Tag Archive for 'php algorithms'

PHP Recursion: The Fundamentals

recursionNailing Down Recursion

On more than one occasion on this blog, I’ve wandered off into the land of operations to examine recursion. In looking over past recursion posts and recursion discussions on the Web, I found a lot of bad information, partial information and helpful information. Likewise, I consulted some programming books, especially Robert Sedgewick’s and Kevin Wayne’s 2011 edition (4th) of Algorithms. Also, I found a great and detailed article on recursion by David Matuszek. Robert Sedgewick and Kevin Wayne are professors at Princeton and David Matuszek is a professor at the University of Pennsylvania. Not surprisingly, their focus is on larger conceptual and mathematical issues surrounding computer programming and the role that recursion plays in that context.

However, I also wanted to get a non-academic view from developers, and I found two very good posts on Martin Fowler’s Refactoring site by Dave Whipp and Ivan Mitrovic. For those of you not familiar with Martin Fowler, he’s written books and articles on computing and is one of the founders of the Agile Movement in program development. He is a primary consultant to businesses and institutions who want to optimize their programs for efficiency and effectiveness. Among PHP developers the Agile approach to programming is quite popular.

What is Recursion?

In a nutshell,

Recursion in computer programming occurs when a method calls itself.

While that definition is a starting point, a more detailed and useful one is provided by Sedewick and Wayne. They spell out three features in a good recursive method:

  1. A recursion begins with a conditional statement with a return. This is called the base case (or halting case), and it supplies the criterion which will stop the recursive operation.
  2. Recursive calls to sub-problems converge to the base case. Each call must bring the values in use closer to the halting conditions.
  3. Called sub-problems should not overlap.

You can find a lot of discussions and debate about the definition and use of recursive methods in programming, but the fundamental fact remains that recursion is one of the central ideas in computer science. As a professional programmer, you need to know about recursion and you should use it. This does not mean you have to use it all the time, but you need to understand what you can do with it and its limitations and advantages. Start off with the following implementations and download the code:
PlayDownload

In PHP and other computer programs, recursion and the need for it arise all the time. So you should have some sense of how to use it and when. Like other computing concepts, you may not use it all the time, but when you need it, you really need it.

World’s Easiest Recursive Function

To get started we’ll look at a simple recursive call. It is a version of what kids do when you take them on a trip. (And what you did when you were a kid on a trip…) You’d ask the reasonable question,

Are we there yet?

If you kept calling the same query as soon as you’d received a negative response, it has recursive-like qualities. The “No!” is the base case, and the car moving to the objective (“there”) is the change that occurs between each call to the query, “Are we there yet?”

//Recursion.php
< ?php
class Recursion
{
    private $ask="<span style='font-family:sans-serif;'>Are we there yet?<br />";
    private $answer="No!<br />";
    private $counter=0;
    public function __construct()
    {
        $this->thereYet();
    }
    private function thereYet()
    {
        //Base case (also called 'halting case')
        if($this->counter < =10)
        {
            echo $this->ask;
            echo $this->answer;
            $this->counter++;
            //Recursive call
            return $this->thereYet();
        }
        else
        {
            echo "<p></p>" . $this->ask;
            $this->answer="<strong><em>Yes! We are there!</em></strong>" ;
            echo $this->answer;
        }
    } 
}
?>

The return value calls for a recursive event inside the thereYet() method. With each call, the counter variable’s value moves toward convergence with the base case. After 10 calls, the counter variable exceeds the base case and no more self-calls are made by the thereYet() method.

While that example could be handled by iteration in a loop; it provides another way to accomplish a task. It’s easy to understand and meets the criteria set up for recursion. (Click below to see more.)
Continue reading ‘PHP Recursion: The Fundamentals’

Share

PHP OOP & Algorithms II: What to Use

algorithm2What Are the Guidelines?

Like just about everything else in computing, there’s a certain amount of empirical testing. More importantly, however, are the general principles derived by both empirical and mathematical calculations. In the first post on algorithms and OOP, you saw an example that showed how many operations were necessary to find “Waldo,” a name near the end of an array of over 1,000 elements. One used a straight up loop and the other a binary search algorithm. Both were pretty quick; in fact it was hard to tell the difference in speed, and without seeing the little dots, you would not have seen the number of iterations required in each algorithm. You also saw that quadratic algorithms grew at rates that can quickly reach a point where processing comes to a crawl.

