{"title":"Bucket Sort","description":"","content":"Bucket Sort is a distribution-based sorting algorithm that divides elements into multiple buckets, sorts each bucket individually, and then combines the results to produce the final sorted array.\n\nThe idea is to spread elements across buckets based on their value range so that each bucket contains a small subset of the data. If the data is uniformly distributed, this leads to near-linear time complexity.\n\nThe performance of Bucket Sort depends heavily on how the buckets are defined and how evenly the elements are distributed. In practice, it is often combined with another sorting algorithm, such as Insertion Sort, to sort individual buckets.\n\nIn this chapter, you will learn how Bucket Sort works, how to design effective bucket strategies, and when it can outperform traditional sorting algorithms.\n\n---\n\n# What Is Bucket Sort?\n\nBucket sort is a distribution-based sorting algorithm. Instead of repeatedly comparing pairs of elements, it divides the input into a fixed number of \"buckets\" (or bins), distributes elements into those buckets based on their value, sorts each bucket individually, and then concatenates the results.\n\nHere is the high-level idea:\n\n1. Create an array of empty buckets.\n2. Scatter: walk through the input and place each element into the bucket that covers its value range.\n3. Sort each bucket, typically with insertion sort or any other comparison sort.\n4. Gather: concatenate all buckets in order to produce the sorted output.\n\n\n```mermaid\nflowchart TD\n A[\"Input Array
[0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51]\"]:::primary\n\n A --> B[\"Create n empty buckets\"]:::teal\n B --> C[\"Distribute elements
into buckets by value\"]:::secondary\n C --> D[\"Sort each bucket
individually\"]:::green\n D --> E[\"Concatenate all
buckets in order\"]:::teal\n E --> F[\"Sorted Array
[0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\"]:::green\n\n classDef primary fill:#00ceff,stroke:#000,color:#000\n classDef secondary fill:#ffa94d,stroke:#000,color:#000\n classDef green fill:#69db7c,stroke:#000,color:#000\n classDef teal fill:#38d9a9,stroke:#000,color:#000\n```\n\n\nYou might notice a resemblance to counting sort. In fact, bucket sort is a generalization of counting sort. Counting sort creates one \"bucket\" per distinct value and only works with integers. Bucket sort creates a fixed number of buckets, each covering a range of values, and works with any data type, including floating point numbers. When bucket count equals the range of integer values, bucket sort reduces to counting sort.\n\nBucket sort works best when the input is uniformly distributed over a known range. If you know your data falls between 0 and 1, or between 0 and 1000, and the values are spread roughly evenly across that range, bucket sort shines. If all values cluster into a single bucket, it degrades to whatever sort you use inside the bucket.\n\n---\n\n# How It Works\n\nLet us walk through the algorithm step by step, using the classic case of sorting floating point numbers in the range [0, 1).\n\n### Step 1: Create n Empty Buckets\n\nGiven n elements, create an array of n empty lists (buckets). Each bucket covers an equal slice of the value range.\n\nFor values in [0, 1) with n = 7 elements, the seven buckets cover these ranges:\n\n\n| Bucket | Range |\n|--------|-------|\n| 0 | [0.00, 0.14) |\n| 1 | [0.14, 0.29) |\n| 2 | [0.29, 0.43) |\n| 3 | [0.43, 0.57) |\n| 4 | [0.57, 0.71) |\n| 5 | [0.71, 0.86) |\n| 6 | [0.86, 1.00) |\n\n\n### Step 2: Distribute Elements into Buckets\n\nFor each element, compute its bucket index using the formula:\n\n\n```shell\nbucketIndex = floor(n * value)\n```\n\n\nFor a general range [minValue, maxValue], the formula becomes:\n\n\n```shell\nbucketIndex = floor(n * (value - minValue) / (maxValue - minValue + 1))\n```\n\n\nThis maps each value to a bucket in O(1) time. You simply multiply, take the floor, and you have the index.