MIT 6.851 Advanced Data Structures

Overview

This project provides functional implementations of advanced data structures covered in MIT's 6.851 graduate course, emphasizing immutability and persistence while maintaining theoretical efficiency bounds.

Implemented Structures

Temporal Data Structures

• Persistent Arrays: Full persistence with O(log n) access
• Retroactive Priority Queues: Support for retroactive operations
• Partial Persistence: Generic transformation techniques

Geometric Data Structures

• Range Trees: Multi-dimensional range queries
• Segment Trees: Interval queries with lazy propagation
• K-d Trees: Spatial partitioning with functional updates

Succinct Data Structures

• Rank/Select: Bit vectors with constant-time operations
• Wavelet Trees: Compressed sequence representation
• Succinct Trees: Tree structures in 2n + o(n) bits

Functional Programming Approach

Each implementation follows functional programming principles:

;; Example: Persistent balanced tree
(define (insert tree key value)
  (match tree
    [(empty) (leaf key value)]
    [(node k v left right)
     (cond
       [(< key k) (balance (node k v (insert left key value) right))]
       [(> key k) (balance (node k v left (insert right key value)))]
       [else (node key value left right)])]))

Educational Value

Clear Implementations

• Each data structure includes detailed comments
• Step-by-step derivation of complexity bounds
• Visualization tools for understanding operations

Comparative Analysis

• Functional vs imperative trade-offs
• Space-time complexity comparisons
• Practical performance measurements

Research Contributions

This work explores: 1. Persistence Techniques: Novel approaches to making imperative structures persistent 2. Cache-Oblivious Algorithms: Functional algorithms that perform well across memory hierarchies 3. Verification: Formal proofs of correctness for key operations

Integration with Other Projects

The data structures developed here serve as building blocks for: - Formal verification experiments - Performance benchmarks for functional languages - Teaching materials for advanced algorithms courses

Future Directions

• GPU-accelerated functional data structures
• Concurrent versions with strong consistency guarantees
• Integration with theorem provers for verified implementations