Review for Midterm Exam

1. Review for Midterm Exam Andreas Klappenecker

2. Topics Covered • Finding Primes in the Digits of Euler's Number • Asymptotic Notations: Big Oh, Big Omega, Big Theta • Time complexity of Insertion Sort • Lower bounds for sorting • Divide-and-Conquer Algorithms • Mergesort • Strassen's method for Matrix Multiplication • Greedy Algorithms, Huffman codes • Greedy Algorithms for Matroids • Matroid Embeddings • Dynamic Programming, Matrix Chain Multiplication • Dynamic Programming, Longest Common Subsequence • Amortized Analysis • Disjoint Sets

3. Asymptotic Notations O(g) = { f:N->R | there exists an integer n0 and a real constant C such that |f(n)| <= C|g(n)| for all n>= n0 } (g) = { f:N->R | there exists an integer n0 and a real constant c such that |f(n)| => c|g(n)| for all n>= n0 }

4. Asymptotic Notation • ½(n2+n+6) = O(n2) • 6n2 = O(n2) • 10765432n2+2n+7= (n2) • ½(n2+n+6) = (n2) • (g) = (g)  O(g) • ½(n2+n+6) = (n2)

5. Sorting • Insertion Sort • Best case running time: linear • Worst case running time: quadratic • Merge Sort O(n log n) • Any comparison based sorting (n log n)

6. Divide-and-Conquer • Mergesort • Quicksort • Strassen’s matrix multiplication algorithm • Recurrence relations • Master theorem (no need to memorize)

7. Greedy Algorithms • Coin change • Huffman codes • Matroids • Kruskal’s algorithm • Matroid embeddings • Prim’s algorithm

8. Dynamic Programming • Matrix chain multiplication • Longest common subsequences • Variations: Edit distance

9. Amortized Analysis • Aggregate Analysis • Accounting Method • Stacks • Counter • Disjoint Sets

10. Exam • Some short questions • Some workout problems • Lectures • Slides • Textbook • Quizzes • Homework