distance optimal target assignment in robotic networks...
TRANSCRIPT
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Distance Optimal Target Assignment in Robotic Networks under Communication and Sensing
Constraints
Jingjin Yu Soon-Jo Chung Petros G. VoulgarisCSAIL @ MIT/MechE @ BU AE @ University of Illinois
Supported by:
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The Stochastic Target Assignment Problem
2
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The Stochastic Target Assignment Problem
2
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem
2
𝑋 = {𝑥1, … , 𝑥𝑛}
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem
2
𝑋 = {𝑥1, … , 𝑥𝑛}
𝑌 = {𝑦1, … , 𝑦𝑛}
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem
2
𝑋 = {𝑥1, … , 𝑥𝑛}
𝑌 = {𝑦1, … , 𝑦𝑛}
Control: 𝑥𝑖 = 𝑢𝑖 , | 𝑢𝑖 | = 1
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem
2
𝑋 = {𝑥1, … , 𝑥𝑛}
𝑌 = {𝑦1, … , 𝑦𝑛}
Control: 𝑥𝑖 = 𝑢𝑖 , | 𝑢𝑖 | = 1
𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem
2
𝑋 = {𝑥1, … , 𝑥𝑛}
𝑌 = {𝑦1, … , 𝑦𝑛}
Control: 𝑥𝑖 = 𝑢𝑖 , | 𝑢𝑖 | = 1
min𝜎,{𝑢𝑖}
𝐷𝑛 =
𝑖
| 𝑥𝑖(𝑡)|𝑑𝑡
𝜎: permutation that pairs 𝑥𝑖 with 𝑦𝜎(𝑖)
𝑄 = 0,1 × [0,1]
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The Stochastic Target Assignment Problem, cont.
3
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
𝑟𝑐𝑜𝑚𝑚
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
𝑟𝑐𝑜𝑚𝑚
𝐺(𝑡)
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
𝑟𝑐𝑜𝑚𝑚
𝐺(𝑡)
Given 𝑟𝑠𝑒𝑛𝑠𝑒 and 𝑟𝑐𝑜𝑚𝑚, how can we guarantee distance optimality?
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The Stochastic Target Assignment Problem, cont.
3
𝑟𝑠𝑒𝑛𝑠𝑒
𝑟𝑐𝑜𝑚𝑚
𝐺(𝑡)
Given 𝑟𝑠𝑒𝑛𝑠𝑒 and 𝑟𝑐𝑜𝑚𝑚, how can we guarantee distance optimality? Performance of decentralized, hierarchical strategies (algorithms)?
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Related Work
4
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Related Work
4
Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009
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Related Work
4
Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009
Treleaven, Pavone, and Frazzoli, Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems, IEEE Trans. Automat Control, vol. 59, no. 9, pp. 2261–2276, 2013.
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Related Work
4
Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009
Treleaven, Pavone, and Frazzoli, Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems, IEEE Trans. Automat Control, vol. 59, no. 9, pp. 2261–2276, 2013.
Penrose, The longest edge of the random minimal spanning tree, Annals of Applied Probability, vol. 7, pp. 340–361, 1997.Penrose, Random Geometric Graphs, 2003
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Related Work
4
Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009
Treleaven, Pavone, and Frazzoli, Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems, IEEE Trans. Automat Control, vol. 59, no. 9, pp. 2261–2276, 2013.
Penrose, The longest edge of the random minimal spanning tree, Annals of Applied Probability, vol. 7, pp. 340–361, 1997.Penrose, Random Geometric Graphs, 2003
Erdős and Rényi, On a classical problem of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 215–220, 1961
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Related Work
4
Smith and Bullo, Monotonic target assignment for robotic networks, IEEE Trans. Automat. Control, vol. 54, no. 9, pp. 2042–2057, 2009
Treleaven, Pavone, and Frazzoli, Asymptotically optimal algorithms for one-to-one pickup and delivery problems with applications to transportation systems, IEEE Trans. Automat Control, vol. 59, no. 9, pp. 2261–2276, 2013.
