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Advanced Algorithms

Dive into COT5405 Network Flow, Men Propose Women Dispose algorithm, Max-Flow Min-Cut Theorem, LPs, Reductions, Optimization vs. decision problems, Computational Geometry, Data Compression, Streaming Algorithms, String Algorithms, Online Algorithms, Parallel Algorithms, and Perceptrons. Enrich your understanding with key concepts, theoretical foundations, and real-world applications. Best of luck!

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Advanced Algorithms

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  1. Advanced Algorithms Piyush Kumar (Lecture 16: Review) Welcome to the end of COT5405

  2. Network Flow • Men Propose Women Dispose Alg. aka Gale-Shapely Algorithm • Produces Man-Optimal Matchings. (Every man gets his best valid partner)

  3. Network Flow • Max Flow Min Cut • Flows as LP • Residual Graphs • Augmenting path algorithm • Max-Flow Min Cut Theorem (Running Time) • Capacity Scaling • Bipartite Matching reduction to Max Flow

  4. Network Flows • Preflow Push Algorithm (weak conservation) (excess) • Transform a preflow into a flow. • Push or Relabel (When one cant push) • Weighted Bipartite Matching

  5. Reductions • Polynomial Time Reductions • Ex: 3-SAT  P INDEPENDENT-SET  P VERTEX-COVER  P SET-COVER. • Optimization Vs decision problems • P,NP, EXP, Certifiers

  6. NP Completeness • First Problem: CIRCUIT SAT • Some examples • Co-NP: Complementary decision problems in NP. • Good characterization. • Bipartite perfect matching • Primes/Factor

  7. Approximation Alg. • c-approximation algorithms • Vertex cover: Greedy/ 2-factor • Hardness: HC / Directed HC / TSP • TSP: 2-factor with triangle inequality • Hardness of approximation of general TSP (Gap TSP)

  8. LP • Standard Form • 2d LPs • Integer Programs and relaxation • Dual of LPs • Primal-Dual picture • Weak Duality: by <= cx • Complementary slackness • VC Approximation using LP Rounding.

  9. Computational Geometry • LP using RIC in fixed dimensions • Convex hulls: Jarvis March/Graham Scan • Range trees, kd-trees.

  10. Data Compression • Huffman Coding • Arithmetic coding • Relationship to Entropy.

  11. D&C • Counting inversions • Closest pair of points • Integer multiplication • Matrix multiplication/transpose • External memory algorithms: DAM Model • Sorting in DAM • Static Searches

  12. D&C • Cache oblivious algorithms: Van EmdeBoas Layout • Introduction to Streaming Algorithms • Fast Fourier Transforms • And applications?

  13. String Algorithms • Dynamic Programming: Edit Distance computation • Short introduction to search engines • Tries • Compressed Tries : Path compression • Suffix Tries

  14. Online Algorithms • Dating Problem • Ski Rental Problem • Monkey looking for food • Competitiveness of LRU.

  15. Parallel Algorithms • PRAM • Prefix Computation • Array Packing • Mergesort • Closest Pair • Planar Convex hulls.

  16. Perceptrons • Feasibility and lp optimization • Perceptrons solve LP Feasibility • Convergence proof.

  17. Good Luck

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