A Table Guide

Table 1 is derived from the 4th Edition of Algorithms (2011, p. 187) by Robert Sedgewick and Kevin Wayne. I’ve summarized it further. (Examples in the book are all in Java.) It is a rough but useful guide, and I have found it handy for a quick look-up.

Table 1: Algorithm Order of Growth

NameOrder of GrowthDescriptionExample
 constant 1statement   2 + 7
 logarithmic log Ndivide in half   binary search
 linear Nloop   find max
 linearithmic N log Ndivide & conquer   mergesort
quadratic N²double loop   check all pairs
cubic N³triple loop   check all triples
exponential2Nexhaustive search   check all subsets

Fortunately, most of the time our algorithms are pretty simple statements, single loops or recursive calls. All of the models in green are algorithms we should try and stick with. You should avoid the reds ones if possible.

What About Recursive Algorithms?

Those who love to profess a little knowledge like to say that recursion is slower than a loop. As indicated, we really don’t want to end up paying attention to small costs. Recursion is important, and using recursive algorithms is cost effective, especially since the difference in running time is negligible.

We often use recursive implementations of methods because they can lead to compact, elegant code that is easier to understand than a corresponding implementation that does not use recursion.
~Robert Sedgewick and Kevin Wayne

(A recursive example of a logarithmic algorithm is not included here, but you can find a recursive binary search in the post on binary searches.) What we need to pay attention to are the order-of-growth issues. For example, quadratic algorithms on the lower end of the scale are not really that bad, but when the N increases at a squaring rate, it’s easy to run into problems. So if you avoid quadratic algorithms, you’ll be better off when the N increases. On the other hand, whether you’re using a recursive method for a binary search, you’ll not see that much of a difference as the N increases compared to non-recursive methods.

A Strategy Design Pattern Handles Algorithms

This blog has lots of posts detailing the Strategy design patterns; so if you’re not familiar with a PHP implementation of the Strategy pattern, you might want to take a quick look at the code. To get started, play the example and download the code.
PlayDownload

In the last post on algorithms, the program used a Factory Method pattern to produce an array, and in this post, the same pattern is used and two additional array products have been added. However, instead of having the algorithm classes be more or less on their own, all of the algorithm classes have been organized into a Strategy design pattern. Figure 1 is a file diagram of the objects in the application:

Figure 1: Object groupings with Strategy and Factory Method patterns

Figure 1: Object groupings with Strategy and Factory Method patterns

If Figure 1 looks a bit daunting, it is only three object groupings. The HTML and PHP Client make a request for one of seven algorithm implementations through a Strategy pattern. The concrete strategy implementations all get their data from a “data factory.” Figure 2 provides an easier (and more accurate) way to understand what’s going on.

Figure 2: Overview of the main purposes of the objects.

Figure 2: Overview of the main purposes of the objects.

So instead of having a complex task, you only have three groupings of classes and interfaces: Those making a request (Clients), those executing operations on data (Algorithms) organized as a Strategy pattern and the data itself organized with a Factory Method (Data.)

Figure 2 shows a simplified version of what the program does. The Context class considerably eases the requesting process. The HTML UI passes the name of the requested concrete strategy to the Client, and the Client uses the name as a parameter in a Context method. The Client is not bound to any of the concrete strategies because the strategy classes are handled by a Context object and method. (Click continue to see the PHP implementations of the algorithms.)
Continue reading ‘PHP OOP & Algorithms II: What to Use’

Share

PHP OOP & Algorithms I: An Introduction

quadAvoiding the Misinformed

Programmers often spend more time un-doing bad information than using good information. One of the comments that set me off recently was someone “explaining” to others on a blog why PHP was not an object oriented language. Then he continued to blather on about the difference between compiled and interpreted languages. Whether or not a language is compiled or not has nothing to do with whether or not it is an object oriented language. Having interfaces, classes and communication between objects are the key criteria of an OOP language, and certainly since PHP5 has been a full-fledged OOP language. (We PHPers should not feel singled out because I recently saw post where a Java programmer pronounced that neither Python nor Perl were OOP, and she was “informed” otherwise by irate Python programmers. Perl has been OOP since V5.) So here I am again wasting time grumbling about people who don’t know what they’re talking about.