\n\n### Step 3: Sort Each Bucket\n\nSort each bucket individually. Insertion sort is the traditional choice here, and for good reason. When elements are uniformly distributed, each bucket holds roughly n/k elements (where k is the number of buckets). With k = n, each bucket holds about 1 element on average, making insertion sort's O(m^2) cost per bucket negligible. Even when a bucket has a few elements, insertion sort is fast for small inputs due to low overhead.\n\n### Step 4: Concatenate All Buckets\n\nWalk through the bucket array from index 0 to n-1, appending each bucket's sorted contents to the output. Since the buckets cover increasing value ranges, this produces a fully sorted array.\n\n\n```mermaid\nflowchart LR\n subgraph Input\n A[\"0.42\"]:::primary\n B[\"0.32\"]:::primary\n C[\"0.23\"]:::primary\n D[\"0.52\"]:::primary\n E[\"0.25\"]:::primary\n F[\"0.47\"]:::primary\n G[\"0.51\"]:::primary\n end\n\n subgraph Buckets\n B1[\"Bucket 1
[0.23, 0.25]\"]:::teal\n B2[\"Bucket 2
[0.32, 0.42]\"]:::teal\n B3[\"Bucket 3
[0.47, 0.51, 0.52]\"]:::teal\n end\n\n subgraph Output\n O[\"[0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\"]:::green\n end\n\n A --> B2\n B --> B2\n C --> B1\n D --> B3\n E --> B1\n F --> B3\n G --> B3\n\n B1 --> O\n B2 --> O\n B3 --> O\n\n classDef primary fill:#00ceff,stroke:#000,color:#000\n classDef teal fill:#38d9a9,stroke:#000,color:#000\n classDef green fill:#69db7c,stroke:#000,color:#000\n```\n\n\nThe diagram above simplifies the bucket indices for readability. In the actual algorithm, several buckets would be empty, and we skip over those during concatenation.\n\n---\n\n# Code Implementation\n\n\n```java\nimport java.util.ArrayList;\nimport java.util.Collections;\nimport java.util.List;\n\npublic class BucketSort {\n public static void bucketSort(float[] arr) {\n int n = arr.length;\n if (n <= 1) return;\n\n // Step 1: Create n empty buckets\n List> buckets = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n buckets.add(new ArrayList<>());\n }\n\n // Step 2: Distribute elements into buckets\n for (float value : arr) {\n int bucketIndex = (int) (n * value);\n // Handle edge case where value == 1.0\n if (bucketIndex == n) bucketIndex = n - 1;\n buckets.get(bucketIndex).add(value);\n }\n\n // Step 3: Sort each bucket\n for (List bucket : buckets) {\n Collections.sort(bucket);\n }\n\n // Step 4: Concatenate all buckets\n int index = 0;\n for (List bucket : buckets) {\n for (float value : bucket) {\n arr[index++] = value;\n }\n }\n }\n\n public static void main(String[] args) {\n float[] arr = {0.42f, 0.32f, 0.23f, 0.52f, 0.25f, 0.47f, 0.51f};\n bucketSort(arr);\n // Output: [0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\n for (float v : arr) {\n System.out.print(v + \" \");\n }\n }\n}\n```\n\n```python\ndef bucket_sort(arr: list[float]) -> list[float]:\n n = len(arr)\n if n <= 1:\n return arr\n\n # Step 1: Create n empty buckets\n buckets = [[] for _ in range(n)]\n\n # Step 2: Distribute elements into buckets\n for value in arr:\n bucket_index = int(n * value)\n # Handle edge case where value == 1.0\n if bucket_index == n:\n bucket_index = n - 1\n buckets[bucket_index].append(value)\n\n # Step 3: Sort each bucket\n for bucket in buckets:\n bucket.sort()\n\n # Step 4: Concatenate all