Applications of WSPD & Visibility Graphs Computational ... · Dr. Tamara Mchedlidze Dr. Darren...

Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Applications of WSPD & Visibility Graphs1

Tamara Mchedlidze · Darren Strash

Computational Geometry Lecture

INSTITUT FUR THEORETISCHE INFORMATIK · FAKULTAT FUR INFORMATIK

Applications of WSPD & Visibility Graphs

25.01.2016

Recall: Well-Separated Pair Decomposition

Def: A pair of disjoint point sets A and B in Rd is calleds-well separated for some s > 0, if A and B can eachbe covered by a ball of radius r whose distance is atleast sr.

≥ sr

r′≥ sr′

Def: For a point set P and some s > 0 an s-well separatedpair decomposition (s-WSPD) is a set of pairsA1, B1, . . . , Am, Bm with

Ai, Bi ⊂ P for all iAi ∩Bi = ∅ for all i⋃m

i=1 Ai ⊗Bi = P ⊗ PAi, Bi s-well separated for all i

Def: For a point set P and some s > 0 an s-well separatedpair decomposition (s-WSPD) is a set of pairsA1, B1, . . . , Am, Bm with

Ai, Bi ⊂ P for all iAi ∩Bi = ∅ for all i⋃m

i=1 Ai ⊗Bi = P ⊗ PAi, Bi s-well separated for all i

Thm 3: Given a point set P in Rd and s ≥ 1 we can constructan s-WSPD with O(sdn) pairs in timeO(n log n + sdn).

Further Applications of WSPD

Euclidean MST

Problem:Given a point set P find a minimum spanning tree(MST) in the Euclidean graph EG(P ).

Euclidean MST

Prim: MST in a graph G = (V,E) can becomputed in O(|E|+ |V | log |V |) time.

Euclidean MST

EG(P ) has Θ(n2) edges ⇒ running time O(n2) :-((1 + ε)-spanner for P has O(n/εd) edges⇒ running time O(n log n + n/εd) :-)

Euclidean MST

How good is the MST of a (1 + ε)-spanner?

Euclidean MST

How good is the MST of a (1 + ε)-spanner?

Thm 5: The MST obtained from a (1 + ε)-spanner of P is a(1 + ε)-approximation of the EMST of P .

Diameter of P

Problem: Find the diameter of a point set P (i.e., the pairx, y ⊂ P with maximum distance).

Diameter of P

brute-force testing all point pairs ⇒ running time O(n2) :-(test distances ||rep(u) rep(v)|| of all ws-pairs Pu, Pv⇒ running time O(n log n + sdn) :-)

Diameter of P

How good is the computed diameter?

Diameter of P

How good is the computed diameter?

Thm 6:The diameter obtained from an s-WSPD of P fors = 4/ε is a (1 + ε)-approximation of the diameter of P .

Closest Pair of Points

Problem: Find the pair x, y ⊂ P with minimum distance.

brute-force testing all point pairs⇒ running time O(n2) :-(test distances ||rep(u) rep(v)|| of all ws-pairs Pu, Pv⇒ running time O(n log n + sdn) :-)

Exercise: For s > 2 this actually yields the closest pair.

Discussion

What are further applications of the WSPD?

Discussion

WSPD is useful whenever one can do without knowing all Θ(n2) exactdistances in a point set and approximate them instead. One example areforce-based layout algorithms in graph drawing, where pairwise repulsiveforces of n points need to be calculated.

Discussion

Why approximate geometrically?

Discussion

On the one hand, this replaces slow computations by faster (but lessprecise) ones; on the other hand, often the input data are imprecise sothat approximate solutions can be sufficient depending on the application.

Discussion

Can we achieve the same time bounds with exact computations?

Discussion

Can we achieve the same time bounds with exact computations?

In R2 this is often true, but not in Rd for d > 2. (e.g. EMST, diameter)

Organizational Information

Oral Exams:

Length: 30 minutesDates: Feb. 23, 24, 25; April 12, 13, 14Times: 9, 9:30, 10, 10:30, 11Doodle: Select all time slots that you have available!

Project presentations:

Next week on Feb. 1 and 3.

Please come to our offices and ask questions!

Motion planning and Visibility Graphs

Robot Motion Planning

Problem:Given a (point) robot at position pstart in a area withpolygonal obstacles, find a shortest path to pgoalavoiding obstacles.

Ideas?

Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Applications of WSPD & Visibility Graphs11-1

First Idea: Shortest Paths in Graphs

pgoalpstart

First compute trapezoidal map

pstart

Remove segments in obstacles

pstart

Nodes in trapezoids and vertical line segments

pstart

Euclidean weighted “dual graph” G with nodes on vertical segments

pstart

Locate start and goal

pstart

Shortest path with Dijkstra in G

pstart

Shortest path with Dijkstra in G

Running time?

pstart

Shortest path with Dijkstra in Gnot the shortest path!not the shortest path!

pstart

Shortest path with Dijkstra in Gnot the shortest path!not the shortest path!

Shortest Paths in Polygonal Areas

Lemma 1: For a set S of disjoint open polygons in R2 and twopoints s and t not in S

Lemma 1: For a set S of disjoint open polygons in R2 and twopoints s and t not in Seach shortest st-path in R2 \

⋃S is a polygonal

path whose internal vertices are vertices of S.

