Bucket Sort Algorithm: A Comprehensive Guide

If you want to visualise Bucket Sorting, think of it as sorting a messy pile of coins by simply dropping them into labelled jars. Essentially, this algorithm is great for handling large datasets, especially when values are uniformly distributed. Unlike traditional comparison-based sorting methods, Bucket Sort Algorithm groups elements into buckets, sorts them individually, and then merges them to produce a sorted array. 

When applied to uniformly distributed data, Bucket Sort Algorithm delivers lightning-fast performance. This blog explores this concept in detail, highlighting its workings and applications. So read on and learn how it makes sorting as easy as tossing items in the right bins! 

Table of Contents 

1) What is a Bucket Sort algorithm?  

2) How Does Bucket Sort Work?  

3) Algorithm of the Bucket Sort  

4) Complexity of Bucket Sort  

5)  Applications of Bucket Sort Algorithm 

6) Variation of Bucket Sort Algorithm 

7) Comparison of Bucket Sort With Other Sorting Algorithms 

8) Which Searching Algorithm is Best Suited for Unsorted Lists? 

9) Why is Bucket Sort O(n+k)?  

10)  Conclusion  

What is a Bucket Sort Algorithm? 

Bucket Sort is a sorting algorithm that distributes different array elements into several "Buckets" based on a predetermined range. Here are some key points to remember: 

1) Each bucket is sorted independently using a suitable sorting algorithm, such as Insertion Sort and recursively applying Bucket Sort. 

2) After sorting, the contents of the buckets are concatenated to form the final sorted array. 

3) This method is particularly efficient for uniformly distributed data across the range. 

4) Bucket Sort minimises comparison operations, enhancing efficiency. 

5) It offers an average time complexity better than traditional comparison-based sorting algorithms, especially for large, uniformly distributed datasets. 
 

  

How does Bucket Sort work? 

Bucket Sort works by distributing the elements of an array into several Buckets and then sorting these Buckets individually before combining them to form the final sorted array. Here's a step-by-step breakdown: 

a) Determine the Number of Buckets: Decide the number of Buckets to use. This can vary based on the dataset. For simplicity, assume 10 Buckets for values ranging from 0 to 99.  

b) Initialise Buckets: Create an array of Buckets where each Bucket is initially empty. These Buckets can be thought of as lists or arrays. 

c) Distribute Elements Into Buckets: For each element in the input array, determine which Bucket it belongs to. This can be done by dividing the element by the range size of the Buckets. Place each element into its corresponding Bucket.  

1)  Example: Consider an array [29, 25, 3, 49, 9, 37, 21, 43]. If we have 10 Buckets for the range 0-99, each Bucket will represent a range of 10. So, 29 goes into Bucket 2 (20-29), 25 goes into Bucket 2 as well, 3 goes into Bucket 0 (0-9), and so on. 

d) Sort Individual Buckets: Sort the elements in each Bucket. This can be done using any sorting algorithm; insertion sort is often used due to its efficiency on small lists. 

e) Concatenate Buckets: Once all Buckets are sorted individually, concatenate them to form a single sorted array. 

Continuing the example: After sorting individual Buckets, we combine them. The sorted Buckets might look like [3, 9], [21, 25, 29], [], [], [37], [], [], [43, 49], [], []. Concatenating these gives the sorted array [3, 9, 21, 25, 29, 37, 43, 49]. 

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Algorithm of the Bucket Sort 

The algorithm of Bucket Sort can be described in a systematic and detailed manner as follows: 

Initialisation 

a) Determine the number of Buckets, ‘k’, to be used, which typically depends on the input data. 

b) Create ‘k’ empty Buckets. Each Bucket can be conceptualised as a list or an array. 

Placing Elements into Buckets 

a) For each element in the input array, determine which Bucket it should be placed in. This can be done using a function or a formula. A common approach is to use the formula ‘Bucket_index = value * k / (max_value + 1)’. 

b) Add each element to the corresponding Bucket based on the calculated index. 

Sorting Individual Buckets 

Sort the elements in each Bucket. This can be done using any type of sorting algorithm. For small Buckets, simple algorithms like insertion sort are commonly used. 