Penrose, The longest edge of the random minimal spanning tree, Annals of Applied Probability, vol. 7, pp. 340–361, 1997.Penrose, Random Geometric Graphs, 2003
Erdős and Rényi, On a classical problem of probability theory, Publ. Math. Inst. Hung. Acad. Sci., vol. Ser. A 6, pp. 215–220, 1961
Karaman and Frazzoli, Sampling-based Algorithms for Optimal Motion Planning. Int. Journal of Robotics Research, vol. 30, no 7, pp. 846-894, 2011
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Main Result
22
Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
Tight asymptotic bounds for high-probability guarantee
Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution
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Main Result
23
Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
Tight asymptotic bounds for high-probability guarantee
Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution
![Page 24: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/24.jpg)
Main Result
24
Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
Tight asymptotic bounds for high-probability guarantee
Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution
![Page 25: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/25.jpg)
Main Result
25
Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
Tight asymptotic bounds for high-probability guarantee
Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution
![Page 26: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/26.jpg)
Main Result
26
Distance optimality guarantee Necessary and sufficient condition for distance optimality (non-stochastic) Non-asymptotic 1 − 𝜖 probabilistic guarantee for 0 < 𝜖 < 1
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
Tight asymptotic bounds for high-probability guarantee
Performance of decentralized, hierarchical strategies Upper bound on the distance cost for arbitrary robot/target distribution 𝑂(1) asymptotic optimality guarantee under the uniform distribution
𝑛 - number of robots
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Distance Optimality Guarantee
6
Theorem (Necessary and Sufficient Conditions for Distance Optimality).Under sensing and communication constraints, distance optimality can be guaranteed if and only if at 𝑡 = 0,
1. Every robot can communicate with every other robot, 2. Each target is observable by some robot.
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Distance Optimality Guarantee
6
Theorem (Necessary and Sufficient Conditions for Distance Optimality).Under sensing and communication constraints, distance optimality can be guaranteed if and only if at 𝑡 = 0,
1. Every robot can communicate with every other robot, 2. Each target is observable by some robot.
𝑟𝑐𝑜𝑚𝑚
![Page 29: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/29.jpg)
Distance Optimality Guarantee
6
Theorem (Necessary and Sufficient Conditions for Distance Optimality).Under sensing and communication constraints, distance optimality can be guaranteed if and only if at 𝑡 = 0,
1. Every robot can communicate with every other robot, 2. Each target is observable by some robot.
𝑟𝑐𝑜𝑚𝑚
![Page 30: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/30.jpg)
Distance Optimality Guarantee
6
Theorem (Necessary and Sufficient Conditions for Distance Optimality).Under sensing and communication constraints, distance optimality can be guaranteed if and only if at 𝑡 = 0,
1. Every robot can communicate with every other robot, 2. Each target is observable by some robot.
𝑟𝑐𝑜𝑚𝑚
𝑟𝑠𝑒𝑛𝑠𝑒
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Non-Asymptotic Optimality Guarantee
7
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Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
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Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
![Page 34: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/34.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
![Page 35: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/35.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
![Page 36: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/36.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
𝑃 𝑛𝑖 = 0 = 1 −1
𝑚
𝑛
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
![Page 37: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/37.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
𝑃 𝑛𝑖 = 0 = 1 −1
𝑚
𝑛
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
< 𝑒−𝑛𝑚
![Page 38: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/38.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
𝑃 𝑛𝑖 = 0 = 1 −1
𝑚
𝑛
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
𝑃
𝑖=1
𝑚
𝐸(𝑛𝑖 = 0) ≤
𝑖=1
𝑚
𝑃(𝑛𝑖 = 0)
< 𝑒−𝑛𝑚
![Page 39: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/39.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
𝑃 𝑛𝑖 = 0 = 1 −1
𝑚
𝑛
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
𝑃
𝑖=1
𝑚
𝐸(𝑛𝑖 = 0) ≤
𝑖=1
𝑚
𝑃(𝑛𝑖 = 0)
< 𝑒−𝑛𝑚
< 𝑚𝑒−𝑛𝑚
![Page 40: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/40.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
𝑟𝑐𝑜𝑚𝑚
𝑞𝑖
𝑃 𝑛𝑖 = 0 = 1 −1
𝑚
𝑛
1 2 …… 𝑚 = ⌈ 5/𝑟𝑐𝑜𝑚𝑚⌉
𝑃
𝑖=1
𝑚
𝐸(𝑛𝑖 = 0) ≤
𝑖=1
𝑚
𝑃(𝑛𝑖 = 0)
< 𝑒−𝑛𝑚
< 𝑚𝑒−𝑛𝑚 = 𝜖
![Page 41: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/41.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
Theorem (Random Geometric Graphs [Penrose ‘97]). For 𝑛 uniformly distributed nodes in the unit square, let 𝐺(0) be the communication graph for a given 𝑟𝑐𝑜𝑚𝑚 at 𝑡 = 0. Then for any real number 𝑐, as 𝑛 → ∞ (i.e., 𝑟𝑐𝑜𝑚𝑚 → 0),
𝑃 𝐺 𝑖𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝜋𝑛𝑟𝑐𝑜𝑚𝑚2 − log 𝑛 ≤ 𝑐) = 𝑒−𝑒
𝑐.
