Quicksort CMPSC 465 Related to CLRS Chapter 7

Transcription

1 Quicksort CMPSC 465 Related to CLRS Chapter 7 I. High-Level Overview The heart of the quicksort algorithm is to partition an array into three regions: Then, recursively, we quicksort both of the subarrays with multiple elements. The base case of the recursion is Some other notes: We don t need any auxiliary storage for this algorithm. In other words, quicksort sorts in place. This fits the divide-and-conquer model. Say we re sorting A[p..r]. Then, the parts are: o Divide: o Conquer: o Combine: II. Partitioning The kind of partitioning we use is called Lomuto partitioning. (This is the CLRS convention and may be different from other versions of quicksort you see.) Here s the gist of partitioning for subarray A[p..r]: We need an element called the, or the element around which to partition. We always select A[r], the last element, as the pivot. We maintain three indices as we partition: last element that s been inspected an is <= the pivot last element that s been inspected and placed element that s currently being inspected This divides the array into four regions that exist while the algorithm is executing. One region the elements that haven t been inspected yet disappears when the algorithm completes. Page 1 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

2 Here s the pseudocode for partitioning: PARTITION(A, p, r) x = A[r] i = p - 1 for j = p to r-1 if A[j] x i = i + 1 swap A[i] and A[j] // pivot: grab last element // index of last element pivot; initially before array // inspect all elements but pivot // move only elements pivot to left region swap A[i+1] and A[r] return i+1 // put pivot in correct place Example: Let s run PARTITION on this small array: 16, 44, 25, 55, 32 Example: Let s run PARTITION on this array A: 8, 1, 6, 4, 0, 3, 9, 5 Page 2 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

3 Problem: Using the output of the previous problem, trace the execution of PARTITION on A[1..4]. III. Analysis of Partitioning First, let s analyze PARTITION S running time. For an n-element subarray Now, let s prove the correctness of PARTITION: Loop Invariant: Initialization: Maintenance: Page 3 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

4 Termination: Page 4 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

5 IV. Back to Quicksort Algorithm Partitioning does most of the work of the algorithm, but, as you can see from our examples, the arrays are not yet sorted. This is because partitioning is only the divide step and we need the conquer part: recursive calls to quicksort on the two regions. To sort A[p..r], we will: Call PARTITION on A[p..r] and get an index q back. That q tells us where the pivot is. Call QUICKSORT on the elements before the pivot: A[.. ] Call QUICKSORT on the elements after the pivot: A[.. ] Remember, there s no combine here, since the subarrays are sorted in place. We also need a base case for the recursion: when we have one-element arrays, so wrap it all inside if p < r And, that s it! Here s the pseudocode: QUICKSORT(A, p, r) if p < r q = PARTITION(A, p, r) QUICKSORT(A, p, q-1) QUICKSORT(A, q+1, r) To sort an entire array, we d make the call. Example: Here s the output of our earlier example of calling partition on A = (8, 1, 6, 4, 0, 3, 9, 5): The output q is 5. Complete the trace of QUICKSORT. Page 5 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

6 Example: Trace QUICKSORT on B = (92, 99, 60, 75, 82, 70, 75). Page 6 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

7 Problem: Trace QUICKSORT on B = (70, 75, 82, 90, 91, 92). Page 7 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465

8 V. Performance of Quicksort We ll analyze a few cases of quicksort s performance (but will forego a complete formal analysis of quicksort). Everything in our analysis depends on how the arrays generated by PARTITION are balanced. A. Worst Case Analysis The worst case behavior for quicksort happens when partitioning leaves us with arrays that are completely unbalanced. Consider the last example. This occurs when So, in such a case, one output array has elements and the other has elements. We get the recurrence Question: How does this compare to insertion sort? B. Best Case Analysis At the opposite end of the spectrum, the best case for quicksort happens when every partitioning yields arrays that are completely balanced. So, each array would have at most elements. We get the recurrence C. On Average On average, it turns out that quicksort performs much closer to the best case than the worst case. Consider a not-so-great split of 9-to-1. So we get the recurrence But this is like one of our unbalanced recurrences from Section 4.4. We get log 10 n full levels and log 10/9 n nonempty levels. But constants don t matter and any split of constant proportionality gets us a recursion tree of depth Θ(lg n). In practice, we ll likely get a mix of good splits and bad splits, but they tend to offset each other. Thus, the intuitive view is that the average case gives Θ(n lg n) time. Homework: CLRS Exercises 7.1-1, 7.1-3, 7.2-1, Page 8 of 8 Prepared by D. Hogan referencing CLRS - Introduction to Algorithms (3rd ed.) for PSU CMPSC 465