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Shape Formation in a Three-dimensional Model for Hybrid Programmable Matter Kristian Hinnenthal, Dorian Rudolph, Christian Scheideler Paderborn University 1 / 20 Outline Introduction Motivation Problem Model 2D 3D Safely Movable Tiles

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Shape Formation in a Three-dimensional Model for Hybrid Programmable Matter

Kristian Hinnenthal, Dorian Rudolph, Christian Scheideler Paderborn University

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SLIDE 2

Outline

Introduction Motivation Problem Model 2D 3D Safely Movable Tiles Algorithm Conclusion

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SLIDE 3

Motiviation I

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SLIDE 4

Motivation II

B. Jenett and D. Cellucci, “A mobile robot for locomotion through a 3D periodic lattice environment,” 2017

IEEE International Conference on Robotics and Automation (ICRA), Singapore, 2017, pp. 5474-5479.

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Problem

◮ Use a single finite automaton robot to reconfigure arbitrary 3D tile structures into a given shape, e.g., a pyramid.

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SLIDE 6

Outline

Introduction Motivation Problem Model 2D 3D Safely Movable Tiles Algorithm Conclusion

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ A finite automaton robot r operates on a set of n hexagonal tiles.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ A finite automaton robot r operates on a set of n hexagonal tiles. ◮ Each node of the triangular lattice is occupied by at most one tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ A finite automaton robot r operates on a set of n hexagonal tiles. ◮ Each node of the triangular lattice is occupied by at most one tile. ◮ r can move on or adjacent to occupied nodes.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ A finite automaton robot r operates on a set of n hexagonal tiles. ◮ Each node of the triangular lattice is occupied by at most one tile. ◮ r can move on or adjacent to occupied nodes. ◮ r may carry at most one tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ A finite automaton robot r operates on a set of n hexagonal tiles. ◮ Each node of the triangular lattice is occupied by at most one tile. ◮ r can move on or adjacent to occupied nodes. ◮ r may carry at most one tile. ◮ Tiles (including the robot if it carries a tile) must remain connected.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles:

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action. ◮ Move: Move to adjacent tile, pickup tile,

r place carried tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action. ◮ Move: Move to adjacent tile, pickup tile,

r place carried tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action. ◮ Move: Move to adjacent tile, pickup tile,

r place carried tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action. ◮ Move: Move to adjacent tile, pickup tile,

r place carried tile.

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2D Model

2D model by Gmyr et al. [DNA 2018]: ◮ r operates in look-compute-move cycles: ◮ Look: Observe adjacent nodes. ◮ Compute: Change state and determine action. ◮ Move: Move to adjacent tile, pickup tile,

r place carried tile.

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3D Model

◮ Same as 2D, but with rhombic dodecahedral tiles.

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3D Model

◮ Same as 2D, but with rhombic dodecahedral tiles.

Tomruen [CC BY-SA 4.0] 8 / 20

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3D Model

◮ Same as 2D, but with rhombic dodecahedral tiles.

Tomruen [CC BY-SA 4.0] TED-43 [CC BY 3.0] 8 / 20

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3D Model

◮ Same as 2D, but with rhombic dodecahedral tiles.

Tomruen [CC BY-SA 4.0] TED-43 [CC BY 3.0] Didier Descouens [CC BY-SA 4.0] 8 / 20

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Lattice

◮ Rhombic dodecahedral tiles are placed in the face-centred cubic (FCC) lattice (i.e., the adjacency graph of the FCC sphere packing)

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Lattice

◮ Rhombic dodecahedral tiles are placed in the face-centred cubic (FCC) lattice (i.e., the adjacency graph of the FCC sphere packing)

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Lattice

◮ Rhombic dodecahedral tiles are placed in the face-centred cubic (FCC) lattice (i.e., the adjacency graph of the FCC sphere packing) ◮ Voronoi cells of the FCC lattice are rhombic dodecahedra

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation.

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation. ◮ Robots can move along the surface while remaining connected.

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation. ◮ Robots can move along the surface while remaining connected.

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation. ◮ Robots can move along the surface while remaining connected.

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation. ◮ Robots can move along the surface while remaining connected.

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Why Rhombic Dodecahedra?

◮ Form a space-filling tesselation. ◮ Robots can move along the surface while remaining connected. ◮ Can be viewed in terms of hexagonal layers.

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Directions

◮ Up: UNE/USE/UW ◮ Same layer: N/NE/SE/S/SW/NW ◮ Down: DNW/DE/DSW

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Safely Movable Tiles

◮ Safely locally movable:

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Safely Movable Tiles

◮ Safely locally movable: ◮ Safely movable:

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Safely Movable Tiles

◮ Safely locally movable: ◮ Safely removable: ◮ Safely movable:

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Safely Movable Tiles

◮ Safely locally movable: ◮ Safely removable: ◮ Safely movable:

Theorem (2D)

A robot that can always find a safely locally movable tile.

