Hardness of Approximation and Greedy Algorithms for the Adaptation Problem in Virtual Environments

1
Hardness of Approximation and GreedyAlgorithms
for the Adaptation Problem inVirtual
Environments

• Ananth I. Sundararaj, Manan Sanghi, John R. Lange
  and Peter A. Dinda
• ais,manan,jarusl,pdinda_at_cs.northwestern.edu
• Prescience Lab
• Department of Electrical Engineering and Computer
  Science
• Northwestern University

2
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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Virtual Machine Grid Computing
Deliver arbitrary amounts of computational power
to perform distributed and parallel computations
Aim
1
New Paradigm
Traditional Paradigm
5
2
Grid Computing using virtual machines
Resource multiplexing using OS level mechanism
Grid Computing
4
3a
6a
3b
Problem1
6b
Virtual Machines What are they?
Complexity from resource users perspective
Solution
Problem2
How to leverage them?
Complexity from resource owners perspective
4
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
Virtual networking ties the machine back to
users home network
Users LAN
Specific hardware and performance
VM
Basic software installation available
Orders a raw machine
Virtuoso continuously monitors and adapts
User
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Virtual Networks
VM traffic going out on foreign LAN
Foreign hostile LAN
X
Users friendly LAN
IP network
Virtual Machine
Host

• 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?

Proxy
VNET A bridge with long wires
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Measurement and Inference
Host and VM
Underlying network

• Size and compute capacities
• Size and compute demands

• Topology
• Bandwidth
• Latency

Application (VTTIF)

• Topology
• Traffic load

Gupta et al. IPDPS 06
Gupta et al. LNCS 05
Application layer
VM layer
Virtual network layer
VNET daemons
Underlying network layer
Physical hosts
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Adaptation Mechanisms
Topology changes
VM Migration

• Overlay links
• Overlay forwarding rules

• Third party migration schemes

Sundararaj et al. LCR 04, HPDC 05
Resource reservation

• Network
• CPU

Lange et al. HPDC 05
Lin et al. GRID 2004
VM Migration
X
VM layer
Topology changes
X
X
VNET daemons
Resource reservation
Physical hosts
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Adaptation in Virtuoso
Any Adaptation Scheme
Infer applications resource demands
input
Algorithm matches demands to resources
Adaptation mechanisms
Defined objective function is optimized
by driving
such that
Measure physically available resources
input
Adaptation in Virtuoso
VTTIF Infers applications demands
input
Algorithm matches demands to resources
VM Migration, Topology, routing, Resource
reservation
Maximizes sum of residual bottleneck bandwidths
that
by driving
Wren measures available resources
input
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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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Results

• 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 dgt 0, it is NP-hard to
  approximate
• MARPVEE within m1/2- d unless PNP.
• EDPP is used to investigate the
  approximability of MARPVEE.
• m is the number of edges in the virtual
  overlay graph

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Greedy Adaptation Algorithms

• Devised two greedy algorithms for mapping VMs to
  hosts
• Finds all mappings in a single pass
  (GreedyMapOne)
• Other takes two passes over input data
  (GreedyMapTwo)
• Adapted Dijkstras 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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Physical Topology
0.89
Virginia, USA 100 Mbit
backplane internal bandwidth 10 MB/sec
IP network
1.76
1.58
Illinois, USA 100 Mbit
backplane internal bandwidth 11 MB/sec
Pittsburgh, USA
Numbers indicate end-to-end available bandwidth
(MB/sec) between the different locations
Application Topology
vm6
vm2
vm1
0.98
Mapping an application topology onto a physical
topology
0.75
0.56
0.7
0.98
1.3
vm3
0.56
0.56
0.56
0.33
0.7
0.59
vm4
vm5
vm7
vm8
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
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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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For More Information

• 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