A Rupestrian Algorithm FUN 2016 Giuseppe A. Di Luna, Paola - - PowerPoint PPT Presentation

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A Rupestrian Algorithm FUN 2016 Giuseppe A. Di Luna, Paola - - PowerPoint PPT Presentation

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A Rupestrian Algorithm FUN 2016 Giuseppe A. Di Luna, Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, Giovanni Viglietta La Maddalena June 8, 2016 A Rupestrian Algorithm A recent archaeological expedition discovered ancient cave

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A Rupestrian Algorithm

FUN 2016 Giuseppe A. Di Luna, Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, Giovanni Viglietta La Maddalena – June 8, 2016

A Rupestrian Algorithm

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A recent archaeological expedition discovered ancient cave paintings on Budelli island, attributed to the semi-nomadic Astracinca civilization.

A Rupestrian Algorithm

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The cave graffiti depict a shamanistic ritual that the Astracinca tribes executed periodically to create new settlements.

A Rupestrian Algorithm

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Each member of the tribe had to drink from a bowl filled with liquid from one of two preparations.

A Rupestrian Algorithm

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The first potion, containing poppy milk, was a strong narcotic that induced a deep state of unconsciousness and catatonia.

A Rupestrian Algorithm

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Those who drunk the second potion, which contained amanita muscaria and cannabis, were inebriated but still mildly functional.

A Rupestrian Algorithm

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The ritual took place on a square grid where the villagers were initially arranged along a line. Each villager was administered one

  • f the two potions, chosen at random. /

A Rupestrian Algorithm

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The inebriated villagers rearranged themselves along a new line and left the village to form a new settlement; the catatonic ones would continue their life in the old village./

A Rupestrian Algorithm

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In some cases, the settlers were unable to reach an agreement on a

  • direction. This could happen if the initial configuration was

symmetric and the settlers kept moving in a symmetric fashion./

A Rupestrian Algorithm

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Hence, occasionally, the outcome of the ritual was the formation of two oppositely oriented lines of equal size. In this case, two new settlements were created./

A Rupestrian Algorithm

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The hallucinatory and dissociative states induced by amanita muscaria and cannabis, as well as the short-term deterioration of memory, are well known.

A Rupestrian Algorithm

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Hence, the ritual requires each villager to remember only a small amount of information, which does not depend on the number of participants./

A Rupestrian Algorithm

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S

Hence, the ritual requires each villager to remember only a small amount of information, which does not depend on the number of participants./

A Rupestrian Algorithm

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S1

2

S3 S

Moreover, the ritual only requires each villager to be able to see, communicate, and make decisions based on his immediate neighbors on the grid./

A Rupestrian Algorithm

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S1

2

S3 S

At regular intervals, each settler moves to an empty node of the grid, based on the states of the currently visible settlers. Each step

  • f the ritual is executed synchronously by all settlers./

A Rupestrian Algorithm

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2

S3 S S1

2

S

f(

, ,

)

S3

At regular intervals, each settler moves to an empty node of the grid, based on the states of the currently visible settlers. Each step

  • f the ritual is executed synchronously by all settlers./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

On the first turn, if a Settler has only one neighbor, he becomes a Marker./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

On the first turn, if a Settler has only one neighbor, he becomes a Marker./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

Then the Marker sends a special message to his neighbor, which is forwarded along the line./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

Then the Marker sends a special message to his neighbor, which is forwarded along the line./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If a Settler receives two messages at the same time, he becomes a Unique Leader./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If a Settler receives two messages at the same time, he becomes a Unique Leader./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If a Settler receives two messages at the same time, he becomes a Unique Leader./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

The Unique Leader moves in some direction while the other Settlers become Followers./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

The Unique Leader moves in some direction while the other Settlers become Followers./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If there is an even number of Settlers, two Leaders are created and form two equal lines./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If there is an even number of Settlers, two Leaders are created and form two equal lines./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If there is an even number of Settlers, two Leaders are created and form two equal lines./

A Rupestrian Algorithm

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Ritual: checking for no Sleepers

If there is an even number of Settlers, two Leaders are created and form two equal lines./

A Rupestrian Algorithm

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Ritual: determining the Explorers

On the first turn, if a Settler has a neighbor who is a Sleeper, he becomes an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

On the first turn, if a Settler has a neighbor who is a Sleeper, he becomes an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

The Explorers will eventually abandon the initial line and scout their surroundings./

A Rupestrian Algorithm

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Ritual: determining the Explorers

On the second turn, if a Settler sees an Explorer, he sends a message to his other neighbor. /

A Rupestrian Algorithm

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Ritual: determining the Explorers

On the second turn, if a Settler sees an Explorer, he sends a message to his other neighbor. /

A Rupestrian Algorithm

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Ritual: determining the Explorers

On the second turn, if a Settler sees an Explorer, he sends a message to his other neighbor. /

A Rupestrian Algorithm

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Ritual: determining the Explorers

If the same Settler does not receive a similar message from his neighbor, he becomes an Explorer too./

A Rupestrian Algorithm

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Ritual: determining the Explorers

Finally, all the Explorers move sideways in a random direction. /

A Rupestrian Algorithm

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Ritual: determining the Explorers

