CSC 380: Design and Analysis of Algorithms Dr. Curry Guinn

Quick Info Dr. Curry Guinn CIS 2015 guinnc@uncw.edu www.uncw.edu/people/guinnc 962-7937 Office Hours: MWF: 10:00am-11:00m and by appointment

Today Interpolation Search Decrease-and-Conquer Divide-and-Conquer QuickSelect Divide-and-Conquer Mergesort Quicksort

Interpolation Search Searches a sorted array similar to binary search but estimates location of the search key in A[l..r] by using its value v. Specifically, the values of the array’s elements are assumed to grow linearly from A[l] to A[r] and the location of v is estimated as the x-coordinate of the point on the straight line through (l, A[l]) and (r, A[r]) whose y-coordinate is v: x = l + (v - A[l])(r - l)/(A[r] – A[l] ) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Analysis of Interpolation Search Efficiency average case: C(n) < log2 log2 n + 1 worst case: C(n) = n Preferable to binary search only for VERY large arrays and/or expensive comparisons Has a counterpart, the method of false position (regula falsi), for solving equations in one unknown (Sec. 12.4) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Selection Problem Find the k-th smallest element in a list of n numbers k = 1 or k = n median: k = n/2 Example: 4, 1, 10, 9, 7, 12, 8, 2, 15 median = ? The median is used in statistics as a measure of an average value of a sample. In fact, it is a better (more robust) indicator than the mean, which is used for the same purpose. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

QuickSelect

Algorithms for the Selection Problem The sorting-based algorithm: Sort and return the k-th element Efficiency (if sorted by mergesort): Θ(nlog n) A faster algorithm is based on the array partitioning: Assuming that the array is indexed from 0 to n-1 and s is a split position obtained by the array partitioning: If s = k-1, the problem is solved; if s > k-1, look for the k-th smallest element in the left part; if s < k-1, look for the (k-s)-th smallest element in the right part. s all are ≤ A[s] all are ≥ A[s] A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Two Partitioning Algorithms There are two principal ways to partition an array: One-directional scan (Lomuto’s partitioning algorithm) Two-directional scan (Hoare’s partitioning algorithm) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Lomuto’s Partitioning Algorithm Scans the array left to right maintaining the array’s partition into three contiguous sections: < p,  p, and unknown, where p is the value of the first element (the partition’s pivot). On each iteration the unknown section is decreased by one element until it’s empty and a partition is achieved by exchanging the pivot with the element in the split position s. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Lomuto’s Partitioning Algorithm A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example of Lomuto Partition

Quickselect A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Efficiency of Quickselect Average case (average split in the middle): C(n) = C(n/2)+(n+1) C(n)  Θ(n) Worst case (degenerate split): C(n)  Θ(n2) A more sophisticated choice of the pivot leads to a complicated algorithm with Θ(n) worst-case efficiency. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Divide-and-Conquer The most-well known algorithm design strategy: Divide instance of problem into two or more smaller instances Solve smaller instances recursively Obtain solution to original (larger) instance by combining these solutions A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Divide-and-Conquer Technique (cont.) a problem of size n subproblem 1 of size n/2 subproblem 2 of size n/2 a solution to subproblem 1 a solution to subproblem 2 a solution to the original problem A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Divide-and-Conquer Examples Sorting: mergesort and quicksort Binary tree traversals Multiplication of large integers Matrix multiplication: Strassen’s algorithm Closest-pair and convex-hull algorithms A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

General Divide-and-Conquer Recurrence T(n) = aT(n/b) + f (n) where f(n)  (nd), d  0 Master Theorem: If a < bd, T(n)  (nd) If a = bd, T(n)  (nd log n) If a > bd, T(n)  (nlog b a ) Note: The same results hold with O instead of . Examples: T(n) = 4T(n/2) + n  T(n)  ? T(n) = 4T(n/2) + n2  T(n)  ? T(n) = 4T(n/2) + n3  T(n)  ? A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Mergesort Split array A[0..n-1] in two about equal halves and make copies of each half in arrays B and C Sort arrays B and C recursively Merge sorted arrays B and C into array A as follows: Repeat the following until no elements remain in one of the arrays: compare the first elements in the remaining unprocessed portions of the arrays copy the smaller of the two into A, while incrementing the index indicating the unprocessed portion of that array Once all elements in one of the arrays are processed, copy the remaining unprocessed elements from the other array into A. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Pseudocode of Mergesort A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Pseudocode of Merge A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Mergesort Example Go over this example in detail, then do another example of merging, something like: (1 2 5 7 9) (3 4 6) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Analysis of Mergesort All cases have same efficiency: Θ(n log n) Number of comparisons in the worst case is close to theoretical minimum for comparison-based sorting: log2 n! ≈ n log2 n - 1.44n Space requirement: Θ(n) (not in-place) Can be implemented without recursion (bottom-up) even if not analyzing in detail, show the recurrence for mergesort in worst case: T(n) = 2 T(n/2) + (n-1) worst case comparisons for merge A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Quicksort Select a pivot (partitioning element) – here, the first element Rearrange the list so that all the elements in the first s positions are smaller than or equal to the pivot and all the elements in the remaining n-s positions are larger than or equal to the pivot (see next slide for an algorithm) Exchange the pivot with the last element in the first (i.e., ) subarray — the pivot is now in its final position Sort the two subarrays recursively p A[i]p A[i]p A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Hoare’s Partitioning Algorithm A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Quicksort Example 5 3 1 9 8 2 4 7 A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Analysis of Quicksort Best case: split in the middle — Θ(n log n) Worst case: sorted array! — Θ(n2) Average case: random arrays — Θ(n log n) Improvements: better pivot selection: median of three partitioning switch to insertion sort on small subfiles elimination of recursion These combine to 20-25% improvement Considered the method of choice for internal sorting of large files (n ≥ 10000) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

For Next Class, Wednesday Class cancelled on Monday, February 12 Quiz 5 AND Homework 4 due Tuesday night!