Decidability of distributed complexity of locally checkable problems - - PowerPoint PPT Presentation

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Decidability of distributed complexity of locally checkable problems - - PowerPoint PPT Presentation

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Decidability of distributed complexity of locally checkable problems on paths Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, el Rabie , Jukka Suomela Mika ANR DESCARTES/ESTATE Tuesday, April 2 Mika el RABIE Decidability

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SLIDE 1

Decidability of distributed complexity of locally checkable problems on paths

Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Mika¨ el Rabie, Jukka Suomela

ANR DESCARTES/ESTATE

Tuesday, April 2

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 1 / 19

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SLIDE 2

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 3

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 4

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 5

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 6

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 7

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 8

LCL Problems on Paths

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 9

LCL Problems on Paths

1 28 37 52 8 1 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 10

LCL Problems on Paths

1 1 2 28 3 37 2 52 3 8 1 32 2 46 1 47 3 73 2 5 1 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 11

LCL Problems on Paths

1 1 2 28 3 37 2 52 3 8 1 32 2 46 1 47 3 73 2 5 1 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 12

LCL Problems on Paths

1 1 2 28 3 37 2 52 3 8 1 32 2 46 1 47 3 73 2 5 1 3 3 Coloring.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 13

LCL Problems on Paths

1 1 2 28 1 37 2 52 1 8 2 32 1 46 2 47 1 73 2 5 1 3 3 Coloring. 2 Coloring.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 14

LCL Problems on Paths

1 1 28 37 1 52 8 1 32 46 1 47 73 5 1 3 3 Coloring. 2 Coloring. Maximal Independent Set.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 15

LCL Problems on Paths

1 28 37 1 52 8 32 46 47 1 73 5 3 3 Coloring. 2 Coloring. Maximal Independent Set. Independent Set.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

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SLIDE 16

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 17

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 18

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 37 1 52 8 32 46 47 1 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 19

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 37 1 52 8 32 46 47 1 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 20

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 2 37 1 52 2 8 32 46 2 47 1 73 2 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 21

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 28 2 37 1 52 2 8 32 46 2 47 1 73 2 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 22

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 1 28 2 37 1 52 2 8 1 32 1 46 2 47 1 73 2 5 1 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 23

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

1 1 28 2 37 1 52 2 8 1 32 1 46 2 47 1 73 2 5 1 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 24

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

2 1 1 28 2 37 1 52 2 8 3 32 1 46 2 47 1 73 2 5 1 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 25

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

21 22 23 24 25 26 27 28 29 30 31

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 26

The log∗n Complexity 3-coloring a Path

3 Coloring a Path

21 22 23 24 25 26 27 28 29 30 31 Worst case communication complexity : Θ(n).

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

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SLIDE 27

The log∗n Complexity 3-coloring a Path

3-coloring in O(log∗ n) Communications

Cole, Vishkin (1986)

There exists a LCL algorithm to 3-color a path in O(log∗ n) communications.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 4 / 19

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SLIDE 28

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 102 36

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 29

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 101010 102 1100110 36 100100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 30

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 101010 102 1100110 36 100100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 31

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 3#0 101010 102 1100110 36 100100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 32

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 3#0 101010 102 1100110 36 100100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 33

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 3#0 101010 102 2#1 1100110 36 2#0 100100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 34

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

42 3#0 110 102 2#1 101 36 2#0 100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 35

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

6 3#0 110 5 2#1 101 4 2#0 100

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 36

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

6 3#0 110 5 2#1 101 4 2#0 100 n colors ⇒ log n bits ⇒ 2 log n new colors ⇒ log log n + 1 bits

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 37

The log∗n Complexity 3-coloring a Path

From n colors to log n colors

6 3#0 110 5 2#1 101 4 2#0 100 n colors ⇒ log n bits ⇒ 2 log n new colors ⇒ log log n + 1 bits After log∗ n iterations, O(1) bits.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

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SLIDE 38

The log∗n Complexity log∗ Lower Bound

Coloration Lower Bound

Linial (1992)

An algorithm which colors the n-cycle with three colors requires time at least 1

2(log∗ n − 3). The same bound holds also for randomized algorithms.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 6 / 19

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SLIDE 39

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 40

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. T T

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 41

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 42

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 43

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T ∀id ≤ n T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 44

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T ∀id ≤ n SL ∈ 2k T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 45

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T SL ∀id ≤ n T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 46

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T SL SR ∀id ≤ n T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 47

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T SL#SR ∈ 2k × 2k T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 48

