Placement of Integration Pointsin Multi-hop Community Networks

Ranveer Chandra (Cornell University)

Lili Qiu, Kamal Jain and Mohammad Mahdian (Microsoft Research)

2

MotivationCommunity 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

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Outline

• Motivation and Related Work

• Problem Formulation– Three link models with increasing realism

• Placement Algorithms

• Advanced Features

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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

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Reducing Search SpaceThe 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

H3

E1E2

E3

E4

E5

E6E7 Since E7 covers all the houses,

prune all other equivalence classes

Use a node for each remaining equivalence class

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Problem FormulationGiven:

– A community with N houses– House demand: dh h– Link capacity : Cape e– House capacity: Caph h– ITAP capacity : Capi i

Goal: Minimize num ITAPs to serve all demands

A

B

C

CapA

CapB

CapAC

CapBC

CapCD

CapC

ITAPX

CapX

Internet

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Simple Interference ModelsIdeal link model:

– Throughput unaffected by path length (# hops)– Possible by using smart antennas, multiple radios

1 2 3 4

Flow from 1 to 4, f bps < Cap12

f f f

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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

Simple Interference ModelsIdeal link model:

– Throughput unaffected by path length (# hops)– Possible by using smart antennas, multiple radios

1 2 3 4

Flow from 1 to 4, f bps < Cap12 and thresh = 4

f f f

1 2 3 4

Flow from 1 to 4, f bps < Cap12 and thresh = 2

0 0 0

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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

Simple Interference ModelsIdeal link model:

– Throughput unaffected by path length (# hops)– Possible by using smart antennas, multiple radios

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

1 2 3 4

Flow from 1 to 4, f bps < Cap12

f/3 f/3 f/3

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Outline

• Motivation and Related Work

• Problem Formulation

• Placement Algorithms– Placement algorithms: Ideal Link Model– Placement algorithms: General Link Model

• Advanced Features

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Ideal Link ModelGoal

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

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Greedy Algorithm

A

B

C

Set of houses

1 211 2

Opened ITAP locations

All possible ITAP locations

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. Add selected ITAP to set of opened ITAPs

4. Loop through steps 2 and 3 until all demands satisfied

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Greedy AlgorithmMain 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. Add selected ITAP to set of opened ITAPs

4. Loop through steps 2 and 3 until all demands satisfied

A

B

C

Set of houses Opened ITAP locations

All possible ITAP locations

1 211 2

2

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Greedy StepCan be mapped to a max flow min cut problem• Handle house demands: Add a virtual source• Handle ITAP capacities: Add a virtual sink

A

B

C

1

2

Cap

AB

Cap

BC

S

dA

dB

dC

CapA1

CapB1

CapC2

T

Cap1

Cap2

CapB2

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CIN

BIN

Greedy StepCan 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

AIN

1

2

Cap

AB

Cap

BC

S

dA

dB

dC

CapA1

CapB1

CapC2

T

Cap1

Cap2

CapB2

AOUT

BOUT

COUT

d A

dB

dC

CapA1

CapB1

CapC2

CapB2

CapA

CapB

CapC

CapBC

CapBA

CapAB

CapCB

Select ITAP that gives max flow from S to T

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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

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Varying communication radius

0

20

40

60

80

100

0 10 20 30 40 50

Communication Radius (m)

# IT

AP

s

greedy cluster random lower bound

100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps

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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

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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 piP 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

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0

20

40

60

80

100

0 5 10 15 20

hop-count threshold

# IT

AP

s

greedy cluster random lower bound

Bounded-hop count100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps, CR 10 m

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Smooth degradation

0

20

40

60

80

100

0 10 20 30 40 50communication radius (m)

# IT

AP

s

greedy random lower bound

100 nodes, Cape 6 Mbps, Capi 100 Mbps, dh 1 Mbps

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Outline

• Motivation and Related Work

• Problem Formulation

• Placement Algorithms

• Advanced Features

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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

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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

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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

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Thank you

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