Tootesort

Tootesort is a simple, non-comparison-based sorting algorithm designed for sorting small arrays of integers with a limited range of values. The algorithm works by counting the occurrences of each unique value in the input array and then reconstructing the sorted array from these counts. It is particularly efficient when the range of possible values (often denoted as *k*) is not significantly larger than the number of elements to be sorted (*n*).

The process begins by initializing an auxiliary array, often called a "count array," with a size equal to the range of values plus one. Each element in the count array is initialized to zero. The algorithm then iterates through the input array, incrementing the corresponding index in the count array for each encountered value. After counting, the algorithm reconstructs the sorted array by iterating through the count array and appending each value the number of times it appeared in the original array.

Tootesort operates in linear time, O(n + k), where *n* is the number of elements and *k* is the range of values. This makes it highly efficient for scenarios where *k* is small relative to *n*. However, its performance degrades significantly when *k* is large, as the space complexity becomes O(n + k), which can be impractical for sparse or widely ranging datasets.

While Tootesort is not commonly used in practice due to its limited applicability, it serves as an educational tool for understanding counting-based sorting techniques. Variations of this approach, such as radix sort, extend its principles to handle larger ranges by processing digits or other discrete units of the input values.