buckets\n result = []\n for bucket in buckets:\n result.extend(bucket)\n\n return result\n\narr = [0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51]\nprint(bucket_sort(arr))\n# Output: [0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\n```\n\n```cpp\n#include \n#include \n#include \n\nvoid bucketSort(std::vector& arr) {\n int n = arr.size();\n if (n <= 1) return;\n\n // Step 1: Create n empty buckets\n std::vector> buckets(n);\n\n // Step 2: Distribute elements into buckets\n for (float value : arr) {\n int bucketIndex = static_cast(n * value);\n // Handle edge case where value == 1.0\n if (bucketIndex == n) bucketIndex = n - 1;\n buckets[bucketIndex].push_back(value);\n }\n\n // Step 3: Sort each bucket\n for (auto& bucket : buckets) {\n std::sort(bucket.begin(), bucket.end());\n }\n\n // Step 4: Concatenate all buckets\n int index = 0;\n for (const auto& bucket : buckets) {\n for (float value : bucket) {\n arr[index++] = value;\n }\n }\n}\n\nint main() {\n std::vector arr = {0.42f, 0.32f, 0.23f, 0.52f, 0.25f, 0.47f, 0.51f};\n bucketSort(arr);\n // Output: 0.23 0.25 0.32 0.42 0.47 0.51 0.52\n for (float v : arr) {\n std::cout << v << \" \";\n }\n return 0;\n}\n```\n\n```csharp\nusing System;\nusing System.Collections.Generic;\n\npublic class BucketSort {\n public static void Sort(float[] arr) {\n int n = arr.Length;\n if (n <= 1) return;\n\n // Step 1: Create n empty buckets\n List[] buckets = new List[n];\n for (int i = 0; i < n; i++) {\n buckets[i] = new List();\n }\n\n // Step 2: Distribute elements into buckets\n foreach (float value in arr) {\n int bucketIndex = (int)(n * value);\n // Handle edge case where value == 1.0\n if (bucketIndex == n) bucketIndex = n - 1;\n buckets[bucketIndex].Add(value);\n }\n\n // Step 3: Sort each bucket\n foreach (var bucket in buckets) {\n bucket.Sort();\n }\n\n // Step 4: Concatenate all buckets\n int index = 0;\n foreach (var bucket in buckets) {\n foreach (float value in bucket) {\n arr[index++] = value;\n }\n }\n }\n\n public static void Main() {\n float[] arr = {0.42f, 0.32f, 0.23f, 0.52f, 0.25f, 0.47f, 0.51f};\n Sort(arr);\n // Output: 0.23 0.25 0.32 0.42 0.47 0.51 0.52\n Console.WriteLine(string.Join(\" \", arr));\n }\n}\n```\n\n```javascript\nfunction bucketSort(arr) {\n const n = arr.length;\n if (n <= 1) return arr;\n\n // Step 1: Create n empty buckets\n const buckets = Array.from({ length: n }, () => []);\n\n // Step 2: Distribute elements into buckets\n for (const value of arr) {\n let bucketIndex = Math.floor(n * value);\n // Handle edge case where value == 1.0\n if (bucketIndex === n) bucketIndex = n - 1;\n buckets[bucketIndex].push(value);\n }\n\n // Step 3: Sort each bucket\n for (const bucket of buckets) {\n bucket.sort((a, b) => a - b);\n }\n\n // Step 4: Concatenate all buckets\n let index = 0;\n for (const bucket of buckets) {\n for (const value of bucket) {\n arr[index++] = value;\n }\n }\n\n return arr;\n}\n\nconst arr = [0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51];\nconsole.log(bucketSort(arr));\n// Output: [0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\n```\n\n```typescript\nfunction bucketSort(arr: number[]): number[] {\n const n = arr.length;\n if (n <= 1) return arr;\n\n // Step 1: Create n empty buckets\n const buckets: number[][] = Array.from({ length: n }, () => []);\n\n // Step 2: Distribute elements into buckets\n for (const value of arr) {\n let bucketIndex = Math.floor(n * value);\n // Handle edge case where value == 1.0\n if (bucketIndex === n) bucketIndex = n - 1;\n buckets[bucketIndex].push(value);\n }\n\n // Step 3: Sort each bucket\n for (const bucket of buckets) {\n