from a database, usually in form of a list and then search for some value in it. Unfortunately in most of the cases the database orders the returned result set and yet most of the developers perform a consecutive search over the list. Yet again when the list is ordered it is better to use binary search instead of sequential search.
Let’s say we have a CSV file containing the usernames and the names of our users.

Username,Name
jamesbond007,James Bond
jsmith,John Smith
...

Now we fetch these values into an array.

// work case
$arr = array(
    array('name' => 'James Bond', 'username' => 'jamesbond007'),
    array('name' => 'John Smith', 'username' => 'jsmith')
);

Now using sequential search …

// using a sentinel
$x = 'jsmith';
$arr[] = array('username' => $x, 'name' => '');
$index = 0;

while ($arr[$index++]['username'] != $x);

if ($index < count($arr)) {
    echo "Hello, {$arr[$index-1]['name']}";
} else {
    echo "Hi, guest!";
}

As you know sometimes JavaScript can be weird. Let’s see the following example and let’s try to answer the question: does “undefined” equals “undefined”. What do I mean?

First take a look at the following code.

var a;
var b = undefined;

// alerts "false"
alert(b == a);

Both a and b are undefined, but they are NOT equal. Now let’s see where it can become a problem.

We have an object with one member variable that is not defined.

var f1 = function()
{
	this.myvar;
};

var obj1 = new f1();

Now you’d like to know whether the object “b” has the property “myvar”. There are lots of examples online, but what’s the right way?

Wrong

Appearantly the value of b.myvar should be undefined.

alert(obj1.myvar); // undefined

But that doesn’t mean that it is the same value if the object was declared this way:

var f2 = function()
{
	this.myvar = undefined;
};

var obj2 = new f2();

This is a completely different thing. Why? In the first case if we try to check whether the property myvar belongs to obj1 we can go wrong with:

if (obj1.myvar) // false

This is false simply because obj1.myvar is undefined

if (obj2.myvar) // false

Again is false, because obj2.myvar is undefined.

The same thing happen when using the other “famous” approach.

if (typeof obj1['myvar'] != 'undefined') // false
if (typeof obj2['myvar'] != 'undefined') // false

Clearly this is not the right answer, becase even it is undefined myvar belongs to obj2!

Right

Now let’s try with hasOwnProperty supported by almost every modern browser.

if (obj1.hasOwnProperty('myvar')); // false
if (obj2.hasOwnProperty('myvar')); // true

Here you can see that when the member variable is explicitly defined as “undefined” the object “has the property” and hasOwnProperty returns true. Yet again, “undefined” does not always equal to “undefined”.

var a = 5;

function f() { return a; } // returns 5

Thus f will return the value of “a” which equals to 5. You can also change the value of the global variable in the function body.

var a = 5;
function f() { a = 10; return a; }
console.log(a); // equals to 10

Now after we call the function f the value of “a” will equal to 10. This is because we reference the global variable “a” into the function body without using the keyword “var”. This means that if you put the “var” keyword the variable “a” inside the function body is no longer the same variable as the variable defined outside the body. It becames “local” and it’s visible only inside the function.

Objects & JavaScript

Actually in JavaScript functions can be also objects. Any variable defined with the “var” keyword becomes private, because it’s only visible inside the function body, while by using the “this” keyword we can define global variables visible outside the function body. Let’s see na example.

var f = function() 
{
	var a = 10;
	this.b = 5;
}

var myfunc = new f();
myfunc.a; // is undefined because a is private
myfunc.b; // equals to 5 because b is public

Public and Private Methods in JavaScript

As you may know in JS you can define functions inside other functions. Actually this is how you can define classes in JavaScript.

var myClass = function() 
{
	var f = function() {
		return 10;
	}
}

But this in fact defines a local (private) function into myClass and we cannot access it from the outside world. Here’s a fully functional class with one public and one private method.

var myClass = function()
{
	// public method
	this.a = function() {
		return 10;
	}
	
	// private method
	var b = function() {
		return 5;
	}
}

var c = new myClass();
c.a(); // will return 10
c.b(); // is undefined, because is private

Accessing Private Methods from Public Methods

Easily we can access private methods from public ones.

var myClass = function()
{
	// public method
	this.a = function() {
		return b();
	}
	
	// private method
	var b = function() {
		return 5;
	}
}

However accessing public mtehods in private ones is more difficult. That’s because we cannot use “this” into the private methods, just because “this” refers to the private method
itself and not to the global method, which is “myClass” in our case.

var myClass = function()
{
	// public method
	this.a = function() {
		return 10;
	}
	
	// private method
	var b = function() {
		return this.a(); // this will result in an error
	}
}

Accessing Public Methods from Private Methods

To access public methods in private methods you need to define a variable that points to the global “this” object.

var myClass = function()
{
	var self = this; 
	
	// public method
	this.a = function() {
		return 10;
	}
	
	// private method
	var b = function() {
		return self.a(); // this will return 10
	}
}

This variable gives you the bridge between private methods and global “this” pointer.