Check-in Exercise #
Hoare Partitioning. One very fast in-place technique for partitioning is to use a pair of pointers that start at the left and right edges of the array and move towards each other. The left pointer loves small items, and hates equal or large items, where a “small” item is an item that is smaller than the pivot (and likewise for large). The right pointers loves large items, and hates equal or small items. The pointers walk until they see something they don’t like, and once both have stopped, they swap items. After swapping, they continue moving towards each other, and the process completes once they have crossed. In this way, everything left of the left pointer is $\leq$ to the pivot, and everything to the right is $\geq$ to the pivot. Finally, we swap the pivot into the appropriate location, and the partitioning is completed. Unlike our prior strategies, this partitioning strategy results in a sort which is measurably faster than mergesort.
Selection. A simpler problem than sorting, in selection, we try to find the Kth largest item in an array. One way to solve this problem is with sorting, but we can do better. A linear time approach was developed in 1972 called PICK, but we did not cover this approach in class, because it is not as fast as the Quick Select technique.
Quick Select. Using partitioning, we can solve the selection problem in expected linear time. The algorithm is to simply partition the array, and then quick select on the side of the array containing the median. Best case time is $\Theta (N)$, expected time is $\Theta (N)$, and worst case time is $\Theta (N^2)$. You should know how to show the best and worst case times. This algorithm is the fastest known algorithm for finding the median.
Stability. A sort is stable if the order of equal items is preserved. This is desirable, for example, if we want to sort on two different properties of our objects. Know how to show the stability or instability of an algorithm.
Optimizing Sorts. We can play a few tricks to speed up a sort. One is to switch to insertion sort for small problems ($\lt$ 15 items). Another is to exploit existing order in the array. A sort that exploits existing order is sometimes called “adaptive”. Python and Java utilize a sort called Timsort that has a number of improvements, resulting in, for example $\Theta (N)$ performance on almost sorted arrays. A third trick, for worst case N^2 sorts only, is to make them switch to a worst case $N \log N$ sort if they detect that they have exceeded a reasonable number of operations.
Shuffling. To shuffle an array, we can assign a random floating point number to every object, and sort based on those numbers. For information on generation of random numbers, see Fall 2014 61B.
Recommended Problems #
C level #
Problem 3 from my Fall 2014 midterm.
Why does Java’s built-in
Array.sortmethod use Quicksort for
char, or other primitive arrays, but Mergesort for all Object arrays?
B level #
A level #
Given that Quick sort runs fastest if we can always somehow pick the median item as the pivot, why don’t we use Quick select to find the median to optimize our pivot selection (as opposed to using the leftmost item).
We can make Mergesort adaptive by providing an optimization for the case where the left subarray is all smaller than the right subarray. Describe how you’d implement this optimization, and give the runtime of a merge operation for this special case.