Theorem [Xue & Kumar ‘04]. For 𝑛 uniformly distributed nodes in the unit square, the network is asymptotically connected if and only if each node has Θ(log 𝑛) neighbors.
![Page 42: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/42.jpg)
Non-Asymptotic Optimality Guarantee
7
Lemma. Given 𝑟𝑐𝑜𝑚𝑚 and fixing 0 < 𝜖 < 1, 𝐺(0) is connected with probability at least 1 − 𝜖 if
𝑛 ≥5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
.
Theorem (Random Geometric Graphs [Penrose ‘97]). For 𝑛 uniformly distributed nodes in the unit square, let 𝐺(0) be the communication graph for a given 𝑟𝑐𝑜𝑚𝑚 at 𝑡 = 0. Then for any real number 𝑐, as 𝑛 → ∞ (i.e., 𝑟𝑐𝑜𝑚𝑚 → 0),
𝑃 𝐺 𝑖𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝜋𝑛𝑟𝑐𝑜𝑚𝑚2 − log 𝑛 ≤ 𝑐) = 𝑒−𝑒
𝑐.
Theorem [Xue & Kumar ‘04]. For 𝑛 uniformly distributed nodes in the unit square, the network is asymptotically connected if and only if each node has Θ(log 𝑛) neighbors.
![Page 43: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/43.jpg)
Non-Asymptotic Optimality Guarantee, cont.
8
Theorem (Non-Asymptotic Bounds) Fixing 0 < 𝜖 < 1, robots can communicate with each other and all targets are observable at 𝑡 = 0 with probability at least 1 − 𝜖 when
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
![Page 44: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/44.jpg)
Non-Asymptotic Optimality Guarantee, cont.
8
Theorem (Non-Asymptotic Bounds) Fixing 0 < 𝜖 < 1, robots can communicate with each other and all targets are observable at 𝑡 = 0 with probability at least 1 − 𝜖 when
𝑛 ≥
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
log1
𝜖
2
𝑟𝑠𝑒𝑛𝑠𝑒
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 <10𝑟𝑐𝑜𝑚𝑚
5
5
𝑟𝑐𝑜𝑚𝑚
2
log1
𝜖
5
𝑟𝑐𝑜𝑚𝑚
2
, 𝑟𝑠𝑒𝑛𝑠𝑒 ≥10𝑟𝑐𝑜𝑚𝑚
5
𝑛 = Θ(−1
𝑟𝑐𝑜𝑚𝑚2 log 𝑟𝑐𝑜𝑚𝑚) is sufficient and necessary for high probability asymptotic
guarantee on the connectivity of 𝐺 0 .
![Page 45: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/45.jpg)
An Ideal Hierarchical Strategy
9
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An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
![Page 47: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/47.jpg)
An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)
1
𝑚
![Page 48: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/48.jpg)
An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)
1
𝑚
![Page 49: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/49.jpg)
An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)
1
𝑚
![Page 50: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/50.jpg)
An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)
1
𝑚
![Page 51: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/51.jpg)
An Ideal Hierarchical Strategy
9
Ideal: 𝑟𝑐𝑜𝑚𝑚, 𝑟𝑠𝑒𝑛𝑠𝑒 as large as needed
Hierarchical: The unit square is partitioned into 𝑚 small squares(here, 𝑚 = 4)
1
𝑚
![Page 52: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/52.jpg)
Bounding Distance Cost at Lower Hierarchy
10
1
𝑚
![Page 53: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/53.jpg)
Bounding Distance Cost at Lower Hierarchy
10
1
𝑚
![Page 54: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/54.jpg)
Bounding Distance Cost at Lower Hierarchy
10
𝑞𝑖
1
𝑚
![Page 55: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/55.jpg)
Bounding Distance Cost at Lower Hierarchy
10
Theorem [Talagrand ‘92] Let 𝑋 = {𝑥1, … , 𝑥𝑛}, 𝑌 = {𝑦1, … , 𝑦𝑛} be two sets sampled 𝑖. 𝑖. 𝑑. from the same arbitrary distribution on 0,1 2. Then
𝐸 min𝜎
𝑖=1
𝑛
𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,
in which 𝐶 is a universal constant.
