Midterm II Review

• 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
• Sorting
  • Radix Sort:
    • Runtime: O(bn/log n)
  • Counting Sort:
    • Runtime: O(n+k)
  • Decision Trees:
    • Mergesort
    • Insertion sort
    • For a given sorting algorithm
• 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)
• Red black trees:
  • Properties of  Binary Search Trees
  • Properties of Red black trees
    •  If black-height is b and the height of the tree is h, then b<=h<=2b
      • h = b when all the nodes in the tree are black
      • h=2b when you have red and black nodes in alternating levels
  • Sample Questions:
    • Justify whether a given tree is a legal Red Black Tree/Binary Tree/Balanced Search Tree
    • Insertion (show rotations if  needed)
    • Maximum and minimum number of elements in a Red Black Tree with black height b
• B-Trees:
  • A B-tree with minimum dregree k >= 2 has following properties
    • Number of keys in root: 1<= #keys <= 2k-1
    • Number of keys in other nodes: k-1 <=  #keys <= 2k-1
    • Each node will have exactly #keys stored in that node + 1 number of children
    • All leaves will be in same level
    • only root split can increase the height of a B-tree
  • Sample Questions:
    1. For a given tree justify if it is a legal B-tree
    2. For a given set of numbers come up with all possible legal B-trees
    3. Maximum and minimum number of keys possible to store in a B-tree
    4. Insertion in a B-tree (show node split if needed)
• Dynamic Programming (DP):
  • It is a problem solving technique (or algorithm design technique) where
    • each subproblem is solved only once and the solution is stored in a Dynamic Programming Table (DP table)
    • and this stored Value/Solution will be used later whenever needed
  • This approach can be used only if the problem has the following two properties:
    • Overlapping Subproblems
    • Optimal Substructure
  • Sample Questions:
    • Understand a given recurrence, and analyze the time and space needed for a brute-force implementation of the recurrence
    • Design a DP algorithm using a given recurrence and either of the following approaches
      • Bottom-up: this approach uses a loop and fills the whole DP table bottom-up ( Example: Homework 8, Problem 3(a))
      • Memoization: this approach uses recursion and fills only the cells of the DP table which are needed ( Example: Homework 8, Problem 3(c))
    • analyze time and space of your DP algorithm
    • Fill the DP Table for a given input and compute the solution from it
  • Modify a well-known DP algorithm (like LCS, Fibonacci, etc.) or the one you develop or a given one, so that it take less space or time
• Greedy Algorithms
  • Greedy Strategies:
    • repeatedly identify the decision to be made (recursion)
    • make a locally optimal choice for each decision
  • Greedy doesn't always work.
  • Sample Questions:
    • Short questions or true/false about the concepts
    • Come up with a greedy solution to a problem or with a counter-example why a greedy solution does not work.