Instead of frothing at the mouth over the misinformed, I decided to spent more time with the well-informed. To renew my acquaintance with algorithms I began reading Algorithms 4th Ed. (2011) by Sedgewick and Wayne. Quickly, I learned some very basic truths about algorithms that had been only vaguely floating around in my head. First and foremost are the following:

Bad programmers worry about the code.
Good programmers worry about data structures and their relationships.
Linus Torvalds (Creator of Linux)

Since we’ve been spending time on this blog acting like good programmers, that was reassuring. In this post, I’d like to look at two things that are important for developing algorithms: 1) What to count as a “cost” in developing algorithms, and 2) Identifying good and bad algorithmic models. First, though, play and download the example. Using two different algorithms, a logarithmic and a linear (both pretty good ones), I’ve added “dots” to loop iterations to visually demonstrate the difference between a logarithmic algorithm (binary search) and a linear algorithm (loop). The “expense” of the algorithm can be seen in the number of dots generated.
PlayDownload

The example is a pretty simple one. However, since this blog is about PHP Design Patterns, I added a little Factory Method. The two algorithm classes act like clients making requests through the factory for a big string array with over 1,000 first names. Figure 1 shows the file diagram:

Figure 1: File diagram for use of Factory Method by two algorithm clients.

Figure 1: File diagram for use of Factory Method by two algorithm clients.

In looking at the file diagram, you may be thinking, “Why didn’t you use a Strategy pattern coupled with that Factory Method?” I thought about it, but then decided you could do it yourself. (Why should I have all the fun?)

Lesson 1: Leave the Bag of Pennies

The first lesson I learned in Bank Robbery 101 was to leave the bag of pennies. They’re just not worth it. Speed is everything in a bank robbery, and so you pay attention to how to get the most with the least time. The same thing applies to analyzing algorithms. For example, an object (as compared to an integer, boolean or string) has an overhead of 16 bytes. I have actually seen posts intoning, “objects are expensive…” Just to be clear,

Objects are not expensive. Iterations are expensive, quadratic algorithms are expensive.

In evaluating an algorithm you need to see how many operations must be completed or the size and nature of the N. An N made of up strings is different than an N made up of Booleans or integers. A quadratic (N²) and cubic (N³) algorithm are among the worst. They’re gobbling up kilo- or megabytes, and so that 16 bytes seems pretty silly to worry about. So instead of seeing an algorithm weight expressed as N² + 84 bytes, you’ll just see it expressed as ~N². (When you see a ~ (tilde) in an algorithm, it denotes ‘approximately.’) Another way of understanding the ~ is to note, They left the bag of pennies.

Lesson 2: Watch out for Nested Loops; they’re Quadratic!

I’ve never liked nested loops, and while I admit that I’ve used them before, I just didn’t like them. They were hard to unwind and refactor, and they always seemed to put a hiccup in the run-time. Now I know why I don’t like them; they’re quadratic.

Quadratic algorithms have the following feature: When the N doubles, the running time increases fourfold.

An easy way to understand the problem with quadradics is to consider a simple matrix or table. Suppose you start with a table of 5 rows and 5 columns. You would create 5² cells—25 cells. Now if you double the number to 10, 10² cells = 100. That’s 4 x 25. Double that 10 to 20 and your have 20² or 400. A nested loop has that same quality as your N increases. If both your inner and outer loop N increases, you’re heading for a massive slowdown.

Algorithms, OOP and Design Patterns are Mutually Exclusive

An important fact to remember is that good algorithms do not guarantee good OOP. Likewise, good OOP does not mean good algorithms. Good algorithms make your code execute more efficiently and effectively. Good OOP makes your programs easier to reuse, update, share and change. Using them together is the ultimate goal of a great program.

Share