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 Revisited Earlier Results Covering on a Circle – p. 2/9
Layout of the Presentation Problem Revisited Earlier Results New Results Covering on a Circle – p. 2/9
Layout of the Presentation Problem Revisited Earlier Results New Results Brief Idea Behind Results Covering on a Circle – p. 2/9
Layout of the Presentation Problem Revisited Earlier Results New Results Brief Idea Behind Results Future Work Covering on a Circle – p. 2/9
Problem Definition Characteristics of Antenna ( B = 1 , θ, R ) . Covering on a Circle – p. 3/9
Problem Definition Characteristics of Antenna ( B = 1 , θ, R ) . User i has bandwidth requirement of b i . Covering on a Circle – p. 3/9
Problem Definition Characteristics of Antenna ( B = 1 , θ, R ) . User i has bandwidth requirement of b i . Output: Orientation of Antenna j , and Covering on a Circle – p. 3/9
Problem Definition Characteristics of Antenna ( B = 1 , θ, R ) . User i has bandwidth requirement of b i . Output: Orientation of Antenna j , and List of users assigned to antenna j say B ( j ) such that � b i ≤ 1 i ∈ B ( j ) Covering on a Circle – p. 3/9
Problem Definition Characteristics of Antenna ( B = 1 , θ, R ) . User i has bandwidth requirement of b i . Output: Orientation of Antenna j , and List of users assigned to antenna j say B ( j ) such that � b i ≤ 1 i ∈ B ( j ) Objective: Minimize some criteria. Covering on a Circle – p. 3/9
More About these Antennas Directional Antenna can direct radiated power in certain direction for few µ sec. Covering on a Circle – p. 4/9
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
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
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
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
Previous Results Summarized Objective: Minimizing the Number of Antennas (MinANT) Covering on a Circle – p. 5/9
Previous Results Summarized Objective: Minimizing the Number of Antennas (MinANT) NP-Hardness and simple 2 approximation algorithm. Covering on a Circle – p. 5/9
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
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 of Antennas. Covering on a Circle – p. 5/9
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 of Antennas. NP-Hardness follows. Covering on a Circle – p. 5/9
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 of Antennas. NP-Hardness follows. Simple (2 , 1) -Bi criteria Approximation. Covering on a Circle – p. 5/9
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 of 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
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) Covering on a Circle – p. 6/9
New Problems Considered Contd... 1. MAX-MIN Fair Allocation of Bandwidth: Assume users have zero lower bounds on their BW requirement b i = 0 . Output: An allocation of users to antenna, such that bandwidth assignment is lexicographically best. Formally given two n -tuples, B 1 = { b 1 , . . . , b n } and B 2 = { b ′ 1 , . . . , b ′ n } each in non-decreasing order, we say that B 1 lexicographically dominates over B 2 if B 1 = B 2 , or there is some index l for which b l > b ′ l and b i = b ′ i for all i < l . Covering on a Circle – p. 7/9
Summary of New Results For MinANT:One can achieve OPT antennas by violating each user requirement by at most 1 / 3 . Covering on a Circle – p. 8/9
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
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
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
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
Idea behind Results For Factor 1.5, we use clever Dynamic Programming + Greedy. Covering on a Circle – p. 9/9
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. Covering on a Circle – p. 9/9
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
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
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
Future work Finishing the Max-Min Fairness. Covering on a Circle – p. 10/9
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
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
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
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