Parallel Algorithms for Geometric Graph Problems Grigory Yaroslavtsev 361 Levine STOC 2014, joint work with Alexandr Andoni, Krzysztof.

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Presentation transcript:

Parallel Algorithms for Geometric Graph Problems Grigory Yaroslavtsev 361 Levine STOC 2014, joint work with Alexandr Andoni, Krzysztof Onak and Aleksandar Nikolov.

Theory Seminar: Fall 14

“The Big Bang Data Theory”

Round Complexity Information-theoretic measure of performance Tools from information theory (Shannon’48) Unconditional results (lower bounds) Example today: Approximating Geometric Graph Problems

Approximation in Graphs s: Given a graph and an optimization problem… Transportation Problem: Tolstoi [1930] Minimum Cut (RAND): Harris and Ross [1955] (declassified, 1999)

Approximation in Graphs 1960s: Single processor, main memory (IBM 360)

Approximation in Graphs 1970s: NP-complete problem – hard to solve exactly in time polynomial in the input size “Black Book”

Approximation in Graphs

The New: Approximating Geometric Problems in Parallel Models s to 2014

The New: Approximating Geometric Problems in Parallel Models

Polynomial time (“easy”) Minimum Spanning Tree Earth-Mover Distance = Min Weight Bi-chromatic Matching NP-hard (“hard”) Steiner Tree Traveling Salesman Clustering (k-medians, facility location, etc.) Geometric Graph Problems Arora-Mitchell-style “Divide and Conquer”, easy to implement in Massively Parallel Computational Models Need new theory!

MST: Single Linkage Clustering [Kleinberg, Tardos]

Earth-Mover Distance Computer vision: compare two pictures of moving objects (stars, MRI scans)

Computational Model S space

Computational Model

MapReduce-style computations

Models of parallel computation Bulk-Synchronous Parallel Model (BSP) [Valiant,90] Pro: Most general, generalizes all other models Con: Many parameters, hard to design algorithms Massive Parallel Computation [Feldman-Muthukrishnan- Sidiropoulos-Stein-Svitkina’07, Karloff-Suri-Vassilvitskii’10, Goodrich-Sitchinava-Zhang’11,..., Beame, Koutris, Suciu’13] Pros: Inspired by modern systems (Hadoop, MapReduce, Dryad, … ) Few parameters, simple to design algorithms New algorithmic ideas, robust to the exact model specification # Rounds is an information-theoretic measure => can prove unconditional lower bounds Between linear sketching and streaming with sorting

Previous work VS.

Large geometric graphs

Wrong representative: O(1)-approximation per level

Wrong representative: O(1)-approximation per level

2 1

2 1

⇒

“Solve-And-Sketch” Framework

Thank you!