bucket.sort((a, b) => a - b);\n }\n\n // Step 4: Concatenate all buckets\n let index = 0;\n for (const bucket of buckets) {\n for (const value of bucket) {\n arr[index++] = value;\n }\n }\n\n return arr;\n}\n\nconst arr: number[] = [0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51];\nconsole.log(bucketSort(arr));\n// Output: [0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\n```\n\n\n---\n\n# Example Walkthrough\n\nLet us trace through the algorithm with the array `[0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51]`. We have n = 7 elements, so we create 7 buckets.\n\n### Distribution Phase\n\nFor each element, we compute `bucketIndex = floor(7 * value)`:\n\n\n```shell\nvalue = 0.42 -> floor(7 * 0.42) = floor(2.94) = 2 -> Bucket 2\nvalue = 0.32 -> floor(7 * 0.32) = floor(2.24) = 2 -> Bucket 2\nvalue = 0.23 -> floor(7 * 0.23) = floor(1.61) = 1 -> Bucket 1\nvalue = 0.52 -> floor(7 * 0.52) = floor(3.64) = 3 -> Bucket 3\nvalue = 0.25 -> floor(7 * 0.25) = floor(1.75) = 1 -> Bucket 1\nvalue = 0.47 -> floor(7 * 0.47) = floor(3.29) = 3 -> Bucket 3\nvalue = 0.51 -> floor(7 * 0.51) = floor(3.57) = 3 -> Bucket 3\n```\n\n\nAfter distribution, the buckets look like this:\n\n\n| Bucket | Contents |\n|--------|----------|\n| 0 | [] |\n| 1 | [0.23, 0.25] |\n| 2 | [0.42, 0.32] |\n| 3 | [0.52, 0.47, 0.51] |\n| 4 | [] |\n| 5 | [] |\n| 6 | [] |\n\n\n### Sorting Phase\n\nSort each non-empty bucket individually:\n\n\n| Bucket | Before Sort | After Sort |\n|--------|------------|------------|\n| 1 | [0.23, 0.25] | [0.23, 0.25] |\n| 2 | [0.42, 0.32] | [0.32, 0.42] |\n| 3 | [0.52, 0.47, 0.51] | [0.47, 0.51, 0.52] |\n\n\nBucket 1 was already sorted. Bucket 2 needed one swap. Bucket 3 required a small rearrangement. This is what makes bucket sort fast: sorting many small groups is cheaper than sorting one large group.\n\n### Concatenation Phase\n\nWalk through buckets 0 through 6 in order, collecting all elements:\n\n\n```shell\nBucket 0: (empty)\nBucket 1: 0.23, 0.25\nBucket 2: 0.32, 0.42\nBucket 3: 0.47, 0.51, 0.52\nBucket 4: (empty)\nBucket 5: (empty)\nBucket 6: (empty)\n\nResult: [0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\n```\n\n\n\n```mermaid\nflowchart TD\n subgraph Distribution\n I[\"[0.42, 0.32, 0.23, 0.52, 0.25, 0.47, 0.51]\"]:::primary\n end\n\n subgraph Buckets After Sort\n B0[\"Bucket 0: empty\"]:::secondary\n B1[\"Bucket 1: [0.23, 0.25]\"]:::teal\n B2[\"Bucket 2: [0.32, 0.42]\"]:::teal\n B3[\"Bucket 3: [0.47, 0.51, 0.52]\"]:::teal\n B4[\"Bucket 4-6: empty\"]:::secondary\n end\n\n subgraph Result\n R[\"[0.23, 0.25, 0.32, 0.42, 0.47, 0.51, 0.52]\"]:::green\n end\n\n I --> B0\n I --> B1\n I --> B2\n I --> B3\n I --> B4\n\n B0 --> R\n B1 --> R\n B2 --> R\n B3 --> R\n B4 --> R\n\n classDef primary fill:#00ceff,stroke:#000,color:#000\n classDef secondary fill:#ffa94d,stroke:#000,color:#000\n classDef teal fill:#38d9a9,stroke:#000,color:#000\n classDef green fill:#69db7c,stroke:#000,color:#000\n```\n\n\n---\n\n# Complexity Analysis\n\nThe performance of bucket sort depends heavily on how evenly the input data is distributed across buckets.