How do you handle duplicate elements in randomized quicksort?

1 Use three-way partitioning

One way to deal with duplicate elements is to use a variant of quicksort called three-way partitioning, also known as Dutch flag algorithm. This method divides the input array into three regions: elements less than the pivot, elements equal to the pivot, and elements greater than the pivot. This way, the equal elements are grouped together and do not affect the recursion. To implement three-way partitioning, you need to maintain two pointers, one for the left boundary of the equal region and one for the right boundary, and swap elements accordingly as you scan the array.

2 Use median-of-medians as pivot

You can also use a more robust pivot selection strategy, such as the median-of-medians algorithm. This algorithm guarantees that the pivot is close to the median of the input, which ensures a balanced partition and reduces the recursion depth. The median-of-medians algorithm works by dividing the input array into groups of five elements, finding the median of each group, and recursively finding the median of the medians. This process takes O(n) time and can be combined with quicksort to achieve an O(n log n) expected running time and a O(n) worst-case space complexity.

4 Use insertion sort for small subarrays

Another way to handle duplicate elements is to use insertion sort for small subarrays, instead of recursing until the base case of one element. Insertion sort is faster and more stable than quicksort for small inputs, as it has less overhead and fewer comparisons. You can choose a threshold value, such as 10 or 20, and switch to insertion sort when the subarray size is below that value. This can improve the overall performance and stability of quicksort, especially when there are many duplicate elements.

5 Use a hybrid approach

You can also use a hybrid approach that combines different methods based on the input characteristics. For example, you can use three-way partitioning for inputs that have a high percentage of duplicate elements, median-of-medians for inputs that have a low percentage of duplicate elements, and hashing or random sampling for inputs that have a large size. You can also use insertion sort for small subarrays in any case. This can help you adapt to different scenarios and optimize your quicksort algorithm.