Hardness of Approximation and GreedyAlgorithms for the Adaptation Problem in

Virtual Environments

Ananth I. Sundararaj, Manan Sanghi, John R. Lange and Peter A. Dinda

{ais,manan,jarusl,pdinda}@cs.northwestern.edu

Prescience Lab

Department of Electrical Engineering and Computer Science

Northwestern University

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Summary

• Virtual execution environments provide opportunities for dynamic adaptation

• Important components are– Resource monitoring and inference– Application independent adaptation mechanisms– Efficient adaptation algorithms

• In this work– Formalize the adaptation problem– Show that it is NP-hard– Prove that it is NP-hard to approximate– Evaluate greedy heuristics

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Aim

Grid Computing

New Paradigm

Traditional Paradigm

Deliver arbitrary amounts of computational power to perform distributed and parallel computations

Problem1:

Grid Computing using virtual machines

Problem2:

Solution

How to leverage them?

Virtual Machines What are they?

6b

6a

5

4

3b3a

2

1

Resource multiplexing using OS level mechanism

Complexity from resource user’s perspective

Complexity from resource owner’s perspective

Virtual Machine Grid Computing

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Virtual Machines

Virtual machine monitors (VMMs)

•Raw machine is the abstraction

•VM represented by a single image

•VMware GSX Server

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The Simplified Virtuoso Model

Orders a raw machine

User

Specific hardware and performance

Basic software installation available

User’s LAN

VM

Virtual networking ties the machine back to user’s home network

Virtuoso continuously monitors and adapts

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User’s friendlyLAN

Foreign hostile LAN

Virtual Machine

VNET: A bridge with long wires

Host

Proxy

X

Virtual NetworksVM traffic going out on foreign LAN

IP network

A machine is suddenly plugged into a foreign network. What happens?

• Does it get an IP address?• Is it a routeable address?• Does firewall let its traffic through? To any port?

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Measurement and Inference

Application (VTTIF)• Topology

• Traffic load

Underlying network layer Physical hosts

Virtual network layerVNET daemons

Application layerVM layer

Host and VM • Size and compute capacities

• Size and compute demands

• Topology

• Bandwidth

• Latency

Underlying network

[Gupta et al. LNCS 05]

[Gupta et al. IPDPS 06]

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Adaptation Mechanisms

Resource reservation• Network

• CPU

Resource reservationPhysical hosts

Topology changesVNET daemons

VM MigrationVM layer

Topology changes • Overlay links

• Overlay forwarding rules

VM Migration• Third party migration schemes

X

XX

[Sundararaj et al. LCR 04, HPDC 05]

[Lange et al. HPDC 05]

[Lin et al. GRID 2004]

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Infer application’s resource demands

Measure physicallyavailable resources

Algorithm matches demands to resources

Adaptation mechanisms

Defined objective

function is optimized

Any Adaptation Scheme

Adaptation in Virtuoso

VTTIF Infers application’s demands

Wren measures available resources

Algorithm matches demands to resources

VM Migration,Topology, routing,

Resource reservation

Maximizes sum of residual bottleneck

bandwidths

input

input

by driving

that by driving

such that

input

input

Adaptation in Virtuoso

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Optimization Problem• Given the

– network traffic load matrix of the application – computational intensity in each VM– topology of the network– load on its links, routers and hosts

• What is the – mapping of VMs to hosts– overlay topology connecting the hosts– forwarding rules on that topology– required CPU and network reservations

• That – maximizes the application performance?

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Problem Formulation (MARPVEE)Mapping and Routing Problem in Virtual Execution Environments

Measured data

Application demands

VM to host mapping

Constraints

Objective function

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• Theorem 1: MARPVEED (decision version) is NP-complete.

• The NP-hardness is established by reduction from the Edge Disjoint Path Problem (EDPP)• A simpler problem with only the routing component

(RPVEED) is shown to be NP-complete

• Theorem 2: For any δ> 0, it is NP-hard to approximate MARPVEE within m1/2- δ unless P=NP.

• EDPP is used to investigate the approximability of MARPVEE.

• m is the number of edges in the virtual overlay graph

Results

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Greedy Adaptation Algorithms

• Devised two greedy algorithms for mapping VMs to hosts

1. Finds all mappings in a single pass (GreedyMapOne)2. Other takes two passes over input data (GreedyMapTwo)

• Adapted Dijkstra’s shortest path algorithm • Finds widest path for an unsplittable network flow

(GreedyRouting)

• MARPVEE involves both, mapping and routing network flows

• First apply the mapping algorithm (either one) followed by the routing algorithm

• Alternatively we can interleave the two

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IP network

Illinois, USA 100 Mbit backplane internal bandwidth 11 MB/sec

0.89

1.58

1.76

Virginia, USA 100 Mbit backplane internal bandwidth 10 MB/sec

Pittsburgh, USA Numbers indicate end-to-end available bandwidth (MB/sec) between the different locations

vm3

vm4vm5

vm2

vm7

vm6

vm8

vm1

0.7 0.98

0.7

0.98

0.59

1.3 0.56

0.75 0.56

0.56 0.560.33

An application consisting of two disjoint pieces executing inside of the virtual machines (VMs). The numbers indicate the bandwidth (MB/sec) demand among the communicating pairs

Physical Topology

Application Topology

Mapping an application topology onto a physical topology

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Evaluation• Implemented an evaluator to calculate the residual bandwidth for

multiple test cases

• We evaluated the four algorithm variations in three different settings– randomly generated topologies using BRITE– smaller topologies created by hand– a real world scenario

• All the algorithms were found to perform well in most scenarios.

• For randomly generated topologies we do not see any differences between the different variations.

• However for the topology created by hand and for the real world scenario that result in a clustered setting, the one-pass variation outperforms the two-pass algorithm.

• Further, we did not notice in difference between the interleaved and non-interleaved variations.

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• Please visit

– Virtuoso: Resource Management and Prediction for Distributed Computing using Virtual Machines

• http://virtuoso.cs.northwestern.edu

– Prescience Lab (Northwestern University)• http://plab.cs.northwestern.edu

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