Uniform dispersal of robots with minimum visibility range Attila - - PowerPoint PPT Presentation

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Uniform dispersal of robots with minimum visibility range Attila - - PowerPoint PPT Presentation

Apr 08, 2023 317 likes •817 views

Uniform dispersal of robots with minimum visibility range Attila Hideg 1 and Tams Lukovszki 2 1 Budapest University of T echnology and Economics, Hungary Attila.Hideg@aut.bme.hu 2 Etvs Lrnd University, Budapest, Hungary

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Uniform dispersal of robots with minimum visibility range

Attila Hideg1 and Tamás Lukovszki2

1 Budapest University of T echnology and Economics, Hungary Attila.Hideg@aut.bme.hu 2 Eötvös Lóránd University, Budapest, Hungary lukovszki@inf.elte.hu

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Filling

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  • n robots with restricted capabilities

>Limited sensing range >No communcation >O(1) bits memory

  • 2D plane setting

>Orthogonal area (2D grid) >Unknown, connected

  • Robots enter at the Door (or multiple Doors)

>When a Door becomes empty, a robot is placed immediately

  • They have to cover the area

https://ssr.seas.harvard.edu/

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Anonymous Restricted Robots

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  • Identical and anonymous
  • Silent
  • 1 hop visibility
  • Common notion of
  • up-down and left-right
  • Orthogonal area

>Decomposed into square-cells

  • Robots can move to

neighboring cells

>i.e. cells sharing a side

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Anonymous Restricted Robots

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  • Identical and anonymous
  • Silent
  • 1 hop visibility
  • Common notion of
  • up-down and left-right
  • Orthogonal area

>Decomposed into square-cells

  • Robots can move to

neighboring cells

>i.e. cells sharing a side

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Synchronous Look-Compute- Move (LCM)

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  • Look: take a snapshot
  • Compute: calculate the destinatin
  • Move: move to the destination

Look Compute Move

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Lower bounds

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  • Visibility: 1 hop
  • W(n) running time
  • Memory: O(1) bits [Barrameda et al. 2008]
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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State of the art

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  • Visibility range: # hops
  • Communcation range: # hops
  • Memory: # bits

Method Doors Visibility Comm. Memory Area DFLF [Hsiang et al. 2004]] Single 2 2 2 Arbitrary TALK [Barrameda et al 2013] Single 2 2 4 Orthogonal MUTE [Barrameda et al 2013] Single 6 9 Orthogonal MULTIPLE [Barrameda et al 2008] Multiple 3 4 Orthogonal Single Door (new) Single 1 13 Orthogonal Multiple Door (new) Multiple 1 13 Orthogonal

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Our Contribution (1)

  • Single Door
  • Requirements

>1 hop visibility >No communcation >13 bits memory >Common top-down, left-right directions

  • Running time

>O(n) rounds: 12n LCM cycles

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Our Contribution (2)

  • Multiple Door
  • Requirements

>1 hop visibility >No communcation >13 bits memory >Common top-down, left-right directions

  • Running time

>O(n) rounds: 12n LCM cycles

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method

  • Mimimcs a DFS traversal of the area
  • Main tasks to solve:

>Prevent collisions >Fill the whole area

  • Main ideas:

>Timing of movements >Follow the Leader method

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Filling the area

  • Mimimcs a DFS traversal of the area
  • DFS tree

>Unkown for the robots >T raversed by the robots >Branches are fjlled

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – States

  • Leader-Follower method
  • Four possible states

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Invariants

  • There is at most one Leader
  • When the Leader stucks,

>the Leadership is transfered

  • Followers only follow their predecessor

>Predecessor is either in a neighboring cell or >It moved away, then in the next step the follower moves to its prevIous position

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

  • Timing of the movements
  • Step

>Four possible directions (North, East, South, West)

  • Round

>4 step long time window >Odd and Even rounds for the robots >Odd: observing, Even: moving

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

  • Example

>North step >r’ moves

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

Look phase:

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

Compute phase:

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

Move phase:

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Method – Preventing Collisions

Look phase:

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: Follower

  • T

ask: Follow its predecessor >One hop visibility

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: Follower

  • T

ask: Follow its predecessor >One hop visibility

  • Odd rounds

>Observes the predecessor

l and empty neighbors l Stores which direction the

predecessor moves >(known from the timing!)

