Computers and Algorithms are highly sophisticated machines.

They are capable of processing, analyzing, and storing huge amounts of data faster than a human being.

While the brain and nervous system are much more complex than computers, they cannot match up to the high computer speeds that have been created over the years. However, this does not mean that computers don’t have their limitations. 

They have boundaries and parameters that cannot be surpassed unless a more sophisticated technology is developed. You can try to figure out the general power of a computer by checking out its processing speed which is usually measured in GHz. For instance, a 2GHz processor can handle 2 billion basic instructions every second, with a basic instruction being something like assigning values or adding numbers in the memory’s system.

Codes like Java or Python get transformed into information that the machine can read before executing. And a small section of human code can turn into several instructions for the computer to handle. Therefore, when a computer is reading several codes and handling different programs and processes simultaneously, machines need to be effective when it comes to data processing and carrying out a wide range of functions. Algorithmic complexity kicks in here.

What is algorithmic complexity?

This topic might sound complicated. However, with the help of write my dissertation service,  algorithmic complexity is quite simple. Algorithms are a list or sequence of steps. Tasks become more complex when more steps are involved. In most cases, a short algorithm will be less complex than a long one. So, how do you evaluate algorithmic complexity? In general, there are several ways we can go about this. We can write an algorithm and try to figure out the number of steps or sequences needed for it to run.

How do you go about computing what makes up for a single step? Is it a single line of code? If this were the case, we would just look at the code and count the lines to figure out the algorithm complexity. But this can lead to several problems since code can be written in different ways. And long sections can be easily condensed from eight lines to two or three lines in different situations.

Fortunately, you don’t have to count the lines of code to calculate algorithmic complexity. What of machine code lines? Would you manage to count these and calculate the complexity of an algorithm? This is quite possible on paper. However, it is one of the most difficult things to do. Different computer systems have a wide range of machine codes that they can easily read. And each is represented differently thus resulting in anomalies that give no true evaluations or conclusions whatsoever.

What counts as an algorithm step?

The most common method that people use to calculate algorithmic complexity is to look at code blocks which are sections of code represented by one operation relevant to input data. It can be a single line of code or multiple lines which run from start to finish without loops. This is one of the best ways to define algorithm steps. However, you can go a step ahead to avoid looking at code. Experienced programmers can look at code and translate it back effectively into algorithm form by analyzing individual blocks. And then view the algorithm is simple, basic English that can be understood. At this point, it’s quite easy for programmers or experts at best essay writing services to look at the algorithm and figure out how complex it is.

Big O Notation

Algorithm and the Big O notation is something most junior students and industry experts have a hard time understanding. However, it’s not as difficult or complex as it sounds. The Big O Notation shows us the level of scaling of an algorithm. In most cases, the algorithmic complexity and the Big O have a similar definition.

Motivation

There are several tools to measure the speed of a program. Programs known as profilers are used to measure running time in seconds and milliseconds. They can help you optimize your code by taking note of where the bottlenecks are. While this is an important tool, it is not relevant when it comes to calculating the complexity of algorithms. 

Algorithm complexity is designed to compare multiple algorithms at the idea level – ignoring other small details such as the hardware the algorithm runs on, the programming language used to implement, instructions given to the CPU. You should compare algorithms for what they are: Ideas of the computation of something. Counting milliseconds won’t help you in this. A bad algorithm written using a low-level language may run quicker than a good algorithm written with high-level languages such as Ruby or Python. So, what does a better algorithm mean?

Since algorithms are programs that perform a wide range of computations; and not other tasks that computers usually do such as user input or networking tasks, complexity analysis allows you to measure the speed of a program as it performs several computations. Some of the common examples that are pure computation include numerical operations such as multiplication and addition, determining the artificial intelligence character path, searching a database that fits in RAM for a particular value, or running an expression pattern matching on a string. Computation is quite common in computer programs.

Complexity analysis is a tool that will help you explain the behavior of an algorithm as the input grows. If it is given a different input, how does the algorithm behave? If an algorithm takes one second to run for an input size of 1000, how will it behave if the input size is doubled? Will it run twice as fast or four times slower? In the programming world, it is important since it enables you to predict the behavior of your algorithm when the input data becomes large. 

For example, if you’ve developed an algorithm web application that operates smoothly with one thousand users, using algorithm complexity analysis will help you have a clear idea of what happens when 2000 users get on board. When it comes to algorithmic complexity, complexity analysis allows you to understand how long a code runs for large test cases used to test the correctness of a program. Therefore, if you’ve measured the program’s behavior for small input, you can get a clear idea of how it behaves when the input becomes large. Let’s talk about how to find the maximum element in an array.

