Midterm Review for CMPS 6610/4610, Fall 2016

Relevant Material:

• All material from 8/30/16 until 10/4/16 (inclusive).
• This includes material covered in the lectures, and homeworks 1 - 4.

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.
3. Heaps and Binary Search Trees (Ch. 6, 12.1, 12.2, 13.2)
   • Binary search tree definition; BST property
   • Rotations
   • In-order tree traversal
   • 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
4. Divide & Conquer (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
   • 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
5. Quicksort and Order Statistics (5.2, C.3, 7, 9)
   • 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)
   • Order Statistics
     • Select the i-th smallest element
     • Randomized selection: Worst-case runtime O(n^2), expected runtime O(n)
     • Deterministic selection: Select median of medians; worst-case runtime O(n)
6. Dynamic Programming (Ch. 15.2-15.4):
• Applies only if the problem has the following two properties:
  • Optimal substructure (recursion)
  • Overlapping subproblems
• In dynamic programming each subproblem is solved only once, and its solution is stored in a DP table. This stored value/solution will be used later whenever needed.
• LCS, matrix chain multiplication
• (Bottom-up) DP: Use one or more nested loops and fill the whole DP table bottom-up.

Practice Problems from the Book:

1. Analyzing algorithms:
   • page 22: 2.1-2, 2.1-3
   • page 39: 2.3-3,4,5,6,7
   • page 39: 2-1,4
2. Asymptotic notation:
   • page 52: 3.1-1,2,3,4,5
   • page 61: 3-1,2,3,4
3. Heaps and binary search trees:
   • page 153: all exercises
   • page 156: 6.2-3, 6.2-4
   • page 160: 6.4-3
   • page 167: 6-2
   • page 289: 12.1-1,2,3,4
   • page 314: 13.2-3
4. Divide & conquer:
   • page 87: all exercises
   • page 92: all exercises
   • page 96: 4.5-1,2,3
   • page 107: 4-1,3
5. Quicksort and order statistics:
   • 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
   • page 219: 9.2-3,4
6. Dynamic programming:
   • page 389: 15.3-1,2,3,4
   • page 396: 15.4-1,2,3,4,5
   • page 406: 15-5