Final Review CMPS 6610, Fall 2018

The Final exam is Monday, December 10th, 9am-noon in the regular classroom (GI 325)

Relevant Material:

• All material from 8/27/18 until 12/5/18 (inclusive).
• This includes material covered in the lectures and homeworks 1 - 10.

Topics from Test 1

1. Analyzing Algorithms (Ch. 2.2)
• Best case and worst case runtimes


2. Asymptotic Notation (Ch. 3, A)
• О, Ω, Θ, ο
• Prove by definition (find a value for c>0 and n_0 > 0 so that for all n≥n_0 the inequality is satisfied)
  • f(n) ∈ O(g(n))
  • f(n) ∈ Ω(g(n))
  • f(n) ∈ Θ(g(n)) [prove both O and Ω]
• Know how to use the limit theorem.
• Analyze code snippets.
• Know summations: Arithmetic series, geometric series (finite and infinite), Harmonic number.
• Practice problems from the book:
  • page 22: 2.1-2, 2.1-3
  • page 39: 2.3-3,4,5,6,7
  • page 39: 2-1,4


3. Heaps (Ch. 6)
   • Heap definition; max-heap property
   • Using array implementation, findMin takes O(1) time, and extractMin and decreaseKey take O(log n) time
   • Heapsort: Repeatedly extract min in min-heap; O(n log n) time
   • Practice problems from the book:
     • page 153: all exercises
     • page 156: 6.2-3, 6.2-4
     • page 160: 6.4-3
     • page 167: 6-2
   
   
4. Divide & Conquer (Ch. 2.3, 4.3, 4.4, 4.5, 4.6)
   • You can call an algorithm divide-and-conquer only if the size of subproblems can be written as n/b where b>1
   • Mergesort, binary search, recursive squaring
   • Not: Convex hull
   • Find the runtime recurrence for a recursive algorithm given in pseudocode
   • Solving Recurrences: Solve a runtime recurrence (i.e., find an upper bound for a recursively defined T(n)):
     • Recursion Tree: Find a guess what a (runtime) recurrence could solve to using recursion trees
       • Given T(n) = aT(n/b) + f(n)
       • a = #of subproblems = #of children at each node
       • n/b = subproblem size
       • Height of the tree = log_b (n) [log of n base b]
     • Big-Oh Induction
       • Prove your guess/claim using big-Oh induction (substitution method): Find conditions on constants c and n_0 in inductive step. (No need to handle base case.)
     • Master Theorem
       • Compute n^(log_b (a)) and compare with f(n)
       • Clearly write which case of the Master theorem applies, and give the values for ε, k, and c:
         • For case 1: Give the value of ε>0
         • For case 2: Give the value of k≥0
         • For case 3: Give the value of ε>0, check the regularity condition and give the value of c<1
       • Proof of the master theorem
   • Practice problems from the book:
     • page 87: all exercises
     • page 92: all exercises
     • page 96: 4.5-1,2,3
     • page 107: 4-1,3
   
   
5. Quicksort (Ch. 5.2, C.3, 7)
   • Randomized Algorithms
     • Probability and expectation
     • Expected runtime vs. average runtime, best-case runtime, and worst-case runtime
   • Quicksort
     • Deterministic quicksort:
       • Best-case runtime O(n log n) [When each pivot partitions the array into two roughly equal pieces]
       • Worst-case runtime: O(n^2) [When the array is already sorted either increasing or decreasing order]
     • Randomized quicksort: Expected runtime O(n log n)
   • Practice problems from the book:
     • page 1200: C.3-1,2,3,4
     • page 122: 5.2-3,4,5
     • page 143: 5-2
     • page 173: all exercises
     • page 178: 7.2-1,2,3

Topics from Test 2

See Test 2 Review

Other Topics

6. Network Flow (Ch. 26.1-26.3)
   • Definitions of flow network, flow, cuts
   • Definitions of residual network, augmenting path. Edges with zero residual capacity do not exist in the residual network (alternatively, an augmenting path over them would have capacity zero).
   • Max-flow min-cut theorem
   • Ford-Fulkerson:
     • Chooses an arbitrary graph in the residual network to augment
     • Runtime: O(|E| |f*|)
   • Edmonds-Karp:
     • Chooses a shortest augmenting path in the residual network. The length of the path is measured by the number of edges.
     • Runtime: O(|V| |E|^2)
   • Maximum bipartite matching
   • Practice problems from the book:
     • page 712: 26.1-1,6
     • page 730: 26.2-2,3,4,10,11
   
   
7. More Randomized Algorithms (Extra material on slides/pictures page)
   • Global Min-Cut
   • Las Vegas vs. Monte Carlo algorithms
   • High-probability bound for Quicksort (turn Las Vegas algorithm into Monte Carlo)
   • Practice problems:
     • Jeff Erickson's Min-Cut notes (page 6): 1, 3, 4
   
   
8. P and NP (Ch. 34, 35.1, 35.2
   • Definition of P, NP, NP-hard, NP-complete, Co-NP
   • Reductions
   • NP-complete problems (SAT, Clique, TSP, Hamiltonian Cycle, Vertex Cover, Independent Set)
   • Approximation algorithms (Vertex Cover, TSP)
   • Practice problems from the book:
     • page 1060: 34.1-1
     • page 1065: 34.2-1,2,3,6,7,8,9,10
     • page 1100: 34.5-1,5,6,7
     • page 1111: 35.1-1,2,4
     • page 1122: 35.3-2