categorizesort

Categorizesort is a sorting technique that partitions a collection of items into categories using a user-supplied categorization function, sorts the items within each category, and then concatenates the results in category order. Its design blends ideas from bucket sort and conventional comparison sorts, aiming to exploit strong separation of keys or attributes before performing finer ordering.

Algorithm: For each element, determine its category by a function f(element) that yields an ordered category label among m categories. Collect elements into m buckets corresponding to their categories. Sort each bucket with a chosen stable or unstable sort or with a specialized subroutine appropriate to the data. Concatenate the buckets in category order to form the final sequence.

Performance and properties: Performance depends on the distribution of elements across categories and the cost of the category function. Let n be the number of elements and m the number of categories. If the buckets are balanced and the per-bucket sort dominates, the overall time is O(n log n) in typical comparison-based settings; if the per-category data are small or the bucket sort within buckets is linear, the overall time approaches O(n). In the degenerate case where all items fall into one category, categorizesort behaves like a standard sort. Space usage is O(n + m) for the buckets; in-place variants exist with extra care.

Variants and notes: Variants include dynamic or adaptive categorization, multi-pass categorization to refine ordering across categories, or using counting/radix-like techniques within buckets for integer keys. The method is particularly suitable when a natural categorical separation exists and where the cost of categorization is low relative to sorting within categories. It can be stable if the chosen per-bucket sort is stable, and can be implemented to be cache-friendly by processing buckets sequentially.

See also: bucket sort, radix sort, counting sort, distribution sort.