Algorithms for Planar Graphs

Algorithms for Planar Graphs

Jeff Erickson University of Illinois, Urbana-Champaign

CIMAT, Guanajuato, Mexico September 10, 2018

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Planar Graphs

‣ A graph is planar if it can be drawn in the plane without edge crossings

▹ Vertices = points

▹ Edges = simple disjoint curves

2. PLANAR GRAPHS

planar embeddingplanar graphplane graphstereographic

projectionface!of a planar

embeddingouter face

2.2 Planar Graphs 1

Embeddings 2

A planar embedding of a graph G is a continuous injective function from the topological 3

graph G> to the plane. More explicitly, a planar embedding maps the vertices of G to 4

distinct points in the plane and maps the edges of G to simple paths in the plane between 5

the images of their endpoints, such that the paths do not intersect except at common 6

endpoints. A planar graph is an abstract graph that has at least one planar embedding. 7

Somewhat confusingly, a planar embedding of a planar graph is also called a plane 8

graph. 9

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A planar graph G. A planar embedding of G.

For many proofs, it is actually more natural to consider graph embeddings on the 10

sphere S2 = {(x , y, z) | x2 + y2 + z2 = 1} instead of the plane. Consider the standard 11

stereographic projection map st: S2\(0, 0, 1)! R2, where st(x , y, z) :=� x

1�z , y1�z

�. The 12

projection st(p) of any point p 2 S2 \ (0,0,1) is the intersection of the line through p 13

and the “north pole” (0, 0, 1) with the x y-plane. Given any spherical embedding, if we 14

rotate the sphere so that the embedding avoids (0,0, 1), stereographic projection gives 15

us a planar embedding; conversely, given any planar embedding, inverse stereographic 16

projection immediately gives us a spherical embedding. Thus, a graph is planar if and 17

only if it has an embedding on the sphere. 18

Stereographic projection.

The components of the complement of the image of a planar or spherical embedding 19

are called the faces of the embedding. The Jordan curve theorem implies that every 20

face of a spherical embedding of a connected graph is homeomorphic to an open disk. A 21

planar embedding of a connected graph has a single unbounded outer face, which is 22

6

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Figure 1: Main stages of our method for mesh reconstruction from scattered data.

Figure 2: The Stanford Lucy model is reconstructed as a mesh frompoint scattered data. Left: a triangle mesh approximating the Lucystatue. Right: zoomed parts of the Lucy mesh (the head and and afoot) are shown in order to demonstrate a high quality of the mesh.

Figure 3: Left: ten range scans of the Stanford bunny model arerendered as a cloud of points. Middle: a zoomed part of bunny' sear, noise is visible. Right: the same part on the mesh reconstructedby our method, noise is eliminated, small features are accuratelyreconstructed, and the mesh triangles have good aspect ratios.

Often scattered points are assigned with a certain con� dence mea-sure [Turk and Levoy 1994; Curless and Levoy 1996]. If ci ∈ [0; 1]

Figure 4: Meshing of noisy point data sampled from a model withsharp features. The model (octa-� ower implicit surface) was poly-gonized with by the marching cubes algorithm and then corruptedby random noise. Left: a triangle mesh connecting the originalpoints accurately. Right: our method reconstructs the sharp edgesand corners of the model using a much smaller number of verticesthan the original marching cubes mesh has.

indicates the con� dence of pi ∈ P , we assign the weight

wi = ci ai (2)

to point pi, i = 1; : : : ; N. We set ci = 1 if no con� dence measure isattributed to point pi.To approximate P by a mesh we need to estimate unit normalsN = f n1 ; ¢ ¢ ¢ ; nN g at the points of P . Often scattered points arealready equipped with unit normals which are directly estimatedfrom range scans. If normals N are not provided, we estimatethem using a standard covariance-based technique [Gumhold et al.2001]. In our approach, the normals need not to be consistent withthe surface orientations.

3 Adaptive Spherical Cover and Mesh Gen-eration

In this section, we explain the core part of our method: creating anadaptive spherical cover ofP and generating a meshM approxi-mating scattered pointsP .

3.1 Control for Overlapping

To build an adaptive spherical cover ofP we use a simpli� ed ver-sion of the approach proposed Wu and Kobbelt [Wu and Kobbelt2004] for generating an approximately minimal set of splats cover-ing the whole surface. Our spherical cover consists of spheres cen-tered at f c1 ; : : : ; cM g ⊂ P and the radii of the spheres f r1 ; : : : ; rM gare chosen adaptively.

