Placement of Integration Points in Multihop Community Networks

1
Placement of Integration Pointsin Multi-hop
Community Networks

• Ranveer Chandra
• (Cornell University)
• Lili Qiu, Kamal Jain and Mohammad Mahdian
  (Microsoft Research)

2
Motivation

• Community networks
• (Houses cooperate in multi-hop network for
  Internet access)

Internet
ITAP (Expensive!)
How many ITAPs will satisfy demands of a
neighborhood?
3
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

4
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

5
Outline

• Motivation and Related Work
• Problem Formulation
• Three link models with increasing realism
• Placement Algorithms
• Advanced Features

6
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

7
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

H1
H2
E3
E1
E2
Since E7 covers all the houses, prune all other
equivalence classes
H3
E7
E6
E4
E5
Use a node for each remaining equivalence class
8
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
CapAC
CapA
CapCD
CapX
C
ITAP X
CapBC
B
CapC
CapB
Goal Minimize num ITAPs to serve all demands
9
Simple Interference Models

• Ideal link model
• Throughput unaffected by path length ( hops)
• Possible by using smart antennas, multiple radios

f
f
f
1
2
3
4
Flow from 1 to 4, f bps lt Cap12
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 lt thresh,
  else 0

f
f
f
1
2
3
4
Flow from 1 to 4, f bps lt Cap12 and thresh 4
0
0
0
1
2
3
4
Flow from 1 to 4, f bps lt Cap12 and thresh 2
11
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 lt thresh,
  else 0

• Smooth Degradation Model
• Throughput degrades by 1/k for path of length k

f/3
f/3
f/3
1
2
3
4
Flow from 1 to 4, f bps lt Cap12
12
Outline

• Motivation and Related Work
• Problem Formulation
• Placement Algorithms
• Placement algorithms Ideal Link Model
• Placement algorithms General Link Model
• Advanced Features

13
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 PNP

14
Greedy Algorithm

• Main Idea
• Initial set of opened ITAPs is null
• Iterate over all ITAPs, and apply greedy step
• Select ITAP satisfying the greatest demand
• 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
1
2
C
Opened ITAP locations
Set of houses
15
Greedy Algorithm

• Main Idea
• Initial set of opened ITAPs is null
• Iterate over all ITAPs, and apply greedy step
• Select ITAP satisfying the greatest demand
• 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
1
2
C
2
Opened ITAP locations
Set of houses
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

A
CapA1
dA
CapAB
1
Cap1
CapB1
dB
B
S
T
CapB2
Cap2
dC
2
CapBC
CapC2
C
17
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
AIN
AOUT
CapA1
CapA1
CapAB
dA
dA
CapAB
CapBA
1
Cap1
CapB1
CapB1
dB
dB
CapB
S
BIN
T
BOUT
CapB2
CapB2
Cap2
dC
dC
CapBC
2
CapBC
CapCB
CapC2
CapC2
CIN
COUT
CapC
Select ITAP that gives max flow from S to T
18
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

19
Varying communication radius
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps
20
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

21
Greedy Step

• Bounded hop-count
• Modify Ford-Fulkerson method for max-flow
• ignore augmenting paths gt 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

22
Bounded-hop count
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps,
CR 10 m
23
Smooth degradation
100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps
24
Outline

• Motivation and Related Work
• Problem Formulation
• Placement Algorithms
• Advanced Features

25
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

26
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

27
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

28

• Thank you

29
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