Computer Science

Part 1: Introduction and Code Bucket Sort is an efficient sorting algorithm when input values are uniformly distributed over a range. It works by distributing elements into different “buckets”, sorting each bucket, and then concatenating the results. Here’s a typical Python implementation where each bucket is sorted with Insertion Sort: def insertion_sort(bucket): for i in range(1, len(bucket)): key = bucket[i] j = i - 1 while j >= 0 and bucket[j] > key: bucket[j + 1] = bucket[j] j -= 1 bucket[j + 1] = key def bucket_sort(arr): n = len(arr) buckets = [[] for _ in range(n)] # Put array elements in different buckets for num in arr: bi = int(n * num) # assuming input numbers are in [0,1) buckets[bi].append(num) # Sort individual buckets using insertion sort for bucket in buckets: insertion_sort(bucket) # Concatenate all buckets into arr[] index = 0 for bucket in buckets: for num in bucket: arr[index] = num index += 1 Why Insertion Sort? Insertion sort is simple and efficient for small or nearly sorted lists. Since each bucket contains only a fraction of the input, sorting each bucket with insertion sort is fast. ...

Differentiation is a key concept in machine learning, especially when optimizing functions like loss functions in neural networks. It helps us find the minimum of these functions, which is crucial for tasks like training a model. But have you ever wondered how popular libraries like TensorFlow and PyTorch perform differentiation? Let’s break it down! 1. Manual Differentiation: The Old-School Method In school, we learn how to manually compute derivatives using calculus. You apply a set of rules to functions to find how they change with respect to their inputs. For example, given a simple function like: ...

For most developers, QuickSort is a fast and efficient sorting algorithm with a time complexity of O(nlogn). This makes it significantly better than other common sorting algorithms, like Selection Sort or Bubble Sort, which have a time complexity of O(n²). However, the question remains: Why is the average time complexity of QuickSort O(nlogn)? In this blog, we will delve deep into the mathematical and probabilistic principles that explain this efficiency, helping you understand the underlying reasons why QuickSort is faster than other algorithms on average. ...

Part 1: Motivation How do we check if something is in a set — fast? The simplest way is a List: if x in items: ... But this is O(n) — too slow for large-scale systems. A HashSet improves to O(1) lookups on average, but it stores the full elements, requiring more memory than raw data — especially for strings or objects. So what if we trade a little accuracy for massive savings? What if a structure could: ...