Algorithm Complexity and Data Structure: Types of Time Complexity

An algorithm is a limited set of rules that, when followed, carry out a certain task. It is not language-specific; we can represent instructions using symbols and any language. An essential component of computational algorithm complexity is algorithm analysis. The complexity theory provides the theoretical estimations for the resources required by an algorithm to perform any computational task. Analyzing an algorithm involves determining how well it can solve problems regarding the amount of time and space needed. For a better understanding of algorithm complexity, enroll in a Professional Certificate Program in Data Science and Business Analytics. 

What is Complexity?

The number of steps an algorithm needs to take to solve a particular problem is measured by the phrase algorithm complexity. It assesses an algorithm’s count of operations about the volume of input data. Instead, then counting the exact steps, the order of the operations is always taken into consideration when determining the complexity. 

Time complexity order is characterized by the number of times an algorithm must be executed depending on the size of the input. Because operating systems, programming languages, and processor power are all taken into account, time complexity is not a measurement of how long it takes to execute a certain method. 

What Are Big-O Notations?

O(f) notation, often known as “Big O” notation or asymptotic notation, is a way of expressing how complex an algorithm is. The size of the function, in this case, denoted by the letter f, matches the size of the input data. The procedure is expressed as a function of the amount of the input data, and the difficulty of the asymptotic calculation O(f) defines the order in which resources, such as memory, CPU time, etc., are utilized by the algorithm. 

• Big-O notation is a theoretical tool for analyzing an algorithm’s performance and complexity. 
• Big-O notation analyses an algorithm’s worst-case behavior or its upper bound of performance. 
• Asymptotic algorithm behavior, or how a technique performs as the size of the input increases drastically, is also considered by Big-O notation. 
• Based on the size of the input data (CPU time, RAM, etc.), the complexity of the computation asymptote O(f) measures the order of resources used.

Let us now look at various types of time complexities.

Various Time Complexities

Five categories of time complexity cases exist:

• O(1) for Constant Time Complexity
• O(log n) is the logarithmic time complexity
• Time Complexity in Linear Form: O(n) 
• O(n log n) Complexity of Time
• O(n2) for Quadratic Time Complexity

Constant Time Complexity

The algorithm is considered to have O(1) complexity if the method’s execution time does not change and stays the same as the size of the input increases. The size of the input data has no bearing on the algorithm. A specific operation requires a predetermined number of steps to be completed, and this number is unaffected by the volume of the input data. 

Logarithmic Time Complexity

For each iteration (log n), a method with a logarithmic complexity (where n is enormous) splits the problem into manageable pieces. A specific procedure on N items requires log(N) steps to complete, with the logarithm base typically being 2. The logarithm base is typically disregarded because it has no bearing on the sequence of the operation count. 

Time Complexity in Linear Form 

An algorithm is deemed to have O complexity (n) when it runs n steps for input sizes n (where the constant is huge), and the amount of time required to complete the process varies linearly as the input size increases. It takes roughly the same number of steps to operate on N items as there are elements. A correlation between the number of components and the number of steps is known as linear complexity. 

Complexity of Time

The problem is broken down into manageable chunks for each iteration by an algorithm with an O (n log n) complexity (where n is really large), which then takes each of the smallest parts and stitches them back together (n log n). It requires N*log(N) steps to perform a specific operation on N objects. 

Quadratic Time Complexity 

An algorithm’s time varies, and a technique for input sizes n (where n is huge) conducts approximately twice as many steps (n2). According to the method’s description, complexity rises quadratically as input size rises (n2). It follows an order of N2 steps for a specific operation, where N is the size of the input information. Quadratic complexity is achieved when the quantity of steps is proportionate to the input data size.

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What Notations Are Used to Represent Algorithmic Complexity? 

The most common method for expressing algorithmic complexity is called Big-O notation. It provides a complexity upper bound and, as a result, represents the algorithm’s worst-case performance. Such a notation makes it simple to compare various algorithms because it shows how the method scales as the input size grows. The order of growth is another name for this.

Continuous runtime is represented by O (1), linear growth by O ( n ), logarithmic growth by O ( l o g n ), log-linear growth by O ( n l o g n ), quadratic growth by O ( n 2 ), exponential growth by O ( 2 n ), and factorial growth by O ( n! ).                                            

What Does It Mean To State the Best-Case, Worst-Case, and Average Time Complexity of Algorithms?

