Final answer:
Ternary quick-sort uses three-way partitioning to sort arrays and is efficient when dealing with many duplicate elements. It improves on classic quicksort by reducing unnecessary comparisons and moves of duplicate keys, with average time complexity of O(n log n) and decreased probability of the worst-case O(n^2) scenario.
Step-by-step explanation:
The logic behind ternary quick-sort, also known as 3-way quicksort, revolves around partitioning the input array into three parts instead of the usual two parts as seen in the classic quicksort. This algorithm is particularly efficient when dealing with arrays that contain many duplicate elements. In ternary quicksort, the array elements are divided into three groups: elements less than a chosen pivot, elements equal to the pivot, and elements greater than the pivot. This way, the equal elements are not included in the subsequent recursive calls, which reduces the amount of unnecessary sorting.
As for its efficiency, ternary quick-sort has a time complexity that closely resembles that of the classic quicksort in the average case, which is O(n log n), but due to its 3-way partitioning method, it can perform much better in cases with many duplicate keys. The worst-case time complexity is still O(n^2), but it's much less likely to occur compared to the 2-way partitioning quicksort, especially in practical scenarios with duplicates.
To give an example, imagine an array with a large number of duplicate values. In a classic quicksort, these duplicates would be moved around unnecessarily in each partition step. In ternary quicksort, once a duplicate is in its correct position, it is never touched again, thereby saving computational time and increasing the algorithm's efficiency in such cases.