Mutual Visibility with an Optimal Number of Colors Gokarna Sharma, - - PowerPoint PPT Presentation

Mutual Visibility with an Optimal Number of Colors

Gokarna Sharma, Costas Busch, and Supratik Mukhopadhyay

Louisiana State University

Algosensors 2015

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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…

Move – towards computed destination

Autonomous robots

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new target

Robots are:

• Dimensionless Points
• Anonymous (no unique identifiers)
• Execute the same algorithm
• Autonomous (no external control)
• Oblivious (no memory of past events)
• Silent (no explicit communication)
• No common coordinate system
• no common unit distance, compass, or notion of clockwise

direction

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

Reach a configuration in which no three robots are collinear

The Mutual Visibility Problem

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Collisions should be avoided – no path crossings and no position sharing

Reach a configuration in which no three robots are collinear

The Mutual Visibility Problem

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Typically the solution has the form of a convex hull (we also provide convex hull)

Convex hull

• 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

• 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

Benefits:

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

when color set size = 1

Robots with Lights Model

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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] (the fully synchronous setting)

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

(possibility of collisions and chirality assumption)

Literature

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Di Luna et al. 2015

MUTUAL VISIBILITY solved for:

(a) SSYNCH robots under RIGID moves with 2 colors; (b) SSYNCH robots under NON-RIGID moves with 3 colors; (c) ASYNCH robots under RIGID moves with 3 colors; (d) ASYNCH robots under NON-RIGID moves with 3 colors, if the robots agree on the direction of

• ne axis.

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Our Contribution

MUTUAL VISIBILITY solved for:

(a) SSYNCH robots under RIGID moves with 2 colors; (b) SSYNCH robots under NON-RIGID moves with 2 colors; (c) ASYNCH robots under RIGID moves with 2 colors; (d) ASYNCH robots under NON-RIGID moves with 2 colors, if the robots agree on the direction of

• ne axis.

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Di Luna et al. algorithm for 3 colors

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Initial State

Di Luna et al. algorithm for 3 colors

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Convex hull robots get red

Di Luna et al. algorithm for 3 colors

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Internal Depletion

Di Luna et al. algorithm for 3 colors

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Interior Depletion

Di Luna et al. algorithm for 3 colors

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Interior Depletion

Di Luna et al. algorithm for 3 colors

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Corner robots move inside preserving convex hull

Di Luna et al. algorithm for 3 colors

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

Di Luna et al. algorithm for 3 colors

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

Di Luna et al. algorithm for 3 colors

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Final configuration, nodes terminate

Our algorithm for 2 colors

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Interior Depletion Phase:

• Internal Robots move to edges of

convex hull Side Depletion Phase:

• Side Robots move outside
• May cause new internal depletion phase

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Initial State

Our algorithm for 2 colors

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

Our algorithm for 2 colors

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Convex hull robots are marked as external (red color)

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Interior Depletion Internal robots move to convex hull edges

Our algorithm for 2 colors

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Interior Depletion

Our algorithm for 2 colors

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Interior Depletion All robots are marked as external (red color)

Our algorithm for 2 colors

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Side Depletion: side robots move out

Our algorithm for 2 colors

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

Our algorithm for 2 colors

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Interior Depletion - again

Our algorithm for 2 colors

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Side Depletion - again

Our algorithm for 2 colors

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Final Configuration

Our algorithm for 2 colors

Convergence

Termination detection by corner robots:

No observed internal node No observed collinear nodes

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Corner robots do not move

Convergence

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In each Interior Depletion (ID) phase: all internal nodes become external (red) In each Side Depletion (SD) phase: at least one external robot becomes corner ID SD ID SD ID SD ID … 1 1 1 New corner robots Eventually all robots become corner

Internal Depletion Target Choice

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Possibly multiple targets r

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Internal Depletion Target Choice

However collisions may occur r

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Check there is no other internal robot in parallel half plane to target edge

Internal Depletion Target Choice

r

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OK Internal Depletion Target Choice

r

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Side Depletion Target Choice

?

r

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Side Depletion Target Choice

Safe area computation r

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Side Depletion Target Choice

Safe area computation r

¼ angle ¼ angle

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Side Depletion Target Choice

Safe area computation r Move anywhere inside safe area

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Side Depletion Target Choice

r At least one of r or r’ will become corner despite what happens in other edges r’

Remarks

We presented a Mutual Visibility algorithm with 2 colors Separate algorithm is needed for ASYNCH NON-RIGID robots (with common axis) The algorithm can also solve the CIRCLE FORMATION problem with 2 or 3 colors (improves previous work by 1 color)

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Thank You!