Merging Buckets 

a) Concatenate all the Buckets to form the final sorted array. 

b) The concatenation should maintain the order of the Buckets and the elements within each Bucket. 

For Example: Consider an example to illustrate these steps. Suppose we have an array ‘[0.78, 0.17, 0.39, 0.26, 0.72, 0.94, 0.21, 0.12, 0.23, 0.68]’ and we choose to use 10 Buckets. 

a) Initialisation: Create 10 empty Buckets. 

b) Placing Elements: Each number is placed in a Bucket based on value. For instance, 0.78 goes into Bucket 7, 0.17 goes into Bucket 1, and so on. 

c) Sorting Buckets: Sort each Bucket. For example, if Bucket 1 has [0.17, 0.12], it becomes [0.12, 0.17] after sorting. 

d) Merging: Merge all Buckets to obtain the sorted array. 

Note: 

a) The efficiency of Bucket Sort largely depends on how the elements are distributed in the Buckets. If the elements are uniformly distributed, the sort can be very efficient, often approaching linear time complexity. 

b) The choice of a sorting algorithm for individual Buckets can also affect the overall efficiency. 

c) Bucket Sort is generally used when the data is uniformly distributed and ranges over a known interval. 

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Complexity of Bucket Sort 

Bucket Sort Algorithm comes with two main complexity types, namely time complexity and space complexity. Let’s explore the two with examples: 

1) Time Complexity 

Factors affecting Time Complexity: 

a) Distribution of Input Data: The efficiency of Bucket Sort largely depends on how the input data is distributed. If the data is uniformly distributed, Bucket Sort can be very efficient. 

b) Number of Buckets (k): The choice of how many Buckets to use is crucial. Ideally, ‘k’ should be chosen such that the distribution of elements in the Buckets is as uniform as possible. 

c) Sorting Algorithm for Buckets: The sorting algorithm for individual Buckets also influences the overall time complexity. Typically, insertion sort is used for its efficiency on small lists. 

Time Complexity Analysis: 

a) Best Case (O(n + k)): This occurs when the elements are uniformly distributed, and each Bucket gets one element. The sorting of each Bucket is O(1), as each Bucket contains only one element. Thus, the time complexity is linear regarding the number of elements, ‘n’, and the number of Buckets, ‘k’. 

b) Average Case (O(n + n^2/k + k)): Typically, when elements are uniformly distributed, the average time complexity involves the time to insert elements into Buckets (O(n)), the time to sort all Buckets (O(n^2/k) if using insertion sort), and the time to merge the Buckets (O(k)). 

c) Worst Case (O(n^2)): The worst case occurs when all elements are placed into a single Bucket. The overall performance becomes dependent on the sorting algorithm used for the Buckets. If insertion sort is used, the time complexity becomes O(n^2). 

Practical code example 

Consider an example in Python to illustrate Bucket Sort:


In this code, ‘insertion_sort’ is used for sorting individual Buckets, which are divided based on the range of the input elements. 

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2) Space Complexity 

Bucket Sort's space complexity is primarily influenced by the number of Buckets used and the space needed to store the input array.   

a) Number of Buckets: The space required for the Buckets themselves is a key component. If you have ‘k’ Buckets, you need space to store these ‘k’ Buckets. 

b) Size of input array: The space required to hold the input array elements in the Buckets is another factor. In the worst case, all elements might end up in a single Bucket. 

Analysing Space Complexity: 

The space complexity of Bucket Sort can be dissected as follows: 

a) Space for Buckets: If there are ‘k’ Buckets, and each Bucket is an array or list, the space required is proportional to ‘k’. 

b) Space for Elements: In the worst case, all ‘n’ elements of the input array could be in the same Bucket. Hence, the space required to store these elements is proportional to ‘n’. 

c) Total space Complexity: Combining the above, the total space complexity of Bucket Sort is O(n + k), where ‘n’ represents the number of elements in the input array and ‘k’ represents the number of Buckets. 

Practical Code Example: 

Consider the Python code example from the previous explanation of Bucket Sort. In this case, the space complexity is determined by the space required for the ‘Buckets_list’ and the space required to hold the ‘input_list’. 

