Complexity Analysis of Data Structures & Algorithms

04 Jan 2023
Beginner
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Introduction

We have learned there are multiple ways to obtain a solution to a particular problem, and it is possible to design algorithms for each of these ways. Based on the criteria and understanding of the problem, a programmer chooses one of the ways to develop a program. However, this does not ensure the cost-efficiency and seamless adoption of the program in its ecosystem. Hence, a programmer should have a sufficient understanding of the performance of an algorithm regarding the resources consumed, the time required to run the algorithm, and the memory used up.

The complexity of an algorithm

In Computer science, computational complexity (analysis) is a unit of measurement of the number of resources used by the algorithm. Algorithmic analysis is a mechanism to evaluate the speed of execution of an algorithm, regardless of its language, technical platform, and hardware.

Complexity indicates the performance of an algorithm with an increase in the count of inputs, while algorithmic analysis emphasizes storage locations, number of disk operations, etc., in an algorithm. Computational analysis is an approximate estimation of the number of steps required to execute an algorithm. The lower the complexity of an algorithm, the faster its execution, irrespective of the volume of input data. The algorithm (program) may fail to execute or perform poorly if its complexity is high.

In other words, complexity is a mathematical expression expressing the performance of an algorithm. The function can be linear, exponential, logarithmic, quadratic, cubic, or a constant. Algorithms with higher running speeds either have fixed, logarithmic, or linear complexities independent of the amount of data consumption. Algorithms with exponential complexities are time-consuming and avoided.

Types of Complexities 

Algorithms play a crucial role in the study of data structures. Different users perceive a given problem in various aspects and design solutions accordingly. The complexity of algorithms is a comparative study of these algorithms that facilitates the programmer to choose the algorithm that executes faster and leverages resources.

Time Complexity

The Time Complexity is an indicative number that denotes the number of times an instruction executes. The function represented by (Big Theta (Ɵ)) is an indicator of the growth of an algorithm. It facilitates the knowledge of the time required to execute the algorithm for a particular volume of data.

The rate of growth Big Theta (Ɵ) calculates the minimum and the maximum number of inputs the algorithm can execute and computes the average time required for its execution. The fastest and the slowest times of an algorithm for its execution are its best and worst cases.

Space Complexity

The space complexity of an algorithm is the sum of the spaces required for its fixed elements and changeable components. 

The fixed component comprises data, variables, and constants used in the program, while the changeable space refers to the program size. Thus, space complexity is the total of its auxiliary and input memory storage.

During its execution, an algorithm uses Instruction space to store the instructions compiled by the compiler, environmental stack memory to stock the current values of the variable at the instance of interruption or a function, and the memory locations required based on the size of the variables and constants. Practically, computing space complexity considers only the instruction space consumed.

Why do we need to understand the Complexity?

The performance of an algorithm influences the execution of the program. Hence, understanding the complexity of an algorithm is significant. The complexity of an algorithm helps the user to

  • Utilize the algorithm with the optimized data resources.

  • Estimate the time taken by the algorithm to produce the results.

  • Ensure the results produced by the execution of the algorithm are valid and not outdated.

  • Calculate the space complexity or storage (memory required) for the algorithm.

Asymptotic Analysis

The term asymptote means evaluating the result for a specific set of values for a given limit. The asymptotic analysis (asymptotics) traces the operational capability of the algorithm for a range of input values that gradually tend to infinity. The objective of the asymptotic analysis is to increase the competence of the algorithm. The algorithmic analysis addresses the performance of the algorithm independent of its hardware.

Remember the following points that help to understand the algorithm -

  • Step, Control, Exit: The algorithm begins with a START and ends with an EXIT

  • Comments: Given in brackets to indicate the purpose of each step.

  • Variables: Used as counters or subscripts and are generally given in capitals.

  • I / O: READ is used for input and PRINT is used for output.

The asymptotic analysis considers the best case, with the algorithm running at the fastest speed (with the lowest time complexity). The worst case indicates the algorithm execution is slow, or program may not execute at all, or the results produced may not be valid due to the delay.

Asymptotic notation

Asymptotic notation is an analytical tool used to describe the execution of an algorithm considering its input values. These notations facilitate the comparative analysis of algorithms without considering their constants and input variables.

For instance, imagine an instruction executes for “n2+3n” times in an algorithm. As the value of “n” increases, the value of “3n” decreases significantly compared to n2. In such cases, the instruction asymptotically executes for n2 times. Asymptotic notation is classified based on the number of input values to the algorithm

Summary

What currency is to every individual to sustain, so are algorithmic analysis and complexities to a programmer. The programmer uses the outcome of the algorithmic analysis as a guideline for program development, and the complexities prevent him from the incorrect choices in designing a solution to the program.

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