[PPT] - Altitude Terrain Guarding and Guarding Uni-Monotone Polygons PowerPoint Presentation

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Stephan Friedrichs Valentin Polishchuk Christiane Schmidt

Altitude Terrain Guarding and Guarding Uni-Monotone Polygons

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EuroCG 2018 2

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png https://www.dronerush.com/best-drones-1977/drone.bmp

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EuroCG 2018 2

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EuroCG 2018 2

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png https://www.dronerush.com/best-drones-1977/drone.bmp

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EuroCG 2018 2

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png https://www.dronerush.com/best-drones-1977/drone.bmp

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EuroCG 2018 2

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png https://www.dronerush.com/best-drones-1977/drone.bmp

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EuroCG 2018 3

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png

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EuroCG 2018 3

image source: http://2.bp.blogspot.com/-eUnhAo8SfFQ/VDaLij2_csI/AAAAAAAABPM/847rgT6VpQE/s1600/Screen%2BShot%2B2014-05-01%2Bat%2B00.02.54.png

2.5D

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EuroCG 2018 3

2.5D 1.5D

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EuroCG 2018

Altitude Terrain Guarding Problem

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EuroCG 2018

Altitude Terrain Guarding Problem

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn}

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} Monotonicity

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} Monotonicity ➡ Points on T are totally ordered wrt to x-coordinate: p<q .

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). p ∈ A q ∈ T

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EuroCG 2018

Altitude Terrain Guarding Problem

4

Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). p ∈ A q ∈ T q2 ∈ T

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). p ∈ A q ∈ T q2 ∈ T

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EuroCG 2018

Altitude Terrain Guarding Problem

4

Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). VT(p) is the visibility region of p with VT(p) := {q∈T | p sees q} . p ∈ A q ∈ T

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EuroCG 2018

Altitude Terrain Guarding Problem

4

Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). VT(p) is the visibility region of p with VT(p) := {q∈T | p sees q} . For G⊆A VT(G) := ∪g∈G VT(g). p ∈ A q ∈ T

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EuroCG 2018

Altitude Terrain Guarding Problem

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Terrain T Altitude line A

x-monotone chain of line segments in R2 defined by its vertices V(T) = { v1 , . . . , vn} A point p ∈ A sees or covers q ∈ T if and only if pq is nowhere below T (i.e. pq lies on or above T ). VT(p) is the visibility region of p with VT(p) := {q∈T | p sees q} . For G⊆A VT(G) := ∪g∈G VT(g). p ∈ A q ∈ T Altitude Terrain Guarding Problem (ATGP) ATGP(T,A) Given: a terrain T and an altitude line A . A minimum set of guards that see all of T. (Formally: A guard set G⊂A is optimal w.r.t. ATGP(T,A) if G is feasible, that is, T⊆VT(G) , and |G| = OPT(T,A) := min{|C| | C⊂A is feasible w.r.t. ATGP(T,A)} .)

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon?

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? monotone polygon

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? monotone polygon

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? monotone polygon

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? uni-monotone polygon

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EuroCG 2018

AGP in uni-monotone polygons

5

uni-monotone polygon? uni-monotone polygon

H

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? uni-monotone polygon

H

LC(P)

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? uni-monotone polygon

H

p ∈ P LC(P)

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? uni-monotone polygon

H

p ∈ P VP(p) - visibility polygon LC(P)

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EuroCG 2018

AGP in uni-monotone polygons

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uni-monotone polygon? uni-monotone polygon

H

p ∈ P VP(p) - visibility polygon Formally: Art Gallery Problem (AGP) AGP(G,W) Given: a polygon P and sets of guard candidates and points to cover G,W⊆P. A minimum guard set C⊆G that covers W (that is, W⊆VP(C)). We want to solve AGP(P,P). LC(P)

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EuroCG 2018

AGP in uni-monotone polygons

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If we want to solve the AGP for a uni-monotone polygon, w.l.o.g. we can restrict our guards to be located on H.

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EuroCG 2018

AGP in uni-monotone polygons

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If we want to solve the AGP for a uni-monotone polygon, w.l.o.g. we can restrict our guards to be located on H. Proof: Consider any optimal guard set G, let g∈G be a guard not located on H

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EuroCG 2018

AGP in uni-monotone polygons

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If we want to solve the AGP for a uni-monotone polygon, w.l.o.g. we can restrict our guards to be located on H. Proof: Consider any optimal guard set G, let g∈G be a guard not located on H

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EuroCG 2018

AGP in uni-monotone polygons

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If we want to solve the AGP for a uni-monotone polygon, w.l.o.g. we can restrict our guards to be located on H. Proof: Consider any optimal guard set G, let g∈G be a guard not located on H

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EuroCG 2018

AGP in uni-monotone polygons

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Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G).