If a Settler who sent the message receives a similar message from his neighbor, he does not become an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

If a Settler who sent the message receives a similar message from his neighbor, he does not become an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

If a Settler who sent the message receives a similar message from his neighbor, he does not become an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

If a Settler who sent the message receives a similar message from his neighbor, he does not become an Explorer./

A Rupestrian Algorithm

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Ritual: determining the Explorers

This protocol prevents the formation of gaps of 3 consecutive empty nodes within the initial line./

A Rupestrian Algorithm

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Ritual: determining the Explorers

At the same time, it creates a large-enough number of Explorers who will participate in the next phases of the ritual. /

A Rupestrian Algorithm

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Ritual: creating Markers

The Explorers start walking along the initial line, turning around if they bump into each other./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Markers

If an Explorer finds a sequence of 3 empty nodes on the initial line, it moves there and becomes a Marker./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

Each Marker initially has a scepter, which is given to the first Explorer who passes by./

A Rupestrian Algorithm

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Ritual: creating Chiefs

The Explorer who receives the scepter becomes a Chief, and starts walking in the opposite direction./

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer sees a Marker with no scepter, it becomes a Disciple and stops moving./

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer sees a Marker with no scepter, it becomes a Disciple and stops moving./

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer and a Chief bump into each other, they swap their states (equivalently, they switch positions)./

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer and a Chief bump into each other, they swap their states (equivalently, they switch positions)./

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer sees a Disciple, it becomes a Disciple, as well. /

A Rupestrian Algorithm

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Ritual: creating Chiefs

If an Explorer sees a Disciple, it becomes a Disciple, as well. /

A Rupestrian Algorithm

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Ritual: Unique Leader case

If a Chief reaches a Marker who still has the scepter, he acquires the scepter and becomes Unique Leader./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If a Chief reaches a Marker who still has the scepter, he acquires the scepter and becomes Unique Leader./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If a Chief reaches a Marker who still has the scepter, he acquires the scepter and becomes Unique Leader./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If a Chief reaches a Marker who still has the scepter, he acquires the scepter and becomes Unique Leader./

A Rupestrian Algorithm

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Ritual: Unique Leader case

The Unique Leader goes around the initial line, and all who see him become Followers./

A Rupestrian Algorithm

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Ritual: Unique Leader case

The Unique Leader goes around the initial line, and all who see him become Followers./

A Rupestrian Algorithm

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Ritual: Unique Leader case

The Unique Leader goes around the initial line, and all who see him become Followers./

A Rupestrian Algorithm

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Ritual: Unique Leader case

The Unique Leader goes around the initial line, and all who see him become Followers./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If two villagers try to move to the same spot, only one succeeds (chosen at random)./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If two villagers try to move to the same spot, only one succeeds (chosen at random)./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If two villagers try to move to the same spot, only one succeeds (chosen at random)./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If two villagers try to move to the same spot, only one succeeds (chosen at random)./

A Rupestrian Algorithm

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Ritual: Unique Leader case

If two villagers try to move to the same spot, only one succeeds (chosen at random)./

A Rupestrian Algorithm

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Ritual: Unique Leader case

Each member of the queue waits for the next member to reach him, and then takes another step./

A Rupestrian Algorithm

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Ritual: Unique Leader case

Each member of the queue waits for the next member to reach him, and then takes another step./

A Rupestrian Algorithm

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Ritual: Unique Leader case

Each member of the queue waits for the next member to reach him, and then takes another step./

A Rupestrian Algorithm

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Ritual: Unique Leader case

Each member of the queue waits for the next member to reach him, and then takes another step./

A Rupestrian Algorithm

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Ritual: Unique Leader case

Eventually, all the active villagers become part of the line and can start the migration./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If the Markers give their scepters to two different Explorers, two Chiefs are created./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If two Chiefs meet, they become Opposing Leaders and switch directions./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If two Chiefs meet, they become Opposing Leaders and switch directions./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The Opposing Leaders walk around the initial line collecting everyone, until they meet again on the other side./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

Now their goal is to determine a winner by comparing the sizes of their respective queues./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

To this end, they send a message to their respective Follower, which is forwarded all the way to the tail./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

To this end, they send a message to their respective Follower, which is forwarded all the way to the tail./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

Then the message bounces back and goes to the head again. /

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

Then the message bounces back and goes to the head again. /

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The first Opposing Leader to get the message back is the loser, and becomes a Follower./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

The other Opposing Leader becomes then the Unique Leader, and guides the procession./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

However, if one of the two queues bends, its head is automatically declared the winner, even if that queue is shorter./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

However, if one of the two queues bends, its head is automatically declared the winner, even if that queue is shorter./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

However, if one of the two queues bends, its head is automatically declared the winner, even if that queue is shorter./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

However, if one of the two queues bends, its head is automatically declared the winner, even if that queue is shorter./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If the two queues have the same size and neither of them bends, each Leader turns around and starts guiding his own queue./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If the two queues have the same size and neither of them bends, each Leader turns around and starts guiding his own queue./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If the two queues have the same size and neither of them bends, each Leader turns around and starts guiding his own queue./