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. c c ∈ [1, k] T T SL#SR ∈ 2k × 2k T − 1 T − 1 SL ∩ SR = ∅

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 49

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. SL#SR T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 50

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. SL#SR T − 1 T − 1 SL#SR S′

L#S′ R

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 51

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. SL#SR T − 1 T − 1 SL#SR S′

L#S′ R

SL ∩ SR = ∅ & S′

L ∩ S′ R = ∅

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 52

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. SL#SR T − 1 T − 1 SL#SR S′

L#S′ R

SL ∩ SR = ∅ & S′

L ∩ S′ R = ∅

S′

L ∩ SR = ∅

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 53

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. SL#SR T − 1 T − 1 SL#SR S′

L#S′ R

SL ∩ SR = ∅ & S′

L ∩ S′ R = ∅

S′

L ∩ SR = ∅

SL#SR = S′

L#S′ R

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 54

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 55

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1 T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 56

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1 ∀id ≤ n T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-57
SLIDE 57

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1 ∀id ≤ n SL ∈ 24k T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-58
SLIDE 58

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1 SL ∈ 24k SR ∈ 24k ∀id ≤ n T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-59
SLIDE 59

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. c ∈ [1, 4k] T − 1 T − 1 SL#SR T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-60
SLIDE 60

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. A2 : algorithm that 44k-colors nodes in T − 1 rounds. c ∈ [1, 44k] T − 1 T − 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-61
SLIDE 61

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. A2 : algorithm that 44k-colors nodes in T − 1 rounds. . . . Alog∗ n : algorithm that n-colors nodes in T − log∗n

2

rounds.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

slide-62
SLIDE 62

The log∗n Complexity log∗ Lower Bound

Speed up Algorithm

A : algorithm that k-colors nodes in T rounds. A1 : algorithm that 4k-colors edges in T − 1 rounds. A2 : algorithm that 44k-colors nodes in T − 1 rounds. . . . Alog∗ n : algorithm that n-colors nodes in T − log∗n

2

rounds. ⇒ any 3-coloring algorithm needs at least log∗n

2

rounds.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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SLIDE 63

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 64

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set. Nodes at distance 2 or 3.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 65

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set. Nodes at distance 2 or 3. Reiterate on the Independent Node. Nodes at distance between 4 and 9.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 66

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set. Nodes at distance 2 or 3. Reiterate on the Independent Node. Nodes at distance between 4 and 9. . . . Reiterate k times. Nodes at distance between 2k and 3k.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 67

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set. Nodes at distance 2 or 3. Reiterate on the Independent Node. Nodes at distance between 4 and 9. . . . Reiterate k times. Nodes at distance between 2k and 3k. Cut the segments in sizes between s and 2s for some s ≤ 2k.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 68

The log∗n Complexity Path Shattering

Independent Set at some Distance

From 3−coloring to Maximal Independent Set. Nodes at distance 2 or 3. Reiterate on the Independent Node. Nodes at distance between 4 and 9. . . . Reiterate k times. Nodes at distance between 2k and 3k. Cut the segments in sizes between s and 2s for some s ≤ 2k. Time complexity : O(s log∗ n).

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 8 / 19

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SLIDE 69

Decidability on Paths without Inputs Automata Output

Transition Automata

Node : Sequence of outputs. Edge : Connecting to an admitable next output.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 9 / 19

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SLIDE 70

Decidability on Paths without Inputs Automata Output

Transition Automata

Node : Sequence of outputs. Edge : Connecting to an admitable next output. 1 2 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 9 / 19

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SLIDE 71

Decidability on Paths without Inputs Automata Output

Transition Automata

Node : Sequence of outputs. Edge : Connecting to an admitable next output. 1 2 3 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 9 / 19

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SLIDE 72

Decidability on Paths without Inputs Automata Output

Transition Automata

Node : Sequence of outputs. Edge : Connecting to an admitable next output. 1 2 3 1 00 01 10

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 9 / 19

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SLIDE 73

Decidability on Paths without Inputs Automata Output

Transition Automata

Node : Sequence of outputs. Edge : Connecting to an admitable next output. 1 2 3 1 00 01 10 1 2

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 9 / 19

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SLIDE 74

Decidability on Paths without Inputs Automata Output

Complexity Separation on Paths

Naor, Sotckmeyer (1995)

If the input graph is an unlabeled path or cycle, the time complexity is decidable. The different time complexities are O(1), Θ(log∗ n) and Ω(n).

