I recently went to the Kenya National Library to search for some history books. Let's use this scenario to explain what Big O notation is—a mathematical concept used to describe how the performance of an algorithm changes as the amount of data it needs to handle increases.
1. Constant Time (O(1))
Suppose I knew exactly where the history books were located because I had been to this library many times before. I could walk directly to the right shelf and pick the book I wanted. This scenario represents constant time, or "O(1)"—the time it takes doesn’t change, no matter how many books the library holds.
2. Linear Time (O(n))
However, if I didn’t know where the books were located, I might have to search each shelf systematically until I found the history section. If the library had 100 shelves, and the history books were on the last one, I'd end up checking every shelf. This is what we call linear time, or "O(n)," where my search time increases directly with the number of shelves.
3. Logarithmic Time (O(log n))
Now, If the library sections were organized in a way that each had a label that could guide me to narrow down my search more efficiently—say, the labels indicated general topics and each topic was subdivided clearly—I could use these clues to skip irrelevant sections, much like narrowing down a search by halving the possibilities each time. This method, which reduces the problem size significantly with each step, represents logarithmic time, or "O(log n)."
4. Linearithmic Time (O(n log n))
If I had to sort through a cart of mixed books to organize them back onto their respective shelves, I might use a strategy where I first categorize the books by subject (linear time) and then alphabetize each category (logarithmic time). This combined process is an example of linearithmic time, or "O(n log n)," where the complexity is a product of both linear and logarithmic operations.
5. Quadratic Time (O(n^2))
If I'm looking for all books by Wangari Maathai, to find them, I could take one book from the history section and compare the author's name with Wangari Maathai, then repeat this process for each book.
I'd need to make these comparisons for every single book, which means if there were 100 books, I'd be making nearly 10,000 comparisons (100 x 100). This approach results in quadratic growth, or "O(n^2)," making it extremely time-consuming as the number of books increases. This method becomes impractical with large collections because the effort grows dramatically with the size of the collection.
6. Exponential Time (O(2^n))
If I decide to make a poster display on the history of Kenya using 5 books, each covering a different aspect like say: pre-colonial era, colonial era, independence, post-independence, and modern developments. I want to see which set of books looks best together, so I start exploring every combination:
Just using the first book alone
Just the second book alone
The first and second books together
The first, second, and third books together ...and so on, up to using all five books together.
Each time I add another book to my potential display, the number of combinations more than doubles. This rapid increase in combinations is exponential growth. With each additional book, the total number of combinations I need to consider grows immensely. It's like doubling the number of paths I can take every time I add a new turn—it gets complicated very quickly!
Hope the analogy helped!!:)
Happy hacking!:)