Sketching, Sampling and other Sublinear Algorithms: Algorithms for parallel models Alex Andoni (MSR SVC)

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Presentation transcript:

Sketching, Sampling and other Sublinear Algorithms: Algorithms for parallel models Alex Andoni (MSR SVC)

Parallel Models  Data cannot be seen by one machine  Distributed across many machines  MapReduce, Hadoop, Dryad,…  Algorithmic tools for the models?  very incipient!

Types of problems  0. Statistics: 2 nd moment of the frequency  1. Sort n numbers  2. s-t connectivity in a graph  3. Minimum Spanning Tree on a graph  … many more!

Computational Model

Model Constraints

PRAMs

Problem 0: Statistics IP

Problem 1: sorting

Problem 2: graph connectivity VS

Problems 3: geometric graphs

Problem: Geometric MST [A-Nikolov-Onak-Yaroslavtsev’??]

General Approach  Partition the space hierarchically in a “nice way”  In each part  Compute a pseudo-solution to the problem  Sketch the pseudo-solution with small space  Send the sketch to be used in the next level/round

MST algorithm: attempt 1  Partition the space hierarchically in a “nice way”  In each part  Compute a pseudo-solution to the problem  Sketch the pseudo-solution with small space  Send the sketch to be used in the next level/round quad trees! compute MST send any point as a representative

Troubles  Quad tree can cut MST edges  forcing irrevocable decisions  Choose a wrong representative

MST algorithm: final

MST algorithm: Glimpse of analysis

Finale  Gotta love your models:  Streaming:  sub-linear space  see all data sequentially  Parallel computing:  sub-linear space per machine  data distributed over many machines  communication (rounds) expensive  Algorithmic tools in development!