Scribing: Students take turns. Notes are written in LaTeX, using a provided template.
P-sets: Must be submitted in LaTeX by email, with page limits.
Final Project: Proposal due October 30th, project due last day of reading period.
Grading: Students also take turns being graders.
Course Goals
Increase ability to analyze and create algorithms.
Learn different techniques for analyzing algorithms not covered in CS124.
Focus on theoretical models and efficiency measures.
Key Concepts Discussed
Sorting and Algorithms
Sorting Complexity: It's known that you cannot sort n numbers faster than n log n using comparison-based sorting.
WordRAM Model: Assumes integers fit in a word, W, allowing operations beyond comparisons (e.g., bitwise operations).
Van Emde Boas Trees
A data structure for dynamic predecessor queries.
Structure:
Recursive structure with sub-clusters.
Stores minimum and maximum.
Operations:
Predecessor Query: Uses divide & conquer; time complexity is O(log log U)
Insertion: Uses swaps and recursive insertion; time complexity is O(log log U)
Space Complexity: Originally U space, can be reduced to linear space with hashing.
Fusion Trees
A more advanced data structure for static predecessor queries.
Efficiency: Logarithmic time based on word size.
Uses linear space.
Advanced Sorting
Sorting can be improved in the WordRAM model:
Deterministic: O(n log log n)
Randomized: O(n sqrt(log log n))
Data Structure Models Discussed
Static vs Dynamic: Static doesn't change items, dynamic allows insertions and deletions.
Predecessor Problem: Finding the largest value less than a given number.
Binary Search Trees: Used for dynamic cases with log n time complexity.
Hashing and Dictionaries
Discussed briefly in context of reducing space.
Dictionary problem: storing key-value pairs.
Additional Concepts
Bitwise Operations: Exploited in algorithms for efficiency beyond comparisons.
Indirection Technique: Used to optimize space and time complexity in data structures.
Questions and Clarifications
Discussed potential optimizations and concerns about time/space efficiency.
Addressed how to implement certain operations using bit manipulation and trees.
Conclusion
The lecture concluded with outlining the constraints and potential improvements of the discussed data structures, as well as preliminary discussion on upcoming topics.