Held: Monday, May 1, 2006
Summary:
Today we consider two important divide-and-conquer sorting algorithms, merge sort and Quicksort.
Related Pages:
Notes:
- Are there questions on Exam 3?
- In an attempt at supporting today's protest, I will only hold half a class today.
Overview:
- Review.
- Divide-and-Conquer Sorting.
- Merge Sort.
- Iterative Merge Sort.
- Asymptotic Analysis of Merge Sort.
- Quicksort.
- We've seen three main sorting algorithms so far
- In bubble sort, the key operation is swapping neighboring values that are out of order.
- In insertion sort, the key operation is inserting a value into a sorted subvector.
- In selection sort, the key operation is selecting the largest value.
- All three algorithms are O(n^{2})
- Can we do better? Certainly. We've seen that heap sort is O(nlogn).
- But heap sort is not stable.
- Are there other alternatives to heap sort?
- Certainly: There are at least two other O(nlogn) sorts.
- Both are based on the divide-and-conquer strategy.
- In merge sort, we divide in half by position.
- In Quicksort, we divide in half by value.
- We'll look at both in some depth.
- The merge sort algorithm begins sensibly for a divide-and-conquer algorithm
if the portion of interest has one or fewer elements
we're done
otherwise
divide the portion of interest into two halves
sort each half
...
- For example, given the vector (5, 4, 3, 1, 6, 2)
- We would sort the (5, 4, 3) part, giving us (3, 4, 5)
- We would sort the (1, 6, 2) part, giving us (1, 2, 6)
- What do we do now? We must merge the two halves together
- 1 is smaller than 3, so it is the first value in our result
- 2 is smaller than 3, so it is the second value
- 3 is smaller than 6, so it is the third value
- 4 is smaller than 6, so it is the fourth value
- 5 is smaller than 6, so it is the fifth value
- only 6 remains, so it is the sixth value
- To merge two lists, we repeatedly compare the first value of each list and take the smaller of the two.
- A problem: Where do you put the result of merging?
- Answer: Merge Sort cannot be done in place; you must make a copy.
- Instead of doing merge sort in a top-down method, recursively dividing and conquering, we can work bottom up.
- First, merge neighboring elements into sorted pairs.
- Next, merge neighboring pairs into sorted four-tuples.
- Next, merge neighboring four-tuples into sorted eight-tuples.
- And so on and so forth.
- It's easier to analyze the iterative version.
- Doing all the sets of merges for one size takes O(n) (each step of each merge puts one element in "the right place" in its sublist)
- After logn "level merges", we have a sorted list of size n.
- Hence, merge sort is O(nlogn).
- The naive way to divide elements in merge sort leads to the need to do complex and out-of-place merging.
- Perhaps we can divide the elements more sensibly.
- In Quicksort, we partition the subvector into two parts: the "small" values and the "large" values.
- Every "small" value is smaller than every "large" value.
- The "small" values are at the left (lower indices) and the large values are at the right (higher indices)
- We then sort the small values in place and the large values in place
- The result is therefore sorted.
- For example, given the vector (5, 4, 3, 1, 6, 2)
- We rearrange the elements to (2, 1, 3, 4, 6, 5), where (2,1,3) are the small elements and (4, 6, 5) are the large elements.
- We sort the two halves.
- The result is (1, 2, 3, 4, 5, 6)
- A problem: How do we determine large and small? It turns out that random selections works pretty well.
- Pick some element of the vector and treat it is as the middle value.
- A problem: How do you partition?
- Step through from left to right and right to left, swapping whenever you encounter elements out of place.
- A subtlety: If you can have repeated values, how do you decide when you're done?
- Normal solution: Don't include the pivot (randomly selected element) in recursive sort