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 10 / 19

slide-75
SLIDE 75

Decidability on Paths without Inputs Automata Output

Complexity Separation on Paths

Naor, Sotckmeyer (1995)

If the input graph is an unlabeled path or cycle, the time complexity is decidable. The different time complexities are O(1), Θ(log∗ n) and Ω(n). 1 00 01 10 1 2

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 10 / 19

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SLIDE 76

Decidability on Paths with Inputs

Problem on Paths with Inputs

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 11 / 19

slide-77
SLIDE 77

Decidability on Paths with Inputs

Problem on Paths with Inputs

1 28 37 52 8 32 46 47 73 5 3

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 11 / 19

slide-78
SLIDE 78

Decidability on Paths with Inputs

Problem on Paths with Inputs

1 28 37 52 8 32 46 47 73 5 3 3-color the red nodes. Carry the color through the blue nodes.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 11 / 19

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SLIDE 79

Decidability on Paths with Inputs

Problem on Paths with Inputs

3 1 1 28 2 37 2 52 2 8 2 32 2 46 2 47 2 73 3 5 3 3 3-color the red nodes. Carry the color through the blue nodes.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 11 / 19

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SLIDE 80

Decidability on Paths with Inputs Complexity is Decidable

Complexity Separation on Paths with Inputs

Balliu, Brandt, Chang, Olivetti, Rabie, Suomela (2018)

For any LCL problem on cycle graphs, its complexity is either Ω(n) or O(log∗ n). Moreover, there is an algorithm that decides whether the problem has complexity Ω(n) or O(log∗ n) on cycle graphs ; for the case the complexity is O(log∗ n), the algorithm outputs a description of an O(log∗ n)-round deterministic LOCAL algorithm that solves it. For any LCL problem on cycle graphs, its complexity is either Ω(log∗ n) or O(1). Moreover, there is an algorithm that decides whether the problem has complexity Ω(log∗ n) or O(1) on cycle graphs ; for the case the complexity is O(1), the algorithm outputs a description of an O(1)-round deterministic LOCAL algorithm that solves it.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 12 / 19

slide-81
SLIDE 81

Decidability on Paths with Inputs Decision is PSPACE hard

Complexity Separation on Paths with Inputs

Balliu, Brandt, Chang, Olivetti, Rabie, Suomela (2018)

It is PSPACE-hard to distinguish whether a given LCL problem with input labels can be solved in O(1) time or needs Ω(n) time on globally oriented path graphs.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 13 / 19

slide-82
SLIDE 82

Decidability on Paths with Inputs Decision is PSPACE hard

Turing Machine Encoding

L R q0 L R q1 L 1 R L R L R L R L R q1 L R q1 q1 q1 q1 q1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L R qf 1 1 1 1 1 1

a

L L L R R R 1 T T T F F F F F F F F F F F F F F F F F F F F q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 F

· · · Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 14 / 19

slide-83
SLIDE 83

Decidability on Paths with Inputs Decision is PSPACE hard

Error Detection

a

L L L R R R 1 1 T T T F F F F F F F F F F F F F F F F F F F F q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 F

· · ·

a a a

E2 E E E E E E E E E E E E

a a a a

E2 E2 E2 E2 E2 E2 E2 E2 E2

1 2 3 4 5 6 7 8 9

Input Output Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 15 / 19

slide-84
SLIDE 84

Decidability on Paths with Inputs Decision on Trees without Inputs

Complexity Separation on Trees

Balliu, Brandt, Chang, Olivetti, Rabie, Suomela (2018)

It is PSPACE-hard to distinguish whether a given LCL problem without input labels can be solved in O(1) time or needs Ω(n) time on trees with degree ∆ = 3.

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 16 / 19

slide-85
SLIDE 85

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-86
SLIDE 86

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1 left right left right left right

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-87
SLIDE 87

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1 right left right left right

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-88
SLIDE 88

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1 left right left right

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-89
SLIDE 89

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-90
SLIDE 90

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-91
SLIDE 91

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102 1 1

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-92
SLIDE 92

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding a Number in a Tree

Encoding 6=01102

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 17 / 19

slide-93
SLIDE 93

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding the Input of the Path

a1 a2 a3 a4 a5 a6

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 18 / 19

slide-94
SLIDE 94

Decidability on Paths with Inputs Decision on Trees without Inputs

Encoding the Input of the Path

a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 18 / 19

slide-95
SLIDE 95

Conclusion Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 19 / 19

slide-96
SLIDE 96

Conclusion

T H A N K Y O U !

Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 19 / 19