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Proof sketch:

⋃S is a polygonal

Visibility Graph

Given a set S of disjoint open polygons...

Visibility Graph

...with point set V (S).

Visibility Graph

Def.: Then Gvis(S) = (V (S), Evis(S)) is the visibility graph of S withEvis(S) = uv | u, v ∈ V (S) and u sees v und w(uv) = |uv|.

Visibility Graph

Where u sees v :⇔ uv ∩⋃S = ∅

Visibility Graph

Define S? = S ∪ s, t and Gvis(S?) analogously.

Visibility Graph

Define S? = S ∪ s, t and Gvis(S?) analogously.

Lemma 1⇒ A shortest st-path in R2 avoiding obstacles in S is equivalent

to a shortest st-path in Gvis(S?).

Algorithm

ShortestPath(S, s, t)

Input: Obstacles S, points s, t ∈ R2 \⋃S

Output: Shortest collision-free st-path in S1 Gvis ← VisibilityGraph(S ∪ s, t)2 foreach uv ∈ Evis do w(uv)← |uv|3 return Dijkstra(Gvis, w, s, t)

Algorithm

n = |V (S)|,m = |Evis(S)|

O(n log n + m)

Algorithm

n = |V (S)|,m = |Evis(S)|

O(n log n + m)

Thm 1: A shortest st-path in an area with polygonal obstacleswith n edges can be computed in O(n2 log n) time.

O(n2 log n)

Computing a Visibility Graph

VisibilityGraph(S)

Input: Set of disjoint polygons SOutput: Visibility graph Gvis(S)

1 E ← ∅2 foreach v ∈ V (S) do3 W ← VisibleVertices(v, S)4 E ← E ∪ vw | w ∈W5 return (V (S), E)

Computing Visible Nodes

VisibleVertices(p, S)p

Problem: Given p and S,find in O(n log n) time allnodes that p sees in V (S)!

VisibleVertices(p, S)pr ← p + (k, 0) | k ∈ R+

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

e1 e2 e3 e4

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

T ← balancedBinaryTree(I)e5

e1 e2 e3 e4

w1, . . . , wn ← sort V (S) incyclic order around p

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

e1 e2 e3 e4

r ← p + (k, 0) | k ∈ R+0 r

v ≺ v′ :⇔∠v < ∠v′ or(∠v = ∠v′ and |pv| < |pv′|)

I ← e ∈ E(S) | e ∩ r 6= ∅

e1 e2 e3 e4

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

e1 e2 e3 e4

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

Sweep method with rotation

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

W ← ∅for i = 1 to n do

if Visible(p, wi) thenW ←W ∪ wi

Add to T edges incident to wi: CW from −→pwi+

Remove from T edges incident to wi:CCW from −→pwi−

return W

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

return W

pwi−→pwi

−→pwi−

r ← p + (k, 0) | k ∈ R+0 r

I ← e ∈ E(S) | e ∩ r 6= ∅

return W

pwi−→pwi

−→pwi−

Visibility Case AnalysisVisible(p, wi)

if pwi intersects polygon of wi thenreturn false

if i = 1 or wi−1 6∈ pwi thene← edge of leftmost leaf of Tif e 6= nil and pwi ∩ e 6= ∅ then

return falseelse return true

elseif wi−1 is not visible then

return falseelse

e← find edge in T , that wi−1wi cuts; if e 6= nilthen return falseelse return true

return falseelse

wi−1

return falseelse

wi−1

p wi−1wi

Summary

Proof:Correctness follows directly from Lemma 1.

Summary

Running time:– VisibleVertices takes O(n log n) time per vertex – n callsto VisibleVertices

Summary

Running time:– VisibleVertices takes O(n log n) time per vertex – n callsto VisibleVertices

O(n2) with duality(see exercise or D. Mount [M12] Lect. 31)

Discussion

Robots are not single points...

Discussion

For robots modelled by a convex polygon that cannot rotate, we canresize (grow) the polygons representing the obstacles(→ Minkowski Sums, Ch. 13 in [BCKO08]).

Discussion

Roboter

Discussion

Obstacle

Discussion

Can we compute faster than O(n2 log n)? Obstacle

Discussion

Can we compute faster than O(n2 log n)?

Yes, by use duality and a simultaneous rotation sweepfor all points in the dual. Computing the arrangement,is also in O(n2). Even though Gvis can have Ω(n2)edges, the visibility graph can be constructed

even faster with an output sensitive O(n log n + m)-time algorithm.

[Ghosh, Mount 1987]

Obstacle

Discussion

Can we compute faster than O(n2 log n)?

Yes, by use duality and a simultaneous rotation sweepfor all points in the dual. Computing the arrangement,is also in O(n2). Even though Gvis can have Ω(n2)edges, the visibility graph can be constructed

even faster with an output sensitive O(n log n + m)-time algorithm.

If you search only for one shortest Euclidean st-path, there is analgorithm with optimal O(n log n) time. [Hershberger, Suri 1999]

[Ghosh, Mount 1987]

Obstacle

Applications of WSPD & Visibility Graphs Computational ... · Dr. Tamara Mchedlidze Dr. Darren...

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