Step 1. Set all points inP as uncovered.

62

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Planar embedding

‣ Vertices = points

‣ Edges = simple interior-disjoint curves

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Incidence list

‣ Each edge represented by two directed edges or darts

‣ Each vertex v stores the list of darts into v

‣ Each dart stores successor, reversal, and tail vertex

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Rotation system

‣ Counterclockwise order of edges incident to each vertex

‣ Specifies the embedding of G on the sphere, up to (ambient) isotopy

[Hamilton 1856] [Kirkman 1856] [Cayley 1857] [Heffter 1891] [Brückner 1900]....

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Rotation system

‣ Counterclockwise order of edges incident to each vertex

‣ Specifies the embedding of G on the sphere, up to (ambient) isotopy

[Hamilton 1856] [Kirkman 1856] [Cayley 1857] [Heffter 1891] [Brückner 1900]....

Ordered incidence list

‣ Record the rotation system in the incidence list

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Easy Consequences

‣ For any planar triangulation, E = 3V–6 and F = 2V–4 ▹ Every face has exactly three sides

‣ For any simple planar graph, E ≤ 3V–6 and F ≤ 2V–4 ▹ No loops or parallel edges ▹ Subgraph of a triangulation

‣ Every simple planar graph has a vertex of degree ≤ 5 ▹ ... at least 4 vertices of degree ≤ 5 ▹ ... either a vertex of degree ≤ 3 or a face of degree ≤ 3

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Tarjan’s red-blue rule

‣ For each cycle in G, color the heaviest edge red.

‣ For each cut in G, color the lightest edge blue.

‣ Every edge of G is either red or blue but not both.

‣ The blue edges define the minimum spanning tree of G.

[Tarjan 1983]

Tarjan’s red-blue algorithm

‣While G has at least one edge, either delete any red edge or contract any blue edge.

‣ Contracted edges = minimum spanning tree of G.

[Tarjan 1983]

Textbook MST algorithms

‣ Borůvka: Contract lightest edge incident to each vertex, delete heavy parallel edges, and (if any edges left) recurse.

‣ Jarník/Prim: Fix a vertex v. Contract lightest edge incident to v, delete heavy parallel edges, and (if edges left) recurse.

‣ Kruskal: For all edges in increasing weight order: If the current edge is a loop, delete it; otherwise, contract it.

[Tarjan 1983]

...in planar graphs

‣ For each cycle in G, color the heaviest edge red.

‣ For each cut in G, color the lightest edge blue.

‣ The blue edges define the minimum spanning tree of G.

‣ For each cut in G*, the heaviest edge is red.

‣ For each cycle in G*, the lightest edge is blue.

‣ The red edges define the maximum spanning tree of G*.

Tarjan’s red-blue algorithm

‣While G has at least one edge, either delete any red edge or contract any blue edge.

‣ Contracted edges = minimum spanning tree of G.

‣ Deleted edges = maximum spanning tree of G*.

Borůvka’s algorithm

‣While G has at least one edge ▹ For each vertex v

• Contract the lightest edge incident to v (which must be blue) ▹ Delete any heavy parallel edges (which must be red)

‣ All contractions in the first iteration take O(E) = O(V) time

‣ All deletions in the first iteration take O(E) = O(V) time

‣ Each iteration removes at least half the vertices.

‣ So overall running time is O(V)

[Borůvka 1928]

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‣While G has at least one edge

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▹ If G has a face f with degree at most 3 • If f is incident to a bridge, contract the bridge • Else delete the heaviest edge incident to f

‣ Each iteration takes O(1) time, so total running time is O(V).

[Matsui 1995]

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Straight-line embedding

‣ Every planar embedding of a simple planar graph G is equivalent to an embedding where every edge is a straight line segment.[Steinitz 1916, Wagner 1936, Fáry 1948, Stein 1951, Stojaković 1959]


‣ Suppose G is a triangulation.

‣ Delete any vertex of degree ≤ 5 and retriangulate the hole

‣ Recursively embed the remaining graph ▹ Base case: Embed 3-cycle as a triangle

‣ Reinsert vertex in resulting polygonal hole
















Schnyder wood

‣WLOG suppose G is a triangulation

‣ Color the outer vertices red, green, and blue

‣ Direct and color the interior edges as follows:

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[Schnyder 1990]

Schnyder wood

‣ Recursive construction:

[Schnyder 1990]

Schnyder wood

‣ The edges of each color induce a directed spanning tree of the interior vertices, rooted at the outer vertex of that color.

