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Agenda 1 cs848 Models and Applications of Distributed Data - - PowerPoint PPT Presentation

Aug 21, 2023 183 likes •416 views

n Background n Model n Hamming Distance 1 n Triangle Finding n Matrix Multiplication Agenda 1 cs848 Models and Applications of Distributed Data Processing Systems The Problem Tradeoff between parallelism and communication cost in a

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Agenda

n Background n Model n Hamming Distance 1 n Triangle Finding n Matrix Multiplication

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Background

The Problem

  • Tradeoff between parallelism and communication cost in a

map-reduce computation.

  • The finer we partition the work of the reducers so that more

parallelism can be extracted, the greater will be the total communication between mappers and reducers.

  • Limited bandwidth
  • Limited resources(memory, processing units…)

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Background

Why important

  • Explore the bounds on the cost of map-reduce

computation.

  • Optimize the algorithms for problem.

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Background

Previous Work

  • First work that addresses the tradeoff between reducer size and

communication cost in one round Map-Reduce computations.

  • Theta-join implementation by Map-Reduce: only one special

case.

  • Limit the input size of any reducer: limits consideration to

algorithms that we might think of as truly parallel.

  • …

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Model

  • A model of problems that can be solved in a single round
  • f map-reduce computation.

Two Parameters

  • Replication rate r: average number of key-value pairs to

which each input is mapped by the mappers.

  • Reducer size p: the maximum number of inputs that one

reducer can receive.

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Model

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r = 2 p = 4

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Model

Tradeoff

  • Determine the best algorithm for a problem where:
  • Cost of solving the problem:
  • Replication rate:

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r = f (q)

af (q)+ bq(+cq2)

r = qi I

i=1 p

∑

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Model

Mapping Schemas

  • No reducer is assigned more than q inputs.
  • For every output, there is (at least) one reducer that is

assigned all of the inputs for that output. We say such a reducer covers the output. This reducer need not be unique, and it is permitted that these same inputs are assigned also to other reducers.

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Model

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Model

cs848 Models and Applications of Distributed Data Processing Systems

10 Steps:

Q1: Is this assumption reasonable? Q2: Can be applied to most problems or only several specific problem?

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Model

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Model

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Hamming Distance 1

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proof in technical report: F. N. Afrati, A. D. Sarma, S. Salihoglu, and J. D. Ullman. Upper and lower bounds on the cost of a map-reduce computation. CoRR, abs/1206.4377, 2012.

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Hamming Distance 1

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Hamming Distance 1

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15 Upper Bound: Splitting Algorithm

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Hamming Distance 1

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16 Upper Bound for large q: Replicas on neighboring reducer

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Hamming Distance 1

  • Analysis for Hamming Distance 1 does not generalize

easily to higher distance.

  • Much higher bound for number of outputs covered by a

reducer.

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Triangle Finding

  • We are given a graph as input and want to find all triples of

nodes such that in the graph there are edges between each pair

  • f these three nodes.
  • Alon Class of Sample Graphs: have the property that we can

partition the nodes into disjoint sets, such that the subgraph induced by each partition is either:

  • A single edge between two nodes, or
  • Contains an odd-length Hamiltonian cycle.

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Matrix Multiplication

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Matrix Multiplication

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20 Matrix Multiplication Using Two Phases

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Reference

  • http://www.slideshare.net/tzulitai/upper-and-lower-

bound-on-the-cost-of-a-map-reduce-computation

  • http://shonan.nii.ac.jp/shonan/seminar011/files/2012/01/

ullman.pdf

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Thank you

Q&A

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