SLIDE 1
Covering on a Circle
Shiva P Kasiviswanathan Pennsylvania State University University Park
Covering on a Circle – p. 1/9
Layout of the Presentation
Problem Revisited
Covering on a Circle – p. 2/9
Covering on a Circle Shiva P Kasiviswanathan Pennsylvania State - - PowerPoint PPT Presentation
Covering on a Circle Shiva P Kasiviswanathan Pennsylvania State - - PowerPoint PPT Presentation
Covering on a Circle Shiva P Kasiviswanathan Pennsylvania State University University Park Covering on a Circle p. 1/9 Layout of the Presentation Problem Revisited Covering on a Circle p. 2/9 Layout of the Presentation Problem
SLIDE 2
SLIDE 3
Layout of the Presentation
Problem Revisited Earlier Results
Covering on a Circle – p. 2/9
SLIDE 4
Layout of the Presentation
Problem Revisited Earlier Results New Results
Covering on a Circle – p. 2/9
SLIDE 5
Layout of the Presentation
Problem Revisited Earlier Results New Results Brief Idea Behind Results
Covering on a Circle – p. 2/9
SLIDE 6
Layout of the Presentation
Problem Revisited Earlier Results New Results Brief Idea Behind Results Future Work
Covering on a Circle – p. 2/9
SLIDE 7
Problem Definition
Characteristics of Antenna (B = 1, θ, R).
Covering on a Circle – p. 3/9
SLIDE 8
Problem Definition
Characteristics of Antenna (B = 1, θ, R). User i has bandwidth requirement of bi.
Covering on a Circle – p. 3/9
SLIDE 9
Problem Definition
Characteristics of Antenna (B = 1, θ, R). User i has bandwidth requirement of bi. Output: Orientation of Antenna j, and
Covering on a Circle – p. 3/9
SLIDE 10
Problem Definition
Characteristics of Antenna (B = 1, θ, R). User i has bandwidth requirement of bi. Output: Orientation of Antenna j, and List of users assigned to antenna j say B(j) such that
- i∈B(j)
bi ≤ 1
Covering on a Circle – p. 3/9
SLIDE 11
Problem Definition
Characteristics of Antenna (B = 1, θ, R). User i has bandwidth requirement of bi. Output: Orientation of Antenna j, and List of users assigned to antenna j say B(j) such that
- i∈B(j)
bi ≤ 1
Objective: Minimize some criteria.
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SLIDE 12
More About these Antennas
Directional Antenna can direct radiated power in certain direction for few µ sec.
Covering on a Circle – p. 4/9
SLIDE 13
More About these Antennas
Directional Antenna can direct radiated power in certain direction for few µ sec. Gives lot of potential advantages, i.e., better Route Delivery.
Covering on a Circle – p. 4/9
SLIDE 14
More About these Antennas
Directional Antenna can direct radiated power in certain direction for few µ sec. Gives lot of potential advantages, i.e., better Route Delivery. Gives better distance range than omni-directional antennas.
Covering on a Circle – p. 4/9
SLIDE 15
More About these Antennas
Directional Antenna can direct radiated power in certain direction for few µ sec. Gives lot of potential advantages, i.e., better Route Delivery. Gives better distance range than omni-directional antennas. Wide Deployment in Base Stations for Cellular Networks.
Covering on a Circle – p. 4/9
SLIDE 16
More About these Antennas
Directional Antenna can direct radiated power in certain direction for few µ sec. Gives lot of potential advantages, i.e., better Route Delivery. Gives better distance range than omni-directional antennas. Wide Deployment in Base Stations for Cellular Networks. Also has potential advantages in wireless multihop net- works.
Covering on a Circle – p. 4/9
SLIDE 17
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT)
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SLIDE 18
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm.
Covering on a Circle – p. 5/9
SLIDE 19
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. 2-Approx algorithm used simple Greedy Heuristic.
Covering on a Circle – p. 5/9
SLIDE 20
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. 2-Approx algorithm used simple Greedy Heuristic. Objective: Minimizing other Parameters for given number
- f Antennas.
Covering on a Circle – p. 5/9
SLIDE 21
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. 2-Approx algorithm used simple Greedy Heuristic. Objective: Minimizing other Parameters for given number
- f Antennas.
NP-Hardness follows.
Covering on a Circle – p. 5/9
SLIDE 22
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. 2-Approx algorithm used simple Greedy Heuristic. Objective: Minimizing other Parameters for given number
- f Antennas.
NP-Hardness follows. Simple (2, 1)-Bi criteria Approximation.
Covering on a Circle – p. 5/9
SLIDE 23
Previous Results Summarized
Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. 2-Approx algorithm used simple Greedy Heuristic. Objective: Minimizing other Parameters for given number
- f Antennas.
