Test 2 Review

• 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 the same level
    • only a 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)
• Augmenting Data Structures:
  • Methodology for augmenting data structures: Determine data structure, determine additional information to be stored, maintain new operations in operations that modify the data structure, develop new operations that use the additional information.
  • Dynamic order statistics.
  • Interval trees.
• Range Trees:
  • New BST ordering with keys in the leaves.
  • Definition of one-dimensional range tree, and range search algorithm (using canonical sets = subtrees in between search paths)
  • Different range query types: Existence, counting, reporting.
  • Higher-dimensional (nested) range trees
  • (No fractional cascading)
• 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 recursive solution, and analyze the time and space needed for a brute-force implementation of the recursion.
    • Develop a recursive solution for a problem similar to problems covered in class or on the homework.
    • 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
      • Memoization: this approach uses recursion and fills only the cells of the DP table which are needed
    • Analyze time and space of your DP algorithm
    • Give an algorithm to fill the DP Table for a given recursive solution and compute the solution from it
• 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.
• Amortized Analysis:
• Aggregate analysis, accounting method
• (No potential method)
• Binary counter, dynamic tables
• Union-Find:
• Union-find data structure definition (operations)
• List implementation, tree implemention
• Union by weight/rank, path compression
• Ackermann function, inverse Ackermann function