Here's the same code with comments highlighting space allocation:
 

Key points in space complexity: 

a) Proportionality to ‘n’ and ‘k’: The space complexity is proportional to the number of Buckets and elements. 

b) In-place Sorting: Unlike other sorting algorithms (like merge sort), Bucket Sort does not sort in place; it requires additional space for Buckets. 

c) Temporary Space: The space used for the final output array in the code is temporary and does not add to the overall space complexity, as it's just a reorganisation of the existing input. 

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Applications of Bucket Sort algorithm 

Here are some key applications of the Bucket Sort Algorithm:
 

a) Sorting Large Datasets: Bucket Sort is ideal for sorting large datasets, especially when the data is uniformly distributed over a range. Its ability to distribute data into Buckets and sort them individually makes it efficient for handling vast amounts of data. 

b) Decimal or Floating-point Sorting: It's particularly useful for sorting decimal or floating-point numbers that are uniformly distributed. Bucket Sort can manage the data more efficiently than comparison-based sorting algorithms. 

c) Distributed Systems: In distributed systems, where data is spread across multiple machines, Bucket Sort can sort data locally in each machine (Bucket) before merging. 

d) External Sorting: Bucket Sort can be an effective choice when dealing with data that doesn't fit into memory. It can sort chunks of data (Buckets) individually, which are then merged. 

e) Graphics Rendering: Bucket Sort is used in graphics for depth sorting or the painter's algorithm, where objects are sorted based on depth before rendering. 

Variation of Bucket Sort Algorithm 

Here are the variations of Bucket Sort: 

1) Postman's Sort 

a) The algorithm sorts numbers from the most significant to the least significant digit. 

b) Sorting numbers on multiple digits at a time significantly increases speed. 

c) Postman’s Sort is a Bucket Sort variant that utilises a hierarchical structure of elements, typically described by a set of attributes. 

d) Letter-sorting machines in post offices follow a similar approach: 

i) Mail is first separated into domestic and international categories. 

ii)  Further sorted by state, province, or territory. 

ii) Then sorted by the destination post office. 

iv) Finally, sorted by routes, and so on. 

1) Histogram Sort 

a)  A histogram is a rough representation of numerical data distribution. 

b) The first step in creating a histogram is to bin (or bucket) the range of values: 

c) This involves segmenting the entire range into intervals and determining how many values fall into each interval. 

d) A variant of Bucket Sort, known as Histogram Sort or Counting Sort, follows a specific approach: 

i) A first pass counts the number of elements in each bucket using a count array. 

ii) The array values are then arranged into buckets using a series of exchanges. 

iii) This method eliminates bucket storage overhead. 

2)  Proxmap Sort 

a) ProxMap Sorting takes a unique approach, conceptually similar to hashing. 

b)  This method uses hashing with buckets, but the buckets have varying sizes. 

c) ProxMapSort works similarly to Bucket Sort by dividing an array into subarrays using a "map key" function that maintains a partial ordering of keys. 

d) While each key gets added to its subarray, insertion sort keeps that subarray sorted. 

e) When ProxMapSort finishes, the entire array is in sorted order. 

3) Generic Bucket Sort 

a) The most common Bucket Sort variant processes n numeric inputs ranging from 0 to a maximum value M. 

b) The value range is divided into n buckets, each of size M/n. 

c) If Insertion Sort is used to sort each bucket, the algorithm achieves expected linear time complexity. 

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Comparison of Bucket Sort With Other Sorting Algorithms 

This table summarises the distinctions between the various sorting algorithms: 

Which Searching Algorithm is Best Suited for Unsorted Lists? 

Linear search algorithm is best suited for unsorted lists. It inspects each element of the list sequentially until it locates the target value or reaches the end of the list. 

Why is Bucket Sort O(n+k)? 

Bucket Sort is O(n + k) because it distributes the input elements into a fixed number of buckets (k), and each bucket is sorted individually, usually with another sorting algorithm. The O(n) time complexity arises from distributing elements into buckets and then collecting them to form the sorted array. 

Conclusion 

In essence, the Bucket Sort Algorithm isn’t just another sorting method—it’s like organising a messy drawer into neat compartments, making everything easier to find. By breaking data into smaller buckets, it simplifies the process and boosts performance. While it’s not a one-size-fits-all solution, knowing when and how to use it can save you time and effort.  

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