05 - 18 Jan - The Sorting Wrap-Up
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This document discusses sorting algorithms and finding medians using heaps. It describes how heapify takes linear time overall to turn an array into a heap. To sort an array using a heap, repeatedly extract the maximum element and place it at the end of the array, requiring O(n log n) time. Finding medians can be done by storing the first half of elements in a max-heap and the second half in a min-heap.
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05 Analysis of Algorithms: Heap and Quick Sort - Corrected
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01 - 01 January - Sorting
This document discusses sorting algorithms that are covered in a CS 321 algorithms course. It introduces several sorting algorithms like counting sort, bubble sort, merge sort, quicksort, and insertion sort. For each algorithm, it provides the time complexity. It also mentions that merge sort runs in O(n log n) time due to having log n layers of recursion where each layer takes O(n) work. Quicksort's recursion tree can potentially be skewed, leading to O(n^2) performance in the worst case. References provided direct the reader to a video and website for learning more about sorting algorithms.
14 - 08 Feb - Dynamic Programming
This document discusses dynamic programming techniques for three problems: edit distance, longest increasing subsequence, and matrix chain multiplication. It provides examples of how to calculate the edit distance between two strings by finding the minimum cost of insertions, deletions, and replacements needed to convert one string to the other. It then explains how to find the longest increasing subsequence of a given array using dynamic programming by building up the solution bottom-up. Finally, it briefly mentions finding the most efficient way to multiply a chain of matrices using dynamic programming.
11 - 03 Feb - From Recursion to Dynamic Programming
This document discusses recursion and memoization. It introduces the Fibonacci numbers and the problem of tiling a corridor of length n. It shows the recursive definition of the number of tilings as C(n) = C(n-1) + C(n-2), with base cases C(1)=1 and C(2)=2. It notes that directly implementing this recursion leads to redundant computations. Memoization is proposed as a technique to store computed values in a table to avoid recomputing them, making the algorithm more efficient.
09 - 27 Jan - Recursion Part 1
The document discusses algorithms for solving satisfiability problems on conjunctive normal form (CNF) formulas. It first describes a trivial algorithm that tries all possible assignments, with a runtime of O(2^n*n^3). It then discusses using a recursive approach, but shows this results in an even worse runtime of O(3^n*n^3). Finally, it proposes a recursive algorithm that runs in O(1.8^n) by recursively trying assignments on subsets of variables.
04 - 15 Jan - Heap Sort
This document discusses heap sort and how to implement a min heap. It shows the steps of heap sort by organizing elements into a min heap where the first element is always the minimum. It explains that a min heap can support fast insertion and deletion of minimum elements in O(log n) time by maintaining the heap property that a node is larger than its children. The key aspects are using a complete binary tree structure represented in an array and algorithms for heapify, insert, and delete min that maintain the min heap ordering.
12 - 05 Feb - Dynamic Programming
The document discusses dynamic programming and its application to finding the independent set with the largest total weight on a path. It describes how dynamic programming can be used to solve this problem by breaking it down into subproblems and storing the solution to each subproblem in an array A, such that A[i] stores the maximum weight independent set for the subpath on the first i vertices. The value of each entry A[i] is computed recursively as the maximum of A[i-1] and A[i-2] + the weight of vertex i.
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05 - 18 Jan - The Sorting Wrap-Up
- 1. CS 321. Algorithm Analysis & Design Lecture 5 The Sorting Wrap-Up
- 2. 19 1 2 13 8 5 6 3 12 11 7 16
- 3. 19 1 2 13 8 5 6 3 12 11 7 16 19
- 4. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2
- 5. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6
- 6. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6 3 12 11 7 16
- 7. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6 3 12 11 7 16
- 8. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6 3 12 11 7 16
- 9. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6 3 12 11 7 16
- 10. 19 1 2 13 8 5 6 3 12 11 7 16 19 1 2 13 8 5 6 3 12 11 7 16
- 11. 19 1 2 13 8 56 3 12 11 716 19 1 2 13 8 5 6 3 12 11 7 16
- 12. 19 1 2 13 8 56 3 12 11 716 19 1 2 13 8 5 6 3 12 11 7 16
- 13. 19 1 2 13 8 56 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 14. 19 1 2 13 8 56 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 15. 19 1 2 13 8 56 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 16. 19 1 213 8 56 3 1211 716 19 1 213 8 5 6 3 12 11 7 16
- 17. 19 1 213 85 6 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 18. 19 1 213 85 6 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 19. 19 1 213 85 6 3 1211 716 19 1 2 13 8 5 6 3 12 11 7 16
- 20. 19 1 213 85 6 312 11 716 19 1 2 13 8 5 6 3 12 11 7 16
- 21. 19 1 213 85 6 312 11 716 19 1 2 13 8 5 6 3 12 11 7 16
- 23. Node at height h has to move at most h steps to find its place.
- 24. Node at height h has to move at most h steps to find its place. Number of nodes at height h?
- 25. Node at height h has to move at most h steps to find its place. Number of nodes at height h? 2(log n - h)
- 26. Node at height h has to move at most h steps to find its place. Number of nodes at height h? 2(log n - h) Overall running time (summed over h):
- 27. Node at height h has to move at most h steps to find its place. Number of nodes at height h? 2(log n - h) n(h/2(h)) Overall running time (summed over h):
- 28. Node at height h has to move at most h steps to find its place. Number of nodes at height h? 2(log n - h) n(h/2(h)) Overall running time (summed over h): Heapify: Turns out to be a linear effort overall.
- 29. 19 1 2 13 8 5 6 3 12 11 7 16 19 13 16 12 11 5 6 3 1 8 7 2
- 30. 19 1 2 13 8 5 6 3 12 11 7 16 19 13 16 12 11 5 6 3 1 8 7 2 To sort, repeatedly extract the maximum element to the end of the array. An O(n log n) effort.
- 31. Finding Medians
- 33. 19 13 16 12 11 5 6 3 1 8 7 2 19 13 16 12 11 5 6 3 1 8 7 2
- 35. Store the first (n/2) elements in a max-heap. Store the last (n/2) elements in a min-heap.