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SLIDE 38

Safely Movable Tiles

◮ Safely locally movable: ◮ Safely removable: ◮ Safely movable:

Theorem (2D)

A robot that can always find a safely locally movable tile. No robot can find a safely removable tile in all configurations.

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SLIDE 39

Safely Movable Tiles

◮ Safely locally movable: ◮ Safely removable: ◮ Safely movable:

Theorem (2D)

A robot that can always find a safely locally movable tile. No robot can find a safely removable tile in all configurations.

Theorem (3D)

No robot can find a safely movable tile in all configurations.

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SLIDE 40

Configuration without Safely Locally Movable Tiles

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SLIDE 41

Configuration without Safely Locally Movable Tiles

◮ Solution: The robot initially carries a tile.

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SLIDE 42

Outline

Introduction Motivation Problem Model 2D 3D Safely Movable Tiles Algorithm Conclusion

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Intermediate Structure

◮ Use a line as intermediate structure for shape formation.

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Intermediate Structure

◮ Use a line as intermediate structure for shape formation.

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Intermediate Structure

◮ Use a line as intermediate structure for shape formation.

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Intermediate Structure

◮ Use a line as intermediate structure for shape formation.

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Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Traverse current column and check adjacent nodes. ◮ If tile in direction USE/UNE/UW/NW/SW, move there. ◮ Terminate if the structure is a line.

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SLIDE 48

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Traverse current column and check adjacent nodes. ◮ If tile in direction USE/UNE/UW/NW/SW, move there. ◮ Terminate if the structure is a line.

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SLIDE 49

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Traverse current column and check adjacent nodes. ◮ If tile in direction USE/UNE/UW/NW/SW, move there. ◮ Terminate if the structure is a line.

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SLIDE 50

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Traverse current column and check adjacent nodes. ◮ If tile in direction USE/UNE/UW/NW/SW, move there. ◮ Terminate if the structure is a line.

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SLIDE 51

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path.

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SLIDE 52

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile.

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SLIDE 53

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 54

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 55

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 56

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 57

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 58

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 59

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 60

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 61

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 62

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 63

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 64

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 65

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 66

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 67

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile. ◮ The carried tile maintains connectivity.

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SLIDE 68

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 69

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 70

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 71

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 72

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 73

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 74

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 75

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 76

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 77

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 78

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 79

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 80

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 81

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 82

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 83

Line Building Algorithm

Algorithm works in three phases: ◮ Find column: Find a column (maximal line in N/S direction) without tiles above or west. ◮ Move column down: If found column has neighbors below, move its tiles DE. ◮ Move column east: Otherwise, move its tiles SE. ◮ Place carried tile at first empty node along the path. ◮ Take tile from the column’s northernmost tile. ◮ Switch to find column if ◮ current column merged with column to the S, ◮ or took last tile.

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SLIDE 84

Analysis

Theorem

The robot can build a line in O(n3) rounds.

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SLIDE 85

Outline

Introduction Motivation Problem Model 2D 3D Safely Movable Tiles Algorithm Conclusion

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SLIDE 86

Future Work

◮ Improve runtime of the algorithm.

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SLIDE 87

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2).

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SLIDE 88

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2). ◮ Using multiple robots.

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SLIDE 89

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2). ◮ Using multiple robots. ◮ Explore the structure or find the outer hull.

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SLIDE 90

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2). ◮ Using multiple robots. ◮ Explore the structure or find the outer hull. ◮ Which capabilities are necessary?

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SLIDE 91

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2). ◮ Using multiple robots. ◮ Explore the structure or find the outer hull. ◮ Which capabilities are necessary? ◮ Blum and Kozen (1978)a appears to imply that at least three pebbles are necessary.

aBlum, M., & Kozen, D. (1978). On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science (sfcs 1978). IEEE. 19 / 20

SLIDE 92

Future Work

◮ Improve runtime of the algorithm. ◮ Lower bound Ω(n2). ◮ Using multiple robots. ◮ Explore the structure or find the outer hull. ◮ Which capabilities are necessary? ◮ Blum and Kozen (1978)a appears to imply that at least three pebbles are necessary. ◮ Exploration of undirected graphs requires Θ(log log n) pebbles. b

aBlum, M., & Kozen, D. (1978). On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science (sfcs 1978). IEEE. bDisser, Y., Hackfeld, & J., Klimm, M. (2015). Undirected Graph Exploration with Θ(log log n) Pebbles. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics. 19 / 20

SLIDE 93

Try out the Simulator

◮ https://go.upb.de/3DHybridSim ◮ Browser: Tested with Chrome, Safari does not work

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