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| "source": [ | ||
| "\n", | ||
| "---\n", | ||
| "[next]()" | ||
| "[next](./06-logarithms.ipynb)" | ||
| ] | ||
| } | ||
| ], | ||
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "[back](./05-space-complexity.ipynb)\n", | ||
| "\n", | ||
| "---\n", | ||
| "## `Logarithms`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "- Up until now, we've come across some of the most common complexities, i.e., $O(1)$, $O(n)$, $O(n^2)$\n", | ||
| "- Sometimes, `Big O` expressions involve more complex mathematical expressions.\n", | ||
| "- One that appears more often than some might like is the `logarithm`! 😉" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `What's a log again?`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "So, logarithm is simply the inverse of exponentiation.\n", | ||
| "\n", | ||
| "Just like division and multiplication are a pair, logarithms and exponents are a pair." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "This equation $log_2(8) = 3$ is read as $log$ base $2$ of $8$ is $3$, and what we're really asking or calculating here is to see $2$ to the power *what?* will give us the value $8$, which is $3$ in this case, meaning $2^3 = 8$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "So, the same equation written in syntactical way would be $log_2(value) = exponent \\longrightarrow 2^{exponent} = value$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Similarly like we had $\\frac{1}{2} = 0.5$ we could switch that and say that $2^{0.5} = 1$, meaning exponentiation and logarithms are inverse to one another." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "One more point to keep in mind is that, logarithm is not something that is always worked with base $2$, we could have $log_3$ of something, meaning, $3$ to power of *what?* gives $8$ may be." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "The most common ones though are the binary logarithms, which is $log_2$, then there is also the base $10$, meaning $10$ to the power what gives us some answer etc." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "So, with algorithms, we tend to see the big-picture, so we'll omit the $2$ and going forward, we'll consider $log === log_2$, so as to not keep writing the subscript $2$ all the time." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Also, the other reason being, it actually doesn't really matter at the end-of-the day, because if we're comparing the trend of a constant time - $O(1)$ and a quadratic time - $O(n^{2})$ and a $O(log \\text{ } n)$ time, it doesn't really matter if it's $log_2$ or $log_3$ or $log_{10}$\n", | ||
| "\n", | ||
| "It's going to be the general trend that we care about, but just to be clear, $log$ alone is not a mathematical operation on its own, we just can't take the $log$ of a number, we need to have a base. So ideally we need to say $log_2$ or $log_{10}$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "But, in further sections, around `Big O` notation and in general when we see $log$ it's just the shorthand for conveying $log_2$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `Rule of thumb`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "The `logarithm` of a number roughly measures the number of times we can divide that number by $2$ **before we get a value that's less than or equal to $1$**" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `Example`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "#### `01`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "$\\frac{8}{2} = 4$\n", | ||
| "\n", | ||
| "$\\frac{4}{2} = 2$\n", | ||
| "\n", | ||
| "$\\frac{2}{2} = 1$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "If we have $8$ and we divide it by $2$, we get $4$ and is still greater than $1$, so we divide it by $2$ again, we get $2$ and we divide it by $2$ again and it's $1$ now." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Meaning, we divided it $3$ times, so $log(8) = 3$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "#### `02`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Now considering 25, and this doesn't divide evenly with $2$.\n", | ||
| "\n", | ||
| "$\\frac{25}{2} \\text{ is } 12.5$\n", | ||
| "\n", | ||
| "$\\frac{12.5}{2} \\text{ is } 6.25$\n", | ||
| "\n", | ||
| "$\\frac{6.25}{2} \\text{ is } 3.125$\n", | ||
| "\n", | ||
| "$\\frac{3.125}{2} \\text{ is } 1.5625$\n", | ||
| "\n", | ||
| "and finally,\n", | ||
| "\n", | ||
| "$\\frac{1.5625}{2} = 0.78125$\n", | ||
| "\n", | ||
| "So, this tell us that somewhere between $4$ and $5$, and the answer is $log(25) \\approx 4.64$\n", | ||
| "\n", | ||
| "And the actual calculation is not that important here, rather what it would look on a trend chart." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `Logarithm Complexity`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "So, `logarithm` time complexity is great!, if we have an algorithm with $log n$ time complexity as seen below compared with previously seen `Big O` complexities.\n", | ||
| "\n", | ||
| "<p align=\"center\">\n", | ||
| "<img src=\"../../assets/log_n_with_others.png\" alt=\"Log N with Other Complexities\">\n", | ||
| "</p>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `Who cares about logarithmic complexities?`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "- Certain searching algorithms have `logarithmic` time complexities, which we will see in later sections.\n", | ||
| "- Also, efficient sorting algorithms have `logarithmic` complexities, we'll see these as well.\n", | ||
| "- And finally, `recursions` sometimes involves `logarithmic` **space** complexities and not *time*." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### `Conclusion`" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "\n", | ||
| "---\n", | ||
| "[next]()" | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "JavaScript (Node.js)", | ||
| "language": "javascript", | ||
| "name": "javascript" | ||
| }, | ||
| "language_info": { | ||
| "name": "javascript" | ||
| }, | ||
| "orig_nbformat": 4 | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 2 | ||
| } |