Placement of Integration Points
in Multi-hop Community Networks
Ranveer Chandra
(Cornell University)
Lili Qiu, Kamal Jain and Mohammad Mahdian
(Microsoft Research)
Motivation
Community networks
(Houses cooperate in multi-hop network for Internet access)
Internet
ITAP
(Expensive!)
How many ITAPs will satisfy demands of a neighborhood?
2
Related Work
• Placement of server replicas, proxies
– Web servers, internet measurement, file servers
– Facility location problem
Handles locality without link capacity constraints
Does not consider impact of wireless interference
• Clustering Approach (Bejerano ’02)
Only works for a TDMA MAC
3
Our Contributions
We propose placement algorithms that:
–
–
–
–
–
Are close to optimal
Work with a general MAC
Take wireless interference into account
Are optimized for changing workload
Provide fault tolerance to ITAP and link failures
4
Outline
• Motivation and Related Work
• Problem Formulation
– Three link models with increasing realism
• Placement Algorithms
• Advanced Features
5
Mapping to a Graph
Nodes: houses and possible ITAP locations
Edges determined by either:
– A connectivity graph given by Internet provider
– Supplied signal and propagation characteristics
– Simplified wireless connectivity model
•  edge (i, j) if and only if distance (i, j)  CR,
where CR is the communication range
6
Reducing Search Space
The entire search space for ITAPs is intractable
Our Approach:
• Form equivalence classes
– Locations covering the same houses are equivalent
• Prune redundant classes
– Prune class if another class covers all its houses
E1
E3
E2
H1
E6
H2
E7
E4
Since E7 covers all the houses,
prune all other equivalence classes
H3
E5
Use a node for each remaining equivalence class
7
Problem Formulation
Given:
–
–
–
–
–
A community with N houses
House demand: dh h
Link capacity : Cape e
House capacity: Caph h
ITAP capacity : Capi i
Internet
A
CapA
C
B
CapC
CapCD
ITAP
X
CapX
CapB
Goal: Minimize num ITAPs to serve all demands
8
Simple Interference Models
Ideal link model:
– Throughput unaffected by path length (# hops)
– Possible by using smart antennas, multiple radios
f
1
f
2
f
3
4
Flow from 1 to 4, f bps < Cap12
9
Simple Interference Models
Ideal link model:
– Throughput unaffected by path length (# hops)
– Possible by using smart antennas, multiple radios
General link model:
– Throughput depends path length (# hops)
– Simplifications of current day radios
– Bounded Hop-count Model
• Throughput unaffected if path length < thresh, else 0
f
1
f
2
f
3
4
Flow from 1 to 4, f bps < Cap12 and thresh = 4
0
0
0
1
2
3
4
Flow from 1 to 4, f bps < Cap12 and thresh = 2
10
Simple Interference Models
Ideal link model:
– Throughput unaffected by path length (# hops)
– Possible by using smart antennas, multiple radios
General link model:
– Throughput depends path length (# hops)
– Simplifications of current day radios
– Bounded Hop-count Model
• Throughput unaffected if path length < thresh, else 0
– Smooth Degradation Model
• Throughput degrades by 1/k for path of length k
f/3
1
f/3
2
f/3
3
4
Flow from 1 to 4, f bps < Cap12
11
Outline
• Motivation and Related Work
• Problem Formulation
• Placement Algorithms
– Placement algorithms: Ideal Link Model
– Placement algorithms: General Link Model
• Advanced Features
12
Ideal Link Model
Goal
Find minimum number of ITAPs that satisfies all demands
Results
• The above problem is NP-hard
• The best polynomial approximation algorithm
– ln(N) worst-case unless P=NP
13
Greedy Algorithm
Main Idea
1. Initial set of opened ITAPs is null
2. Iterate over all ITAPs, and apply greedy step
– Select ITAP satisfying the greatest demand
3.
4.
Add selected ITAP to set of opened ITAPs
Loop through steps 2 and 3 until all demands satisfied
A
All possible ITAP locations
B
1
1
2
C
Set of houses
Opened ITAP locations
14
Greedy Algorithm
Main Idea
1. Initial set of opened ITAPs is null
2. Iterate over all ITAPs, and apply greedy step
– Select ITAP satisfying the greatest demand
3.
4.
Add selected ITAP to set of opened ITAPs
Loop through steps 2 and 3 until all demands satisfied
A
All possible ITAP locations
B
1
C
2
Set of houses
Opened ITAP locations
1
2
15
Greedy Step
Can be mapped to a max flow min cut problem
• Handle house demands: Add a virtual source
• Handle ITAP capacities: Add a virtual sink
A
1
S
B
T
2
C
16
Greedy Step
Can be mapped to a max flow min cut problem
• Handle house demands: Add a virtual source
• Handle ITAP capacities: Add a virtual sink
• Handle house capacities: Split the house nodes
CapA
AAOUT
IN
1
S
CapB
BBOUT
IN
T
2
CCOUT
IN
Cap
C
Select ITAP that gives max flow from S to T
17
Ideal Link Model: Algorithms
• Greedy placement
– ln(N) worst-case bound (best possible in worst-case)
• Cluster-based placement
– Partition network nodes into minimum number of disjoint clusters
– Place an ITAP in each cluster
• Random placement
– Randomly open an ITAP iteratively until all demands are satisfied
• Lower bound
– Relax the integer constraints and solve the relaxed LP problem
18
Varying communication radius
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps
100
# ITAPs
80
60
40
20
0
0
10
20
30
40
50
Communication Radius (m)
greedy
cluster
random
lower bound
19
General Link Model
Problem is NP-Hard. Use Greedy heuristic
• Main idea
– iteratively open ITAP to maximize satisfied demand
• The Greedy step
– Formulate a linear program (not efficient)
– Develop better algorithms for two special cases:
• bounded hop-count
• smoothed throughput degradation
20
Greedy Step
• Bounded hop-count
– Modify Ford-Fulkerson method for max-flow
– ignore augmenting paths > hop-count threshold
• Smooth throughput model (throughput = 1/L)
– Goal: max piP 1/|pi|, where
• P is the set of all the augmenting paths in the graph
– Observation: prefer imbalance in path lengths
– Approach: iteratively pick shortest augmenting path
21
Bounded-hop count
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps, CR 10 m
100
# ITAPs
80
60
40
20
0
0
5
10
15
20
hop-count threshold
greedy
cluster
random
lower bound
22
Smooth degradation
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps
100
# ITAPs
80
60
40
20
0
0
10
20
30
40
50
communication radius (m)
greedy
random
lower bound
23
Outline
•
•
•
•
Motivation and Related Work
Problem Formulation
Placement Algorithms
Advanced Features
24
Changing Demands
• Problem
– Place ITAPs to handle changing demands
• User demands exhibit periodicity (e.g. diurnal pattern)
• Greedy algorithm
– max(Xi/ Yi), where
• Xi is satisfied demand in period i, and
• Yi is the total demand in period i
– ln(kN) worst-case bound, where k is number of periods
25
Fault Tolerance Considerations
• Problem
– Ensure Internet connectivity when nodes and link fail
• Approach
– Control parameters
• Number of independent paths: p
• Over-provisioning factor: all paths allocate f/d capacity
– Compute satisfied demands using LP
– Greedy algorithm gives good results
26
Conclusion
• First ITAP placement study for general MAC
• Design ITAP placement algorithms for
– Three wireless throughput models
– handling periodically changing demands
– providing fault-tolerance
• Showed efficiency using simulations, analyses
• Greedy algorithms are near optimal in all cases
27
Thank you
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