Test 1 Review for CS 5633, Spring 2012

Relevant Material:

• All material from 1/17/12 until 2/9/12 (inclusive)
• This includes material covered in the lectures, and homeworks 1 - 4

1. Analyzing Algorithms
• Best case, worst case, and average case runtimes
• Loop invariants
2. Asymptotic Notation (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 satisfies)
• f(n) belongs to O(g(n))
• f(n) belongs to Ω(g(n))
• f(n) belongs to Θ(g(n)) [prove both O and Ω]
• Prove using limit theorem
• Analyze code snippets
3. Recursion:
• Divide & Conquer
• You can call an algorithm Divide and Conquer only if the size of subproblems can be written as n/b where b>1
• Regular Recursion
• subproblems can be of size n-1, n-2, n-3 etc.
• Recursive algorithms and thinking
• Find a recursive solution to a problem; recursive pseudocode
• Fibonacci, recursive squaring, Strassen's algorithm
• Runtime recurrence
• Find the runtime recurrence for a recursive algorithm given in pseudo-cde
4. Recursion Tree: Find a guess what a (runtime) recurrence could solve to using Recursion Tree method
• 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]
5. Solving Recurrence: Solve a runtime recurrence (i.e., find and asymptotic bound for a recursively given T(n)):
• Big-Oh induction (= substitution method)
• you will have a guess/claim
• state your inductive hypothesis for all k<n
• replace k with a value[the size of subproblem in the right hand side of recurrence] < n
• plugin the value you got in previous step into your recurrence
• find out your desired and residual part
• residual part must be<0, so you will end up with a condition most likely on c
• Verify if it works for your base case
• select a value of c(which satisfies the base case and the condition you got from residual<0) and n_0(base case value of n)
• Master Method
• find the value of a and b
• compute n^(log of a base b) and compare with O(f(n))
• please clearly write which case it is and the value of epsilon(for case I and III), k (for case II) and check regularity condition and give the value of c if it is case III
6. Probability, Random variables and Expected values
• Define Sample Space and Random Variables for given problem
• Compute expected values of your random variables using definition of expectation/linearity of expectation
7. Randomized Algorithms
• Expected runtime vs. average runtime, best-case runtime, and worst-case runtime
8. Quicksort
   • Runtime:
     • Best case: O(n log n) [When each pivot partitions the array into two roughly equal pieces]
     • Worst case: O(n^2) [When the array is already sorted either increasing or decresing order]
     • Average Case: O(n log n) [Randomized runtime analysis will not be needed]
   • Sample Questions:
     • For a given Sequence of numbers what will be the runtime when Quicksort, Radixsort or Countingsort is used?
     • Show different steps (pivot, and the partitions) of sorting an array using Quicksort.
     • Deterministic Quicksort
9. Sorting
   • Radix Sort:
     • Runtime: O(bn/log n)
   • Counting Sort:
     • Runtime: O(n+k)
   • Decision Trees:
     • For a given (sorting) algorithm
10. Order Statistics
    • Select Algorithm: to select the i-th smallest element
      • Randomized Select:
        • Runtime:
          • Worst Case: O(n^2)
          • Average Case/Expected Runtime: O(n)
      • Deterministic Select:
        • Runtime: O(n)