Consider the scenario of progressively looking for an item within a list of unsorted things. The item might be at the top of the list if we’re lucky. It might be the last thing on the list if we’re unfortunate. The first is referred to as best-case complexity, whereas the second is referred to as worst-case complexity. The complexity is O (1) if the searched item is always the first one and O (n) if the searched item is always the last one. We may also determine the average complexity, which will be O (n) in size. Worst-case complexity is typically meant when the word “complexity” is used. A Python Programming Bootcamp from upGrad is recommended to get an in-depth idea of the complexity of algorithms. 

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Why Should We Care About an Algorithm’s Performance When Processors Are Getting Faster, and Memories Are Getting Cheaper?

The real execution time, which varies on the processor speed, instructions set, disc speed, compiler, etc., is unimportant to complexity analysis. The identical algorithm will execute quicker in assembly than it will in Python. External factors include hardware, memory, and programming languages. Algorithm complexity refers to how an algorithm uses data to solve a certain problem. The “idea level” software design issue is at issue. 

It is conceivable for an inefficient algorithm to produce a speedy result when run on powerful hardware. Large input datasets, however, will highlight the hardware’s limits. Therefore, it is preferable to focus on algorithm optimization before considering hardware changes. 

Are There Techniques to Figure Out the Complexity of Algorithms? 

Instead of focusing on exact implementation times, we should consider how many high-level instructions there are about the amount of input. Iterating through the input in a single loop is linear. The algorithm is quadratic if there are loops inside of loops, with each successive loop iterating through the input. Whether the loops examine only alternate items or omit a predetermined number of things is irrelevant. Remember that scaling factors and constants are disregarded by complexity. 

Similar to a loop inside a loop, a loop after that is also quadratic because only the dominant term needs to be taken into account. If other actions inside the function do not depend on the input size, a function with recurrence that calls oneself n times is linear. However, a Fibonacci series implementation that is recursive is meant to be exponential. 

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If an Algorithm is Inefficient, Does That Mean We Can’t Use It? 

Algorithms with polynomial complexity of the order O (For c > 1, n c ) would be adequate. They can handle inputs of up to tens of thousands of objects. Anything exponential is probably only viable with inputs under twenty.

Algorithms with the complexity of O (n 2), like Quicksort, seldom encounter worst-case inputs and frequently follow (n l o g n) in actual use. In some circumstances, we can beforehand process the input to prevent worst-case outcomes. Similarly, we can choose less-than-ideal solutions to reduce complexity to polynomial time.

Since Algorithmic Complexity is About Algorithms, is it Relevant To Talk About Data Structures? 

Data structures merely serve as data storage, but when we manipulate them, algorithmic complexity is taken into account. The analysis must be done on processes like insertion, deletion, searching, and indexing. The goal is to choose the appropriate data structures to cut down on complexity. 

The Complexity of Well-known Sorting Algorithms

Bubble sort is perhaps the most straightforward sorting algorithm; however, it is inefficient because it is quadratic in most situations. Better options include Quicksort, Merge sort, Heapsort, and other algorithms having a log-linear complexity. The best-case complexity occurs if the list has already been sorted, in which case Bubble sort, Tim sort, Insertion sort, and Cube sort all finish in linear time.

The best and worst examples don’t happen very often. The average example is built using a model of input distribution that could also be a random sample. Selecting the most appropriate algorithm for a given task can be aided by analysis of these averages or abstract basic processes.

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The Complexity of Some Important Algorithms

Here are some examples of some important algorithms:

• (n log n) is the Fast Fourier Transform symbol.
• Using the Karatsuba algorithm, multiply two n-digit numbers: ( n 1.59 )
• Gaussian elimination: O (n 3), although the operands are O (x 2 n) bit complex.
• Using Euclid’s algorithm, GCD (a, b): ( ( l o g ( a b ) ) but O ( ( l o g ( a b ) )
• When the numbers a and b are big, the bit complexity is 2

Conclusion 

Performance varies between algorithms. Metrics for evaluating algorithm efficiency would be helpful because we always like to choose an efficient method. An algorithm’s efficiency in terms of the volume of data it must process is described by the complexity of the algorithm. The domain, as well as the range of this purpose, often have natural units. If you are interested in a career in programming and data structures and want to work with such complexities, enroll in a Data Analytics 360 Cornell Certificate Program from upGrad today.