Review of course COMP-251B winter 2010

Transcription

1 Review of course COMP-251B winter 2010 Lecture 1. Book Section 15.2 : Chained matrix product Matrix product is associative Computing all possible ways of parenthesizing Recursive solution Worst-case running-time of straightforward recursion Dynamic programming solution, running-time Lecture 2. Book Section 4.3 : Master Method Master Method, Cases 1, 2, 3: comparing log b a to f(n). Lectures 3-4. Book Section : Quicksort Description of Quicksort: running-time analysis Lectures 4-5. Book Sections : Hiring problem and Indicator Random variables Randomized algorithms vs probabilistic analysis, Hiring problem: deterministic vs randomized, Using Indicator random variables in running time analysis, Analysis of randomized Hiring Problem. Lecture 5. Book Section 7.4 : Quicksort Randomized Quicksort Analysis of randomized Quicksort (Constructive) mathematical induction proofs Lecture 6. Book Section 9.1 : Minimum and Maximum Separately: n-1 comparisons Together: (3/2)n comparisons.

2 Lecture 7. Book Section 9.2 : SELECT in expected O(n) time Analysis of randomized SELECT, Relation to QUICKSORT. Lecture 8. Book Section 9.3 : SELECT in worst case O(n) time Description of deterministic SELECT Analysis of deterministic SELECT (Constructive induction) Lecture 9. Book Chapter 8 : n log n lower bound on comparison sorting. Decision trees Worst case running time = height of decision tree n log n lower bound for sorting Lecture 10. Book Chapter 8 : Sorting in O(n) time Numbers bounded by k, Counting sort, O(n) if k is O(n), Stable sorts, Radix sort, O(n) if k is O(n c ), Bucket sort, O(n) if randomly distributed. Lecture 11. Book Chapter 12 : Binary Search Trees Reminder of BST property Operations on BST o Tree-search o Tree-minimum, Tree-maximum o Tree-predecessor, Tree-Successor o Tree-insert o Tree-Delete

3 Lectures Book Chapter 13 : Red-Black Trees Red-black property Operations on RB-T o RB-Tree-search, min, max, pred, succ o RB-Tree-insert and RB-insert-fixup o RB-Tree-Delete and RB-delete-fixup Lecture 14. MID-TERM EXAM! Expect that questions failed on the mid-term may be on the final. Lectures 15,16,19. Book Chapter 15 : Dynamic Programming Avoiding exponential time recursions Optimal sub-structure principle Optimal BST Lecture 18. Book Chapter 16 : Greedy Algorithms. Basics of greedy algorithms Making change Greedy Fractional Knapsack Running time Lectures Book Chapter 21 : Data structures for Disjoint Sets. Finding connected components in a graph Disjoint sets operations o Make-set o Union o Find-set Link list representation Forest representation Union by rank and path compression

4 Lecture Book Chapter 23 : Minimum Spanning Trees Definition Generic MST algorithm Finding safe edges Cuts, edge crossing, light edges, Kruskal s algorithm and Data structure for disjoint sets Prim s algorithm and priority queues Lectures Book Chapter 24 : Single-source shortest paths. Weight of edges and paths Optimal sub-structure Negative-weight edges/cycles Relaxation Properties of Shortest paths and Relaxation Bellman-Ford algorithm Dijkstra s algorithm Running time analysis Lectures Book Chapter 25 : All-Pairs shortest paths. Shortest paths and matrix multiplication Recursive definition Sequential determination of shortest paths Improving running time Floyd-Warshall Johnson s algorithm for sparse graphs Re-weighting Lecture 27. Book Chapter 31 : Number Theoretic Algorithms. Exponentiation mod n RSA cryptosystem Quantum computing and Shor s algorithm

5 Lecture 28. Review of course material in preparation to FINAL EXAM!