A Rupestrian Algorithm

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Ritual: Opposing Leaders case

If the two queues have the same size and neither of them bends, each Leader turns around and starts guiding his own queue./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

If the Chiefs are on opposite sides of the initial line, they eventually meet a Marker with no scepter./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

If the Chiefs are on opposite sides of the initial line, they eventually meet a Marker with no scepter./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

If the Chiefs are on opposite sides of the initial line, they eventually meet a Marker with no scepter./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Then each Chief becomes a Parallel Leader and starts walking in the opposite direction, collecting everyone except the Markers./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Then each Chief becomes a Parallel Leader and starts walking in the opposite direction, collecting everyone except the Markers./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

A Parallel Leader stops when he reaches the opposite Marker. Then the lengths of the two queues have to be compared./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Preliminarily, a Mover leaves each queue and walks to the opposite Marker./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Preliminarily, a Mover leaves each queue and walks to the opposite Marker./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Preliminarily, a Mover leaves each queue and walks to the opposite Marker./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Preliminarily, a Mover leaves each queue and walks to the opposite Marker./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Then the Marker sends a message to both the Mover and the Parallel Leader, who become Probes./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Then the Marker sends a message to both the Mover and the Parallel Leader, who become Probes./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Each Probe walks toward the opposite Marker, waiting one turn each time he meets a Follower./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

When a Marker sees an incoming Probe, he knows which side has the longest queue./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Based on this information, the Probe leading the shortest queue becomes Unique Leader, while everyone else waits./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Based on this information, the Probe leading the shortest queue becomes Unique Leader, while everyone else waits./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

The Unique Leader walks around the initial line, collecting everyone and forming a unique queue./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

The Unique Leader walks around the initial line, collecting everyone and forming a unique queue./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

If a Parallel Leader with no Followers reaches a Marker, there are not enough people to create a Mover and a Probe on that side./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

If a Parallel Leader with no Followers reaches a Marker, there are not enough people to create a Mover and a Probe on that side./

A Rupestrian Algorithm

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Ritual: Parallel Leaders case

Since this cannot happen on both sides, the Parallel Leader with no Followers can safely become Unique Leader and collect everyone./

A Rupestrian Algorithm

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Ritual: termination

All villagers need to know when the ritual is terminated and the actual migration may start./

A Rupestrian Algorithm

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Ritual: termination

The termination detection procedure is started by the head of each queue, as soon as he “suspects” that the ritual may be finished./

A Rupestrian Algorithm

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Ritual: termination

A message is forwarded along the queue from head to tail, and advances only if the queue does not bend in that point./

A Rupestrian Algorithm

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Ritual: termination

A message is forwarded along the queue from head to tail, and advances only if the queue does not bend in that point./

A Rupestrian Algorithm

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Ritual: termination

A message is forwarded along the queue from head to tail, and advances only if the queue does not bend in that point./

A Rupestrian Algorithm

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Ritual: termination

A message is forwarded along the queue from head to tail, and advances only if the queue does not bend in that point./

A Rupestrian Algorithm

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Ritual: termination

A message is forwarded along the queue from head to tail, and advances only if the queue does not bend in that point./

A Rupestrian Algorithm

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Ritual: termination

When the message reaches the tail, the queue is straight. Then the message is sent back to the head./

A Rupestrian Algorithm

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Ritual: termination

When the message reaches the tail, the queue is straight. Then the message is sent back to the head./

A Rupestrian Algorithm

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Ritual: termination

When the message reaches the tail, the queue is straight. Then the message is sent back to the head./

A Rupestrian Algorithm

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Ritual: termination

When the head gets the message, he reaches a Terminated state, and stops./

A Rupestrian Algorithm

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Ritual: termination

Each Follower becomes Terminated too, as soon as he sees someone else in the Terminated state./

A Rupestrian Algorithm

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Ritual: termination

Each Follower becomes Terminated too, as soon as he sees someone else in the Terminated state./

A Rupestrian Algorithm

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Ritual: termination

Each Follower becomes Terminated too, as soon as he sees someone else in the Terminated state./

A Rupestrian Algorithm

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Future work

In the autumnal months, the new flowers of cannabis had greater quantities of THC. This caused the inebriated settlers to experience temporal distortion, as well: Atakan, Morrison, Bossong, Martin-Santos, and Crippa. The effect of cannabis on perception of time: a critical review. Current Pharmaceutical Design, 18(32):4915–22, 2012.

A Rupestrian Algorithm

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Future work

In the autumnal months, the new flowers of cannabis had greater quantities of THC. This caused the inebriated settlers to experience temporal distortion, as well: Atakan, Morrison, Bossong, Martin-Santos, and Crippa. The effect of cannabis on perception of time: a critical review. Current Pharmaceutical Design, 18(32):4915–22, 2012. In those months, a more complicated version of the ritual was used, which took into account the fact that the participants may have a different perception of the passage of time, and may unpredictably slow down in their movements and communication. We are currently studying the paintings depicting the long version

  • f the ritual, which are expected to solve the line recovery problem

under the asynchronous setting.

A Rupestrian Algorithm