𝑞𝑖
1
𝑚
![Page 56: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/56.jpg)
Bounding Distance Cost at Lower Hierarchy
10
Theorem [Talagrand ‘92] Let 𝑋 = {𝑥1, … , 𝑥𝑛}, 𝑌 = {𝑦1, … , 𝑦𝑛} be two sets sampled 𝑖. 𝑖. 𝑑. from the same arbitrary distribution on 0,1 2. Then
𝐸 min𝜎
𝑖=1
𝑛
𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,
in which 𝐶 is a universal constant.
𝑞𝑖
𝐸 𝐷𝑖 ≤𝐶
𝑚𝑛𝑖 log 𝑛𝑖
1
𝑚
![Page 57: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/57.jpg)
𝑖=1
𝑚
𝐸[𝐷𝑖] ≤ 𝐶 𝑚
𝑖=1
𝑚1
𝑚𝑛𝑖 log 𝑛𝑖
≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚
log 𝑖 𝑛𝑖𝑚
≤ 𝐶 𝑛 log 𝑛
Bounding Distance Cost at Lower Hierarchy
57
Theorem [Talagrand ‘92] Let 𝑋 = {𝑥1, … , 𝑥𝑛}, 𝑌 = {𝑦1, … , 𝑦𝑛} be two sets sampled 𝑖. 𝑖. 𝑑. from the same arbitrary distribution on 0,1 2. Then
𝐸 min𝜎
𝑖=1
𝑛
𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,
in which 𝐶 is a universal constant.
𝑞𝑖
𝐸 𝐷𝑖 ≤𝐶
𝑚𝑛𝑖 log 𝑛𝑖
1
𝑚
![Page 58: Distance Optimal Target Assignment in Robotic Networks ...people.csail.mit.edu/jingjin/files/YuChuVou14ICRAslides.pdf · Distance Optimal Target Assignment in Robotic Networks under](https://reader033.vdocument.in/reader033/viewer/2022042007/5e703d3b3d56b57f0b3acf16/html5/thumbnails/58.jpg)
𝑖=1
𝑚
𝐸[𝐷𝑖] ≤ 𝐶 𝑚
𝑖=1
𝑚1
𝑚𝑛𝑖 log 𝑛𝑖
≤ 𝐶 𝑚 𝑖 𝑛𝑖𝑚
log 𝑖 𝑛𝑖𝑚
≤ 𝐶 𝑛 log 𝑛
Bounding Distance Cost at Lower Hierarchy
58
Theorem [Talagrand ‘92] Let 𝑋 = {𝑥1, … , 𝑥𝑛}, 𝑌 = {𝑦1, … , 𝑦𝑛} be two sets sampled 𝑖. 𝑖. 𝑑. from the same arbitrary distribution on 0,1 2. Then
𝐸 min𝜎
𝑖=1
𝑛
𝑥𝑖 − 𝑦𝜎 𝑖 ≤ 𝐶 𝑛 log 𝑛,
in which 𝐶 is a universal constant.
𝑞𝑖
𝐸 𝐷𝑖 ≤𝐶
𝑚𝑛𝑖 log 𝑛𝑖
1
𝑚
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
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𝑞𝑖
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
,
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗
2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗
2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)
𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖
2
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Bounding Distance Cost at Higher Hierarchy
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𝑞𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗
2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)
𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖
2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖
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𝑖=1
𝑚
𝐸[ 𝑆𝑖 ] =
𝑖=1
𝑚
2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛
𝑖=1
𝑚1
𝑚𝑝𝑖 1 − 𝑝𝑖
≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖
𝑚1 − 𝑖=1
𝑚 𝑝𝑖
𝑚= 2𝑛 𝑚 − 1
Bounding Distance Cost at Higher Hierarchy
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𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗
2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)
𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖
2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖
𝑞𝑖
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Bounding Distance Cost at Higher Hierarchy
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𝑃 𝑥𝑗 ∈ 𝑞𝑖 = 𝑃 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖
𝑃 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖 = 𝑃 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖 = 𝑝𝑖(1 − 𝑝𝑖)
𝑍𝑗 =
