Which Of The Following Best Describes Binary Form – Today, with all the information we consume and generate every day, algorithms must be good enough to handle large amounts of data.

In this post, we’ll learn a little more about time complexity, big-O notation, and why we need to worry about it when developing algorithms.

Which Of The Following Best Describes Binary Form

The examples shown in this story are created in Python, so it will be easy to understand if you have at least a basic knowledge of Python, but it is not a prerequisite.

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Computational complexity is a branch of computer science that analyzes algorithms based on the amount of resources required to run them. The amount of resources required varies based on the size of the input, so complexity is generally expressed as a function

It is important to note that when analyzing an algorithm we can consider time complexity and space complexity. Space complexity is basically the amount of memory space required to solve an input-size problem. Although space complexity is important when analyzing algorithms, in this article we will focus only on time complexity.

As you’re reading this story right now, you may have an idea about time complexity, but to make sure we’re all on the same page, let’s understand what time complexity means from Wikipedia. with a brief explanation from

In computer science, time complexity is the computational complexity that describes the amount of time it takes to run an algorithm. The time complexity is usually estimated by counting the elementary operations performed by the algorithm, assuming that each elementary operation takes a certain amount of time to perform.

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When analyzing the time complexity of an algorithm we may find three cases: best case, average case and worst case. Let’s understand what this means.

Suppose we have the following unordered list [1, 5, 3, 9, 2, 4, 6, 7, 8] and we need to find the index of the value in this list using linear search.

Big-O notation, sometimes called “asymptotic notation”, is a mathematical notation that describes the limiting behavior of a function as the argument approaches a certain value or infinity.

In computer science, Big-O notation is used to classify algorithms based on how their running time or space requirements grow as the size of the input (

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) is growing. This notation specifies functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation.

Note that we will focus our study on these common time complications but there are some other complications that you can study later.

As mentioned earlier, we generally use Big-O notation to describe the time complexity of algorithms. The formal definition of notation involves a lot of math, but informally we can assume that Big-O notation gives us the estimated time of the algorithm in the worst case. When using Big-O notation, we describe the efficiency of an algorithm based on the increasing size of the input data (

Now, let’s go through each of these common time complications and look at some example algorithms. Note that I tried to follow the following approach: give a small explanation, show a simple and comprehensible example and show a more complex example (usually from a real-world problem).

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An algorithm is said to have constant time when it does not depend on the input data (

). No matter the size of the input data, the runtime will always be the same. For Example:

Regardless of the size of the input data, it will always have the same running time because it only gets the first value from the list.

An algorithm with constant time complexity is better because we don’t need to worry about input size.

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An algorithm is said to have logarithmic time complexity when it minimizes the size of the input data at each step (it does not need to check all values ​​of the input data), for example:

Algorithms with logarithmic time complexity are usually found in binary tree operations or when using binary search. Let’s take a look at an example of a binary search, where we need to find the position of an element in an ordered list:

It is important to realize that an algorithm that must access all elements of its input data cannot take logarithmic time, such as the time taken to read the input data.

An algorithm is said to have linear time complexity when its running time increases linearly with the size of the input data. This is the best possible time complexity when the algorithm must examine all values ​​of the input data. For Example:

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Let’s take a look at an example of a linear search, where we need to find the position of an element in an unordered list:

Note that in this example, we need to look through all the values ​​in the list to find the value we are looking for.

An algorithm is said to have quasilinear time complexity when every operation on the input data has logarithmic time complexity. This is commonly seen in sorting algorithms (eg mergesort, timsort, heapsort).

For example: For each value in data1 (O(n)), use binary search (O(log n)) to search for the same value in data2.

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Another, more complex example, can be found in the Mergesort algorithm. Mergesort is an efficient, general-purpose, comparison-based sorting algorithm with time complexity, let’s look at an example:

An algorithm is said to have quadratic time complexity when it needs to perform a linear time operation for each value in the input data, for example:

A bubble array is a great example of quadratic time complexity because for each value it needs to compare with all the other values ​​in the list, let’s look at an example:

An algorithm is said to have exponential time complexity when growth doubles with each addition to the input data set. This type of time complexity is commonly seen in brute force algorithms.

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In cryptography, a brute force attack may check all possible elements of a password by systematically iterating through subsets. Using an elaborate algorithm to do this, it becomes surprisingly difficult for a source to crack a short password versus a long one. This is one reason why a long password is considered more secure than a short one.

If you don’t know what a recursive function is, let’s clarify it quickly: A recursive function may be defined as a function that calls itself under certain conditions. As you may have noticed, defining the time complexity of recursive functions is a bit difficult because it depends on how many times the function is called and the time complexity of a single function call.

This makes more sense when we look at the recursion tree. The following recursive tree was created using the Fibonacci algorithm

Note that it will call itself until the pages are reached. When it reaches the leaf it returns the value itself.

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You can find more complete details on the time complexity of the recursive Fibonacci algorithm here on StackOverflow.

An algorithm is said to have factorial time complexity when it grows factorially based on the size of the input data, for example:

A classic example of an algorithm with factorial time complexity is Heap’s algorithm, which is used to generate all possible permutations.

Heap finds a systematic way to choose a pair of elements to change at each step, so that every possible permutation of those elements is produced exactly once.

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Note that it will grow factorially based on the size of the input data, so we can say that the algorithm has factorial time complexity O(n!).

It is important to note that when analyzing the time complexity of an algorithm with multiple operations we need to define the algorithm based on the largest complexity among all operations. For Example:

Even though the operation in ‘my_function’ doesn’t make sense we can see how many times it has complexity: O(1) + O(n) + O(n²). So, when the size of the input data increases, the bottleneck of this algorithm will be the operation that takes O(n²). Based on this, we can describe the time complexity of this algorithm as O(n²).

To make your life easier, here you can find a page with the time complexity of operations on several common data structures.

Did You Solve It? Are You Brainy At Binary?

If after reading this whole story you still have some doubts about the complexity of time and the importance of knowing Big-O notation, let us clarify some points.

Even when working with modern languages, such as Python, which provide built-in functions such as programming algorithms, one day you may need to implement an algorithm to perform some kind of operation on a certain amount of data. have By studying time complexity you will understand the important concept of efficiency and find bottlenecks in your code that should be improved, especially when working with large data sets.

In addition, if you plan to apply for a software engineer position at a large company such as Google, Facebook, Twitter, and Amazon, you will need to answer questions about time complexity using Big-O notation. .

Thank you

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