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: Follower

  • T

ask: Follow its predecessor >One hop visibility

  • Odd rounds

>Observes the predecessor

l and empty neighbors l Stores which direction the

predecessor moves >(known from the timing!)

  • Even rounds

>Moves to the previous position of the predecessor

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: Follower

  • T

ask: Follow its predecessor >One hop visibility

  • Odd rounds

>Observes the predecessor

l and empty neighbors l Stores which direction the

predecessor moves >(known from the timing!)

  • Even rounds

>Moves to the previous position of the predecessor

  • If predecessor did not move

>Switches to Leader state >Leadership transferred

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: Leader

  • T

ask

>Has to move to free cells

  • Odd rounds

>T akes a snapshot, and stores occupied cells

  • Even round

>Moves to the cell corrseponding to the direction of the step if it is unoccupied

  • Switch to Stopped if no free cells around

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

State: None

  • Robot placed at the Door

>After predecessor moved from it >Initial state is None >Knows which step is the currently performed (N,E,S,W)

  • Predecessor moved in even round

>None-state robot must have its odd round in the next round >Idle in the current round and >becomes a Follower in the next round

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Result

Theorem 1: The algorithm fjlls a connected orthogonal area with a single door

  • In O(n) rounds
  • Requirements

>Visibility range of 1 hop >O(1) bits of persistent memory >Common notion of North, South, East, West

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The Leader only moves to free cells

>T akes a snapshot in odd rounds >Knows which ones are free in even rounds

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The Leader only moves to free cells

>T akes a snapshot in odd rounds >Knows which ones are free in even rounds

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The Leader only moves to free cells

>T akes a snapshot in odd rounds >Knows which ones are free in even rounds

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The Leader only moves to free cells

>T akes a snapshot in odd rounds >Knows which ones are free in even rounds

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • Each Follower knows where its

predecessor is

>Odd round:

– knows which neighbor – observes its movement

>Even round: follows

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • At most one Leader is present in the area

>Leadership transferred

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • At most one Leader is present in the area

>Leadership transferred

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • No collisions can occur during the

dispersion

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • No collisions can occur during the

dispersion

>Follower: follows its unique predecessor

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • No collisions can occur during the

dispersion

>Follower: follows its unique predecessor >Leader: free cells

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The proposed method fjlls the area
  • For contradiction: Assume robots terminated and

>the area is not fjlled

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The proposed method fjlls the area

>r : fjrst robot terminating which has an empty neighbor >Before r terminated, r was a Leader

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • The proposed method fjlls the area
  • Contradiction: robots cannot terminate if

unoccupied cells are present

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

k-Door Filling

Multiple Door or k-Door fjlling

  • Assume each Door has
  • enough robots
  • A speed-up by k is possible

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

k-Door Filling

Theorem 2: The algorithm fjlls a connected orthogonal area with multiple Doors

  • In O(n) rounds
  • Requirements

>Visibility range of 1 hop >O(1) bits of persistent memory >Common notion of North, South, East, West

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • A Leader cannot collide with another

Leader

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • A Leader cannot collide with a Follower

>Leaders can determine free cells

  • Paths of difgerent Leaders cannot cross

each other

>Leader moves to free cells >Leaders cannot collide

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • k speed-up is possible
  • Worst-case optimal

>c is a bootleneck >Only robots from D3 are >fjlling the area

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • At most k times worse then the optimal

algorithm

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • At most k times worse then the optimal

algorithm

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Analysis

  • At most k times worse then the optimal

algorithm

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

Summary

  • Solve the Filling problem with robots

having

>1 hop visibility (optimal) >O(1) bits memory (assimp. opt) >O(n) rounds

  • Running time

>Assimp. opt for Single Door >Assimp. worst-case otpimal for Multiple Door

– At most k times the optimal algorithm

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Attila Hideg and T amás Lukovszki Uniform dispersal of robots with minimum visibility range

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Thank you!