−1, 𝑥𝑗 ∉ 𝑞𝑖 , 𝑦𝑗 ∈ 𝑞𝑖1, 𝑥𝑗 ∈ 𝑞𝑖 , 𝑦𝑗 ∉ 𝑞𝑖0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
, 𝑆𝑖 = 𝑍1 +⋯+ 𝑍𝑛
𝐸[𝑆𝑖2] = 𝑛𝐸 𝑍𝑗
2 = 2𝑛𝑝𝑖(1 − 𝑝𝑖)
𝐸 𝑆𝑖 = 𝐸 𝑆𝑖2 ≤ 𝐸 𝑆𝑖
2 ⇒ 𝐸[ 𝑆𝑖 ] ≤ 2𝑛𝑝𝑖 1 − 𝑝𝑖
𝑖=1
𝑚
𝐸[ 𝑆𝑖 ] =
𝑖=1
𝑚
2𝑛𝑝𝑖 1 − 𝑝𝑖 = 𝑚 2𝑛
𝑖=1
𝑚1
𝑚𝑝𝑖 1 − 𝑝𝑖
≤ 𝑚 2𝑛 𝑖=1𝑚 𝑝𝑖
𝑚1 − 𝑖=1
𝑚 𝑝𝑖
𝑚= 2𝑛 𝑚 − 1
𝑞𝑖
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Bounds on Distance Optimality
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Theorem (Performance Upper-Bound of Ideal Hierarchical Strategies) Let𝐷𝑛 be the total distance of an ideal hierarchical strategy with ℎ hierarchies and 𝑚𝑖 regions at hierarchy 𝑖, then for arbitrary distribution on 0,1 2,
𝐸 𝐷𝑛 ≤ 𝐶 𝑛 log 𝑛 + 2 𝑛
𝑖=1
ℎ−1𝑚𝑖+1
𝑚𝑖.
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Bounds on Distance Optimality
12
Theorem (Performance Upper-Bound of Ideal Hierarchical Strategies) Let𝐷𝑛 be the total distance of an ideal hierarchical strategy with ℎ hierarchies and 𝑚𝑖 regions at hierarchy 𝑖, then for arbitrary distribution on 0,1 2,
𝐸 𝐷𝑛 ≤ 𝐶 𝑛 log 𝑛 + 2 𝑛
𝑖=1
ℎ−1𝑚𝑖+1
𝑚𝑖.
Theorem [Ajtai et al. ‘84]. Under the uniform distribution, with high
probability, 𝐶1 𝑛 log 𝑛 ≤ 𝐷𝑛∗ ≤ 𝐶2 𝑛 log 𝑛.
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Bounds on Distance Optimality
12
Theorem (Performance Upper-Bound of Ideal Hierarchical Strategies) Let𝐷𝑛 be the total distance of an ideal hierarchical strategy with ℎ hierarchies and 𝑚𝑖 regions at hierarchy 𝑖, then for arbitrary distribution on 0,1 2,
𝐸 𝐷𝑛 ≤ 𝐶 𝑛 log 𝑛 + 2 𝑛
𝑖=1
ℎ−1𝑚𝑖+1
𝑚𝑖.
Theorem [Ajtai et al. ‘84]. Under the uniform distribution, with high
probability, 𝐶1 𝑛 log 𝑛 ≤ 𝐷𝑛∗ ≤ 𝐶2 𝑛 log 𝑛.
Corollary. With uniform distribution, fixing ℎ and {𝑚𝑖}, as 𝑛 → ∞,
𝐸[𝐷𝑛]
𝐸[ 𝐷𝑛∗]→ 𝑂 1 .
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Bounds on Distance Optimality
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Corollary. With uniform distribution, fixing ℎ and {𝑚𝑖}, as 𝑛 → ∞,
𝐸[𝐷𝑛]
𝐸[ 𝐷𝑛∗]→ 𝑂 1 .
A two-level ideal hierarchical strategy
𝑛 - number of robots
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Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒
13
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Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒
13
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Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒
13
Two-level decentralized hierarchical strategy
𝑛 - number of robots
Two-level ideal hierarchical strategy
2
𝑛 - number of robots
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Incorporating Arbitrary 𝑟𝑐𝑜𝑚𝑚 and 𝑟𝑠𝑒𝑛𝑠𝑒
13
Two-level decentralized hierarchical strategy
𝑛 - number of robots
Two-level ideal hierarchical strategy
2
𝑛 - number of robots
Arbitrary 𝑟𝑠𝑒𝑛𝑠𝑒 can also be handled similarly.
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Summary of Contribution
Guarantee on the distance optimality of the stochastic target assignment problem Necessary and sufficient condition for optimality Non-asymptotic probabilistic bounds Asymptotically tight bounds for high-probability guarantee
Performance of decentralized hierarchical strategies General upper bounds for arbitrary distributions 𝑂(1) approximation algorithm for the uniform distribution
Important takeaway: locally optimal behavior leads to near globallyoptimal behavior