NP-Hardness follows. Simple (2, 1)-Bi criteria Approximation. Hardness Result: Better than 3/2 impossible, unless P=NP .
Covering on a Circle – p. 5/9
SLIDE 24
New Problems Considered
- 1. Consider variant of MinANT problem (MinVARANT) where
antennas are not homogeneous: Antennas have variable range and span. Objective: Again minimizing the Number of Antennas (MinANT)
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SLIDE 25
New Problems Considered Contd...
- 1. MAX-MIN Fair Allocation of Bandwidth:
Assume users have zero lower bounds on their BW requirement bi = 0. Output: An allocation of users to antenna, such that bandwidth assignment is lexicographically best. Formally given two n-tuples, B1 = {b1, . . . , bn} and B2 =
{b′
1, . . . , b′ n} each in non-decreasing order, we say that
B1 lexicographically dominates over B2 if B1 = B2, or
there is some index l for which bl > b′
l and bi = b′ i for all
i < l.
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SLIDE 26
Summary of New Results
For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1/3.
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SLIDE 27
Summary of New Results
For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1/3. Modifying this gives MinANT an 1.5 approximation
- algorithm. (Optimal, under P!=NP).
Covering on a Circle – p. 8/9
SLIDE 28
Summary of New Results
For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1/3. Modifying this gives MinANT an 1.5 approximation
- algorithm. (Optimal, under P!=NP).
Non Homogeneous Antenna:MinVARANT has
3-approximation.
Covering on a Circle – p. 8/9
SLIDE 29
Summary of New Results
For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1/3. Modifying this gives MinANT an 1.5 approximation
- algorithm. (Optimal, under P!=NP).
Non Homogeneous Antenna:MinVARANT has
3-approximation.
Max-Min Fair: Work going on.
Covering on a Circle – p. 8/9
SLIDE 30
Summary of New Results
For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1/3. Modifying this gives MinANT an 1.5 approximation
- algorithm. (Optimal, under P!=NP).
Non Homogeneous Antenna:MinVARANT has
3-approximation.
Max-Min Fair: Work going on.
Covering on a Circle – p. 8/9
SLIDE 31
Idea behind Results
For Factor 1.5, we use clever Dynamic Programming + Greedy.
Covering on a Circle – p. 9/9
SLIDE 32
Idea behind Results
For Factor 1.5, we use clever Dynamic Programming + Greedy. The idea is to split users into two groups Heavy with demand > 1/2 and Light otherwise.
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SLIDE 33
Idea behind Results
For Factor 1.5, we use clever Dynamic Programming + Greedy. The idea is to split users into two groups Heavy with demand > 1/2 and Light otherwise. No two heavy users can be combined in an same antenna.
Covering on a Circle – p. 9/9
SLIDE 34
Idea behind Results
For Factor 1.5, we use clever Dynamic Programming + Greedy. The idea is to split users into two groups Heavy with demand > 1/2 and Light otherwise. No two heavy users can be combined in an same antenna. Do greedy on Light users and combine with Dynamic Programming on heavy users.
Covering on a Circle – p. 9/9
SLIDE 35
Idea behind Results
For Factor 1.5, we use clever Dynamic Programming + Greedy. The idea is to split users into two groups Heavy with demand > 1/2 and Light otherwise. No two heavy users can be combined in an same antenna. Do greedy on Light users and combine with Dynamic Programming on heavy users. Careful case analysis needed: Split into case with span
≥ π and span < π.
Covering on a Circle – p. 9/9
SLIDE 36
Future work
Finishing the Max-Min Fairness.
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SLIDE 37
Future work
Finishing the Max-Min Fairness. Closely related to load balanced scheduling on Multi-processors with conflicts.
Covering on a Circle – p. 10/9
SLIDE 38
Future work
Finishing the Max-Min Fairness. Closely related to load balanced scheduling on Multi-processors with conflicts. Users can be imagined as jobs and antennas are the processors.
Covering on a Circle – p. 10/9
SLIDE 39
Future work
Finishing the Max-Min Fairness. Closely related to load balanced scheduling on Multi-processors with conflicts. Users can be imagined as jobs and antennas are the processors. The conflict graph can be formed by adding an edge if users are “far" apart.
Covering on a Circle – p. 10/9
SLIDE 40
Future work
Finishing the Max-Min Fairness. Closely related to load balanced scheduling on Multi-processors with conflicts. Users can be imagined as jobs and antennas are the processors. The conflict graph can be formed by adding an edge if users are “far" apart. The problem is therefore coloring this conflict graph with m colors (number of processors).
Covering on a Circle – p. 10/9