\n\n\n| Case | Time Complexity | Explanation |\n|------|----------------|-------------|\n| **Best** | O(n + k) | Elements are uniformly distributed, each bucket has ~1 element, no real sorting needed within buckets. k is the number of buckets. |\n| **Average** | O(n + n^2/k + k) | With k = n buckets and uniform distribution, this simplifies to O(n). Each bucket has O(1) elements on average. |\n| **Worst** | O(n^2) | All elements land in a single bucket. The inner sort dominates, and if that is insertion sort, you get quadratic time. |\n| **Space** | O(n + k) | n elements stored across k buckets, plus the bucket array itself. |\n\n\nLet us break down the average case more carefully. If we have n elements and k buckets, and the input is uniformly distributed, each bucket receives approximately n/k elements. Sorting one bucket with insertion sort costs O((n/k)^2). Across k buckets, the total sorting cost is:\n\n\n```shell\nk * O((n/k)^2) = O(n^2 / k)\n```\n\n\nAdding the O(n) distribution step and O(k) bucket creation:\n\n\n```shell\nTotal = O(n + n^2/k + k)\n```\n\n\nWhen k = n (number of buckets equals number of elements), this becomes:\n\n\n```shell\nO(n + n^2/n + n) = O(n + n + n) = O(n)\n```\n\n\nThat is the magic number. Linear time sorting.\n\n### Stability\n\nWhether bucket sort is stable depends on two things: the inner sorting algorithm and how elements are collected during concatenation. If you use a stable sort (like insertion sort) within each bucket and collect elements in bucket order, the overall sort is stable. If you use an unstable inner sort (like quicksort), bucket sort loses stability.\n\n---\n\n# When to Use Bucket Sort\n\nBucket sort is not a general-purpose sorting algorithm. It excels in specific scenarios and falls flat in others.\n\n### Good Use Cases\n\n- **Uniformly distributed floating point numbers.** This is the textbook case. If you are sorting random numbers between 0 and 1, bucket sort is hard to beat.\n- **Data with a known, bounded range.** If you know all values fall between 0 and 10,000, you can set up buckets to cover that range efficiently.\n- **Sorting by a hash or computed key.** When the distribution function produces well-spread keys, bucket sort performs well.\n- **External sorting.** When data does not fit in memory, bucket sort's divide-and-conquer-by-range approach maps naturally to disk-based sorting, where each bucket can be a separate file.\n- **Histogram-based applications.** If you already need to bin data into ranges (statistics, image processing), bucket sort comes naturally.\n\n### Poor Use Cases\n\n- **Skewed or clustered distributions.** If most values cluster in a narrow range, most elements end up in the same bucket, and you are essentially running the inner sort on nearly all the data. Example: sorting ages of college students (mostly 18-22) with buckets spanning 0-100.\n- **Unknown value range.** Without knowing the min and max, you cannot compute bucket indices efficiently. You would need an extra pass to find the range first.\n- **Small arrays.** The overhead of creating and managing buckets is not worth it for small inputs. A simple insertion sort or the language's built-in sort is faster.\n- **Integer data with large range.** If you have integers spread across billions, you would need an impractical number of buckets. Radix sort is a better choice here.\n\n\n```mermaid\nflowchart TD\n Q1{\"Known value range?\"}:::primary\n Q2{\"Uniform distribution?\"}:::primary\n Q3{\"Floating point data?\"}:::primary\n\n Y1[\"Bucket Sort\"]:::green\n N1[\"Consider Radix Sort
or Comparison Sort\"]:::secondary\n N2[\"Consider Counting Sort
or Comparison Sort\"]:::secondary\n N3[\"Comparison Sort
(Merge/Quick Sort)\"]:::red\n\n Q1 -->|Yes| Q2\n Q1 -->|No| N3\n Q2 -->|Yes| Q3\n Q2 -->|No| N1\n Q3 -->|Yes| Y1\n Q3 -->|No| N2\n\n classDef primary fill:#00ceff,stroke:#000,color:#000\n classDef secondary fill:#ffa94d,stroke:#000,color:#000\n classDef green fill:#69db7c,stroke:#000,color:#000\n classDef red fill:#ff8787,stroke:#000,color:#000\n```\n","pageType":"dsa"}

Bucket Sort

Ashish Pratap Singh

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