Fast Complete Visibility for Point Robots with Lights Costas Busch - - PowerPoint PPT Presentation

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Fast Complete Visibility for Point Robots with Lights Costas Busch - - PowerPoint PPT Presentation

Jan 29, 2023 372 likes •1.02k views

Fast Complete Visibility for Point Robots with Lights Costas Busch Louisiana State University Joint work with R. Vaidyanathan, G. Sharma, J. Trahan, S. Rai MAC 2016 1 Autonomous robots Look-Compute-Move cycles in the Euclidean plane 2

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Fast Complete Visibility for Point Robots with Lights

Costas Busch Louisiana State University

Joint work with

  • R. Vaidyanathan, G. Sharma, J. Trahan, S. Rai

MAC 2016

1

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Look-Compute-Move cycles in the Euclidean plane

Autonomous robots

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Look - Sense the positions of other robots

Autonomous robots

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Compute – Determine destination based on sensed positions of other robots

Autonomous robots

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Thinking…

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Move – towards computed destination

Autonomous robots

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new target

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Robots are:

  • Dimensionless Points
  • Anonymous (no unique identifiers)
  • Autonomous (no external control)
  • Oblivious (no memory of past)
  • Silent (no explicit communication)
  • No common coordinate system

Autonomous robots

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Obstructed visibility

Robots do not see through other robots

Autonomous robots

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p

p does not see robots

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Reach a configuration in which no three robots are collinear

The Mutual Visibility Problem

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The Mutual Visibility Problem

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Convex hull The solution is typically a convex hull

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  • Proposed by Peleg, D. (2005)
  • Each robot has an externally visible light
  • Given an identical color set

Robots with Lights Model

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color set

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  • The robots communicate with each other

through colored lights (otherwise silent)

  • The colors of lights are not erased at the end of

the LCM cycle (otherwise oblivious)

Robots with Lights Model

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color set

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Benefits:

  • #robots N does not need to be known
  • nodes always terminate
  • Corresponds to model with no lights when

color set size = 1

Robots with Lights Model

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Fully synchronous all robots are active in a LCM round Semi-synchronous a subset of the robots is active in a LCM round Asynchronous there is no notion of round, each robot has its own notion of time for LCM cycle

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Timing

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Solvability

Di Luna et al. [SSS’14]

  • 6-color algorithm in the semi-synchronous setting
  • 10-color algorithm in the asynchronous setting

Di Luna et al. [Information and Computation 2015]

  • 3-color algorithm

Runtime

Vaidyanathan et al. [IPDPS’2014]

Fully synchronous setting

  • 12-color algorithm with running time O(log n) rounds

(possibility of collisions)

  • Improved to O(1) rounds in [SSS’2016] with no

collisions

Literature

14

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Two Approaches for Convex Hull

Approach 1 [Di. Luna et al 2015]: Interior Depletion Shrink repetitively Gives low number of colors Approach 2 [Di Luna et al 2014]: Interior Depletion Edge Depletion (expand) Gives fast algorithms

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Initial State

Approach 1

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Approach 1

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Convex hull robots get red color

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Approach 1

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Interior Depletion Robots move to edges of convex hull

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Approach 1

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Interior Depletion Robots move to edges of convex hull

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Approach 1

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Interior Depletion Robots move to edges of convex hull

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Approach 1

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Corner robots move inside preserving convex hull

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Approach 1

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Corner robots move inside preserving convex hull Yellow nodes don’t move again

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Approach 1

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Corner robots move inside preserving convex hull Yellow nodes don’t move again

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Final configuration, nodes terminate

Approach 1

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Initial State

Approach 2

All robots are marked as “OFF” (gray color)

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Convex hull corners are marked red

Approach 2

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Interior Depletion Internal robots move to convex hull edges

Approach 2

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Interior Depletion

Approach 2

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Edge Depletion: edge robots move out

Approach 2

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Side Depletion: side robots move out Robots that become internal get “OFF” color

Approach 2

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Move to edge - again

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Edge Depletion - again

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Final Configuration

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Fast Interior Depletion

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Fast Interior Depletion

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free free free

𝑠 Free triangles – no other robot inside

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Fast Interior Depletion

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𝑠 Moves to a free edge

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Fast Interior Depletion

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𝑠 There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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There are many free edges: min 𝑅 2 , 𝑆

perimeter points internal robots

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Fast Interior Depletion

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Total rounds for interior depletion:𝑃 log 𝑂

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Fast Interior Depletion

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For same edge collisions can be avoided pick a different target point according to relative position to edge

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Fast Interior Depletion

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Problem: for different edges collisions may occur

free Path collision

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Fast Edge Depletion

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Fast Edge Depletion

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Leftmost moves up

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Fast Edge Depletion

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Go to the middle

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Fast Edge Depletion

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2 new edges

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Fast Edge Depletion

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2 new edges

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Fast Edge Depletion

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2 new edges Repeat process for new edges Total rounds for edge depletion: 𝑃 log 𝑂

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Total time

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Total rounds for edge depletion: 𝑃 log 𝑂 Total rounds for interior depletion:𝑃 log 𝑂

+

𝑃 log 𝑂

rounds

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Faster: O(1) rounds

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Interior depletion in O(1) rounds Edge depletion in O(1) rounds Avoids collisions

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Interior depletion in O(1) rounds

  • 1. Corner robots move inside

to reveal themselves

  • 2. Interior nodes move toward corner nodes

to avoid collision

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𝑑𝑗 𝑑′ 𝑑′′ 𝑐 𝑏

𝑣 𝑥 𝑧 𝑨

𝑀 𝑀′

Revelation safe area

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𝑑𝑗 𝑑′ 𝑑′′ 𝑐 𝑏

𝑣 𝑥 𝑧 𝑨

𝑀 𝑀′

Revelation safe area Now visible by all interior robots

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Edge depletion in O(1) rounds Edge points move toward an arc

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Convex

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Future Directions

Time analysis for other problems:

  • Robots without Lights
  • Fat robots

Other metrics:

  • Total distance moved

Lower Bounds

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Thank You!