Hardness of Approximation and Greedy
Algorithms 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
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
2
Virtual Machine Grid Computing
1
arbitrary amounts of
AimDeliver
computational power to perform
distributed and parallel computations
Traditional
Paradigm
New
Paradigm
2
Resource multiplexing using
Grid OS level mechanism
Computing
3b
5
Grid Computing
using virtual
machines
4
3a
6a
Problem1:
6b
Complexity from resource
Solution
user’s perspective
Problem2:
Complexity from resource
owner’s perspective
Virtual Machines
What are they?
How to leverage
them?
3
Virtual Machines
Virtual machine monitors (VMMs)
•Raw machine is the abstraction
•VM represented by a single
image
•VMware GSX Server
4
The Simplified Virtuoso Model
User’s
LAN
Virtual networking ties the
machine back to user’s
home network
Orders a raw
machine
VM
Specific hardware and
performance
Basic software
installation available
Virtuoso continuously monitors and adapts
User
5
Virtual Networks
VM traffic going
out on foreign
LAN
Foreign hostile
LAN
User’s friendly
LAN
X
IP network
Host
Proxy
Virtual Machine
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?
VNET: A bridge with long wires
6
Measurement and Inference
Underlying network
Host and VM
• Topology
• Size and compute capacities
• Bandwidth
• Size and compute demands
• Latency
[Gupta et al. IPDPS 06]
Application (VTTIF)
• Topology
• Traffic load
[Gupta et al. LNCS 05]
Application layer
VM layer
Virtual network layer
VNET daemons
Underlying network layer
7
Physical hosts
Adaptation Mechanisms
VM Migration
Topology changes
• Third party migration schemes
• Overlay links
• Overlay forwarding rules
Resource reservation
[Sundararaj et al.
LCR 04, HPDC 05]
• Network [Lange et al. HPDC 05]
• CPU
X
[Lin et al. GRID 2004]
VM Migration
VM layer
X
X
Topology changes
VNET daemons
Resource reservation8
Physical hosts
Adaptation in Virtuoso
Any Adaptation Scheme
Infer application’s
resource demands
input
by driving Adaptation such that Defined objective
Algorithm matches
demands to resources
mechanisms
function is optimized
Measure physically
available resources
input
Adaptation in Virtuoso
VTTIF Infers
application’s demands
input
by driving
Algorithm matches
demands to resources
Wren measures
available resources
input
VM Migration,
that
Topology, routing,
Resource reservation
Maximizes sum of
residual bottleneck
bandwidths
9
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?
10
Problem Formulation (MARPVEE)
Mapping and Routing Problem in Virtual Execution Environments
Measured data
Application
demands
VM to host mapping
Constraints
Objective function
11
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 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
12
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
13
Physical Topology
0.89
Virginia, USA
100
Mbit backplane
internal
bandwidth 10 MB/sec
1.76
IP network
1.58
Illinois, USA
100 Mbit backplane
internal bandwidth 11 MB/sec
Numbers indicate end-to-end available bandwidth
(MB/sec) between the different locations
Pittsburgh, USA
Application Topology
vm1
0.75
vm6
vm2
0.98
Mapping an application topology
onto a physical topology
0.56
1.3
vm3
0.56
0.56
0.7
0.98
0.56
0.33
vm5
0.59
vm4
vm7
0.7
vm8
14
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
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.
15
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
16