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EuroCG 2018

AGP in uni-monotone polygons

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Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G)

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EuroCG 2018

AGP in uni-monotone polygons

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Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G)

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EuroCG 2018

AGP in uni-monotone polygons

7

Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G)

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EuroCG 2018

AGP in uni-monotone polygons

7

Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G)

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EuroCG 2018

AGP in uni-monotone polygons

7

Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G)

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EuroCG 2018

AGP in uni-monotone polygons

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Let P be a uni-monotone polygon, let G be a guard set with g∈H ∀g∈G that covers LC(P), that is, LC(P)⊂VP(G) . Then G covers all of P, that is, P⊆VP(G). Proof: Assume p∈P, p∉LC(P), p∉VP(G) ➜ATGP and AGP for uni-monotone polygons equivalent

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EuroCG 2018

AGP-TGP-ATGP

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Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

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EuroCG 2018

AGP-TGP-ATGP

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Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard

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EuroCG 2018

AGP-TGP-ATGP

8

Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard NP-hard, even in monotone polygons

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EuroCG 2018

AGP-TGP-ATGP

8

Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard NP-hard, even in monotone polygons

Altitude Terrain Guarding Problem (ATGP) Given: a terrain T and an altitude line A . A minimum set of guards that see all of T.

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EuroCG 2018

AGP-TGP-ATGP

8

Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard NP-hard, even in monotone polygons

Altitude Terrain Guarding Problem (ATGP) Given: a terrain T and an altitude line A . A minimum set of guards that see all of T. Art Gallery Problem (AGP) in Uni-Monotone Polygons Given: a uni-monotone polygon P. A minimum guard set that covers P.

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EuroCG 2018

AGP-TGP-ATGP

8

Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard NP-hard, even in monotone polygons

Our Problems: “Inbetween”?!

Altitude Terrain Guarding Problem (ATGP) Given: a terrain T and an altitude line A . A minimum set of guards that see all of T. Art Gallery Problem (AGP) in Uni-Monotone Polygons Given: a uni-monotone polygon P. A minimum guard set that covers P.

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EuroCG 2018

AGP-TGP-ATGP

8

Terrain Guarding Problem (TGP) Given: a terrain T. Find: minimum number of guards located on T. Art Gallery Problem (AGP) Given: a polygon P. A minimum guard set that covers P.

NP-hard NP-hard, even in monotone polygons Both polytime

Our Problems: “Inbetween”?!

Altitude Terrain Guarding Problem (ATGP) Given: a terrain T and an altitude line A . A minimum set of guards that see all of T. Art Gallery Problem (AGP) in Uni-Monotone Polygons Given: a uni-monotone polygon P. A minimum guard set that covers P.

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EuroCG 2018 9

We show:

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EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons

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EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

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EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

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EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

A set W⊂P (W⊂T ) is a witness set if ∀wi≠wj ∈W we have VP(wi)∩VP(wj) = ∅.

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EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

A set W⊂P (W⊂T ) is a witness set if ∀wi≠wj ∈W we have VP(wi)∩VP(wj) = ∅.

SLIDE 57

EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

A set W⊂P (W⊂T ) is a witness set if ∀wi≠wj ∈W we have VP(wi)∩VP(wj) = ∅. A maximum witness set Wopt is a witness set of maximum cardinality, |Wopt| = max{|W|: witness set W}.

SLIDE 58

EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

A set W⊂P (W⊂T ) is a witness set if ∀wi≠wj ∈W we have VP(wi)∩VP(wj) = ∅. A maximum witness set Wopt is a witness set of maximum cardinality, |Wopt| = max{|W|: witness set W}. A polygon class P is perfect if

SLIDE 59

EuroCG 2018 9

We show:

A polytime algorithm for AGTP and AGP in uni-monotone polygons
Uni-monotone polygons are perfect - first non-trivial class

perfect ?

A set W⊂P (W⊂T ) is a witness set if ∀wi≠wj ∈W we have VP(wi)∩VP(wj) = ∅. A maximum witness set Wopt is a witness set of maximum cardinality, |Wopt| = max{|W|: witness set W}. A polygon class P is perfect if cardinality of an optimum guard set = cardinality of a maximum witness set ∀P∈P

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Altitude Terrain Guarding Problem

10

Guards to the left “don’t help”:

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

11

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EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

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EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right

SLIDE 76

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.

SLIDE 77

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)

SLIDE 78

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg

SLIDE 79

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

SLIDE 80

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

SLIDE 81

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

Delete pec and add pe’c to C

SLIDE 82

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

Delete pec and add pe’c to C
Delete e from Eg and add e’

SLIDE 83

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

Delete pec and add pe’c to C
Delete e from Eg and add e’

SLIDE 84

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

Delete pec and add pe’c to C
Delete e from Eg and add e’

SLIDE 85

EuroCG 2018

Sweep Algorithm

12

Start: empty set of guards G = ∅ ; at leftmost point of A ; all edges in E(T) are completely unseen

(Eg=E(T)).

Sweep along A from left to right
Place a guard gi whenever we could no longer see all of an “edge” e if we would move more to the right
first point in C.
Compute VT(gi)
Remove all completely seen edges from Eg
For each edge e = {v,w} partially seen by gi: split edge, keep the open interval that is not yet guarded

—>new “edge” e’⊂e

Delete pec and add pe’c to C
Delete e from Eg and add e’

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EuroCG 2018