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Advanced Algorithms

Advanced Algorithms. Piyush Kumar (Lecture 16: Review). Welcome to the end of COT5405. Network Flow. Men Propose Women Dispose Alg. aka Gale-Shapely Algorithm Produces Man-Optimal Matchings. (Every man gets his best valid partner). Network Flow. Max Flow Min Cut Flows as LP

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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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