Calculating instructions

In this post, we’ll use several programming languages. However, don’t fret if you don’t understand a particular language. Since you know the basics of programming, you should easily read examples without problems even if you aren’t familiar with the programming language that we’ll use here since they will be simple. And we won’t use complex language features. As a student participating in algorithm competitions, you are probably familiar with C++. Therefore, you shouldn’t have an issue following through. You should work on C++ exercises to develop and improve your skills. An array’s maximum element can be analyzed using a piece of code.

The first thing you need to do is count the number of important instructions executed by the piece of code. This happens once most of the time. Therefore, it won’t be essential as we gradually develop our plan. While analyzing this piece, you need to break it up into instructions that can be executed directly by the CPU. We’ll assume that your processor can execute the instructions for complexity algorithms that we’ll discuss:

Assign values to variables

  • Look up the value of an element in an array
  • Compare two values
  • Increase a value
  • Compute basic arithmetic such as multiplication and addition

Given an array A of size n:

var M = A [ 0];

for (var i = 0; i < n; ++i) {

if (A [ i] >= M) {

M = A [ i];

}

}

Our first line is:

var M = A [ 0];

This line has two instructions: One for assigning a value to M and one for looking up A [0]. We are assuming that n is at least 1. These instructions are required by an algorithm regardless of the n value. The loop initialization code also needs to run. With this, we have two extra instructions namely: a comparison and an assignment:

 i = 0

I < n

Before the loop iteration, these will run. After a loop iteration, two more instructions are needed to run, a comparison that ensures that we stay on the loop and an increment of i:

++i

i < n

If you ignore the loop body, the instructions that this algorithm will need are 4 +2n. This is 4 instructions at the beginning of a loop and 2 instructions at the end of every iteration where an n is present. You can easily define the mathematical function f(n), given that n, gives you a set of instructions that the algorithm needs. For an empty body, its f(n) = 4 + 2n.

Analyzing worst-case

When you look at the body, we have a comparison and an array lookup operation that happens every time:

If (A [i] >=M) {…

These are two instructions. But the body may run or fail to run depending on the existing array values. In case it is that A [ i ] > = M, you will run the extra instructions – an array and lookup assignment:

 M = A [ I ]

You cannot define an f (n) easily, because the number of instructions doesn’t depend on n solely, but also on your input. For instance, A = [1, 2, 3, 4], the algorithm will require extra instructions compared to a = [4, 3, 2, 1]. As you analyze algorithms, you should think of the worst-case scenario. What’s the nastiest thing that can happen to our algorithm? When does an algorithm need several instructions to complete? In our case, when you have an increasing order array such as A = [1, 2, 3, 4].

In this case, M should be replaced each time so that it generates extra instructions. Computer experts have a name for this: worst-case analysis. This only involves cases when we consider ourselves unlucky.

Asymptotic behavior

With such a function, you have a clear idea of how fast your algorithm is. However, you won’t need to go through the tiring task of counting the number of instructions in your program. The number of instructions that you need for each programming language statement is dependent on the compiler programming language and the instruction set of the CPU. We’ll now run a function through a filter to discover some of the minor details that most programmers ignore.

In the function 6n + 4, there are two terms: 6n + 4. When analyzing complexity, we prioritize what happens to the function that counts instructions as the input (n) grows. This goes hand in hand with the worst-case scenario that we’ve discussed in the previous section. 

We are interested in the manner in which an algorithm behaves when treated poorly. When it’s required to perform a difficult task. This aspect is extremely useful when comparing different algorithms. If an algorithm manages to beat other algorithms with a large input, it’s probably true that the algorithm will win when given a small and easy input.

Logarithms

If you understand this section, feel free to skip it. The majority of students don’t understand the concept of logarithms. A logarithm is simply an operation applied to a specific number that reduces its size. It’s similar to square root in many ways. Square roots are the inverse of logarithms. Logarithms will help you denote problems using notations. After reading this article, you need to practice logarithms.

Conclusion

Now that you understand the basics of algorithm complexity analysis, it’s time to start developing simple and fast programs that will help you optimize your efforts on things that truly matter instead of minor unimportant things. This will boost your productivity and performance in the long run. We hope that you’ll put this knowledge into practical use.

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