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 Residual Graphs Augmenting path algorithm Max-Flow Min Cut Theorem (Running Time) Capacity Scaling Bipartite Matching reduction to Max Flow

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

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

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

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)

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.

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

Data Compression Huffman Coding Arithmetic coding Relationship to Entropy.

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

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

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

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

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

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

Good Luck