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Schnyder wood

‣ The edges of each color induce a directed spanning tree of the interior vertices, rooted at the outer vertex of that color.

▹ Each tree has V–2 vertices and V–3 edges

▹ So G has 3(V–3) + 3 = 3V–6 edges

▹ We just proved Euler’s formula again! 1

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Separators

‣ A separator of an n-vertex graph G is a subset S of vertices where each component of G\S has at most 2n/3 vertices.

‣ Used in lots of divide-and-conquer algorithms

T(n) ≤ T(n/3) + T(2n/3) + f(n)

⟹ T(n) =O(n) if f(n) = O(n1−ε)O(n log n) if f(n) = Θ(n)O( f(n)) if f(n) = Ω(n1+ε)

Easy separators

‣ Paths: One vertex

Easy separators


‣ Cycles: Two vertices

Easy separators



‣ Trees: One vertex [Jordan 1869]

Easy separators




‣ Outer-planar graphs: Two vertices

Easy separators





‣ √n×√n grid: √n vertices

Easy separators





‣ √n×√n grid: √n vertices

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Fundamental cycle separator

‣ Suppose G is a planar triangulation.



‣ Let (T, C) be an arbitrary tree-cotree decomposition.




‣ Each vertex in C* has degree ≤ 3.









‣ There is an edge e such that each component of C*\e* has at most 2F/3 dual vertices.






‣ Then cycle(T,e) is a (dual) separator of size ≤ 2 depth(T) + 1.













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Extensions

‣ Planar triangulations have cycle separators of size O(√n).[Miller ’84 ’86]

Extensions


‣ Separator hierarchy: Recursively separate both components

Extensions



▹ Cutting across the hierarchy gives us an r-division: n/r pieces, each with O(r) vertices and adjacent to O(√r) shallower separator vertices[Frederickson ’87]

Extensions




▹ For cycle separator hierarchies: Each piece is a disk with O(1) holes.[Klein Subramanian ’08]

Extensions




▹ For cycle separator hierarchies: Each piece is a disk with O(1) holes.[Klein Subramanian ’08]

▹ The entire hierarchy can be computed in O(n) time.[Klein Mozes Sommer ’13]

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Further tricks

‣Multiple-source shortest paths: We can (implicitly) compute distL(S,L) and distR(S,R) in only O(n log n) time.

▹ Yes, even though there are O(n3/2) distances to compute!

▹ More details tomorrow!

[Klein ’05]

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D.6. The SMAWK algorithm

If M[t, S[t]] < M[t, k], then M[t, k] and everything above it is dead, so we cansafely push column k onto the stack (unless we already have t = n, in which case everyentry in column k is dead) and then increment k.

S[1]S[2]S[3] S[4]S[5] S[6] S[7] k

⇤ ⇤

On the other hand, if M[t,S[t]]> M[t, k], then we know that M[t, S[t]] and everythingbelow it is dead. But by the inductive hypothesis, every entry above M[t,S[t]] is alreadydead. Thus, the entire column S[t] is dead, so we can safely pop it o� the stack.

S[1]S[2]S[3] S[4]S[5] k

⇤ > ⇤

In both cases, all invariants are maintained.Immediately after every comparison M[t, S[t]]� M[t, k]? in the R�� algorithm,

we either increment the column index k or declare a column to be dead; each of theseevents can happen at most once per column. It follows that R�� performs at most 2n

comparisons and thus runs in O(n) time overall.Moreover, when R�� ends, every column whose index is not on the stack is

completely dead. Thus, to compute the leftmost minimum element in every row of M , itsu�ces to examine only the t < m columns with indices in the output array S[1 .. t].

Finally, the main SMAWK algorithm first R��s the input array (if it has fewerrows than columns), then recursively computes the leftmost row minima in every other,and finally fills in the rest of the row minima in O(m+ n) additional time. The argument

��

Shortest paths with negative edges

‣ Using several more advanced tricks, we can compute shortest paths in planar graphs with arbitrary edge weights in O(n log2 n / log log n) time.[Klein Mozes Weimann ’10, Mozes Wulff-Nilsen ’10]

▹ Multiple-source shortest paths [Klein ’05]

▹ Structured r-divisions instead of single separators[Klein Mozes Sommer ’13]

▹ Exploit Monge property in Bellman-Ford [Fakcharoenphol Rao ’06]

▹ Careful parameter balancing (r = n·α(n)/log n)

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Algorithms for Planar Graphs

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