Advanced Algorithms

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Advanced Algorithms

• Piyush Kumar
• (Lecture 16 Review)

Welcome to the end of COT5405
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Network Flow

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

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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

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Network Flows

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

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Reductions

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

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NP Completeness

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

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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)

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LP

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

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Computational Geometry

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

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Data Compression

• Huffman Coding
• Arithmetic coding
• Relationship to Entropy.

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DC

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

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DC

• Cache oblivious algorithms Van EmdeBoas Layout
• Introduction to Streaming Algorithms
• Fast Fourier Transforms
• And applications?

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String Algorithms

• Dynamic Programming Edit Distance computation
• Short introduction to search engines
• Tries
• Compressed Tries Path compression
• Suffix Tries

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Online Algorithms

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

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Parallel Algorithms

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

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Perceptrons

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

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Good Luck