Mutual Visibility with an Optimal Number of Colors Gokarna Sharma, Costas Busch, and Supratik Mukhopadhyay Louisiana State University Algosensors 2015 1
Autonomous robots Look-Compute-Move cycles in the Euclidean plane 2
Autonomous robots Look - Sense the positions of other robots 3
Autonomous robots Compute – Determine destination based on sensed positions of other robots Thinking… 4
Autonomous robots Move – towards computed destination new target 5
Autonomous robots 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 6
Autonomous robots Obstructed visibility Robots do not see through other robots p p does not see robots 7
The Mutual Visibility Problem Reach a configuration in which no three robots are collinear Collisions should be avoided – no path crossings and no position sharing 8
The Mutual Visibility Problem Reach a configuration in which no three robots are collinear Convex hull Typically the solution has the form of a convex hull (we also provide convex hull) 9
Robots with Lights Model • Proposed by Peleg, D. (2005) • Each robot has an externally visible light • Given an identical color set color set 10
Robots with Lights Model 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) color set 11
Robots with Lights Model Benefits: #robots n does not need to be known • -nodes always terminate Corresponds to model with no lights when • when color set size = 1 12
Literature 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) 13
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 one axis. 14
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 one axis. 15
Di Luna et al. algorithm for 3 colors Initial State 16
Di Luna et al. algorithm for 3 colors Convex hull robots get red 17
Di Luna et al. algorithm for 3 colors Internal Depletion 18
Di Luna et al. algorithm for 3 colors Interior Depletion 19
Di Luna et al. algorithm for 3 colors Interior Depletion 20
Di Luna et al. algorithm for 3 colors Corner robots move inside preserving convex hull 21
Di Luna et al. algorithm for 3 colors Corner robots move inside preserving convex hull Yellow nodes don’t move again 22
Di Luna et al. algorithm for 3 colors Corner robots move inside preserving convex hull Yellow nodes don’t move again 23
Di Luna et al. algorithm for 3 colors Final configuration, nodes terminate 24
Our algorithm for 2 colors Interior Depletion Phase: • Internal Robots move to edges of convex hull Side Depletion Phase: • Side Robots move outside • May cause new internal depletion phase 25
Our algorithm for 2 colors Initial State All robots are marked as “OFF” (gray color) 26
Our algorithm for 2 colors Convex hull robots are marked as external (red color) 27
Our algorithm for 2 colors Interior Depletion Internal robots move to convex hull edges 28
Our algorithm for 2 colors Interior Depletion 29
Our algorithm for 2 colors Interior Depletion All robots are marked as external (red color) 30
Our algorithm for 2 colors Side Depletion: side robots move out 31
Our algorithm for 2 colors Side Depletion: side robots move out Robots that become internal get “OFF” color 32
Our algorithm for 2 colors Interior Depletion - again 33
Our algorithm for 2 colors Side Depletion - again 34
Our algorithm for 2 colors Final Configuration 35
Convergence Corner robots do not move Termination detection by corner robots: No observed internal node No observed collinear nodes 36
Convergence 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 37
Internal Depletion Target Choice Possibly multiple targets r 38
Internal Depletion Target Choice However collisions may occur r 39
Internal Depletion Target Choice Check there is no other internal robot in parallel half plane to target edge r 40
Internal Depletion Target Choice OK r 41
Side Depletion Target Choice ? r 42
Side Depletion Target Choice Safe area computation r 43
Side Depletion Target Choice Safe area computation ¼ angle ¼ angle r 44
Side Depletion Target Choice Safe area computation Move anywhere inside safe area r 45
Side Depletion Target Choice At least one of r or r’ will become corner despite what happens in other edges r’ r 46
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) 47
Thank You! 48
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