COMP333

Algorithm Theory and Design

Daniel Sutantyo

Department of Computing

Macquarie University

Summary

• Algorithm complexity (running time, recursion tree)
• Algorithm correctness (induction, loop invariants)
• Problem solving methods:
  • exhaustive search
  • dynamic programming
  • greedy method
  • divide-and-conquer
  • algorithms involving strings
  • probabilistic methods
  • algorithms involving graphs

Week 1-2: Algorithms, Complexity, Correctness

• formal statement of a problem, types of problems
• induction
• algorithm complexity:
  • \(O\)-notation, \(\Theta\)-notation, \(\Omega\)-notation
  • recursion tree, master method, and substitution method
• algorithm correctness:
  • proof by induction for recursive algorithms
  • loop invariants for iterative algorithms
• see workshop questions for practice

Week 3: Brute Force Algorithms

• search-space tree, permutations and combinations
• \(O(n!)\) and \(O(2^n)\) 
• recursive backtracking, heuristic
• not very examinable
  • the point of the topic is to give you some intuition on problem solving, and it is largely replaced by the topics in the following weeks
• workshop questions can be 'difficult'

Week 4: Dynamic Programming

• search-space tree
• problems and subproblems (decomposition)
• optimal substructure, cut-and-paste technique
• approach:
  • show there is an optimal substructure
  • show the recursive relation that gives the optimal solution
  • compute value of the optimal solution
  • construct an optimal solution
• workshop: maximum subarray, longest increasing subsequence

Week 5: Greedy Algorithm

Week 6: Divide and Conquer

• majority of time was spent on master theorem, recursion-tree method and substitution method for finding complexity of algorithms (i.e. week 1-2 material)
• workshop questions are very relevant

Week 7: Strings algorithm

• longest common subsequence and edit distance problems
• workshop questions:
  • do not have to worry about Huffman codes and string matching

Week 8: Probabilistic algorithm

• understand what probabilistic algorithm is
  • Monte Carlo vs Las Vega
• difficult topic to examine
  • requires backgound in probability theory
• quicksort, Miller-Rabin test, 
• don't worry about Rabin-Karp string matching (from workshop)

Week 9: Graph Algorithms

• Graph traversal, DFS, BFS
• Maximum flow problem
• Bellman-Ford
• Matrix multiplication method
• Floyd-Warshall algorithm

Exam

• 3 hours exam
• Part A: 135 marks (6 questions), Part B: 45 marks (3 questions)
• Some notes:
  • please revise induction, loop invariants, recursion-tree method, substitution method
  • some questions will be a repeat of workshop questions
  • some questions will be 'new', a problem that you (probably) have never seen before

Exam

• Part A: 6 questions, each worth 15-25 marks, covering:
  • Complexity
  • Correctness
  • Greedy Algorithm
  • Dynamic Programming
  • Probabilistic Method
  • Graph Algorithms
• Subparts of a question may ask questions related to brute-force methods