Graphs & Algorithms: Advanced Topics Planar Separators

Transcription

1 Graphs & Algorithms: Advanced Topics Planar Separators Johannes Lengler, Uli Wagner ETH Zürich

2 Separators Definition Let G = (V, E) be a graph on n vertices, f : N 0 R a function, α (0, 1) a parameter. S V is a (f (n), α)-separator if S f (n) and all connected components of G S are of size αn. Example Every tree contains a (1, 1 2 )-separator, which can be found in O(n) time. Applications For example, divide-and-conquer algorithms (more details later).

3 Planar Separators Theorem (Lipton Tarjan 1979) Every planar graph G on n vertices has a ( 8n, 2 3 )-separator, which can be found in time O(n). Remarks f (n) = Θ( n) is best possible if we want (f (n), α)-separators for all planar graphs, for a constant α < 1. More generally, one can find (Θ( n), 2 3 )-separators in graphs with a fixed forbidden minor H (Alon Seymour Thomas 1990; algorithm runs in time O(n 3/2 ); constants depending on H). Sparsity of the graph (i.e., a linear number of edges) is not enough to guarantee the existence of small separators. E.g., for fixed r 3, with high probability, a random r-regular graph G is vertex expanding, i.e., every subset A V, A n/2 has at least ε A neighbors outside of A, for some constant ε depending on r. Thus, such a G has no (f (n), α)-separator with α < 1 and f (n) = o(n).

4 Divide-and-Conquer method 1. BASE CASE: Solve the problem on constant-size sets by brute force. 2. Otherwise DIVIDE: Find a very small vertex set C fast such that G C falls into two small pieces A and B with no edges in between. 3. CONQUER: Explore all solutions restricted to C (brute force) and solve the corresponding subproblems on A and B recursively. Put together the partial solutions. Here: Small means < βn, where β < 1 is a constant. very small means o(n). O(very small) Outcome: Algorithm with subexponential runtime 2

5 Planar Independent Sets MAXIMUM (PLANAR) INDEPENDENT SET Input: (Planar) graph G Output: Independent set X V with maximum cardinality, that is, X = α(g). Theorem The MAXIMUM PLANAR INDEPENDENT SET problem is NP-hard. Theorem The MAXIMUM PLANAR INDEPENDENT SET problem can be solved in time 2 O( n). Remark We don t know whether it is possible to solve the MAXIMUM INDEPENDENT SET problem in time 2 o(n). In fact, we don t expect that to happen.

6 Algorithm PlanarIndependentSet Input: Plane graph G Output: Maximum independent set I IF V (G) 1 THEN I := V (G) ELSE I := Find a ( 8n, 2 3 )-separator C for G. Let A B = V \ C a partition of V such that A, B 2 3n, E(A, B) =. FOR ALL independent set S C DO I A := PlanarIndSet(G[A \ N(S)]) I B := PlanarIndSet(G[B \ N(S)]) IF S + I A + I B > I THEN I := S I A I B output I

7 Jordan curves Recap (1) A (parametrized) curve is a continuous map γ : [0, 1] R 2. The points γ(0) and γ(1) are called the endpoints of the curve. We will often identify γ with its image γ([0, 1]). A curve is closed if γ(0) = γ(1). A curve is simple if it has no repeated points except possibly first = last. Examples. Line segments, polygonal arcs, circular arcs, Bezier-curves, etc... Warning! Non-simple curves can be weird (space-filling curves)

8 Drawings & Planar graphs Recap (2) A drawing of a multigraph G is a function D defined on V (G) E(G) that assigns to each vertex v a point D(v) R 2, and to each edge e = uv a simple curve D(e) with endpoints D(u) and D(v), such that the images of vertices are distinct, and if D(u) D(e) then D(u) is an endpoint of D(e). drawing plane embedding

9 Jordan Curve Theorem Recap (3) An open set in the plane is a set U R 2 such that for every p U, all points within some small distance (which depends on p) belong to U. A connected region is an open set U that, for any pair u, v U, contains a curve with endpoints u, v. Theorem (Jordan Curve Theorem) A simple closed curve γ partitions R 2 into two connected regions, denoted Int(C) and Ext(C), each having γ as boundary. Not true on the torus!

10 Faces Recap (4) The faces of a plane multigraph are the maximal connected regions of the plane that contain no points used in the embedding. A finite plane multigraph G has one unbounded face (also called outer face).

11 Some Useful Facts About Planar Graphs Recap (5) A (simple) planar graph on n 3 vertices has at most 3n 6 edges and at most 2n 4 faces. Data structure: to store a plane graph, store for each edge four pointers to its clockwise and counterclockwise neighbors at each endpoints; with each vertex, store a pointer to some incident edge. A triangulation is a simple plane graph in which every face is a triangle. Any simple plane graph G can be extended, in time O(n), to a triangulation by adding additional edges to the drawing. Fact. Planarity is preserved under edge contractions. Corollary. Contracting a connected subgraph of a plane graph to a single point preserves planarity. Theorem (Hopcroft Tarjan 1974) Planarity of a graph can be tested, and a drawing be found, in linear time.

12 Spanning Trees of Small Diameter Lemma Let G be a connected planar graph, and let T be a spanning tree of G with diameter s. Then G has an (s + 1, 2 3 )-separator. Given a plane drawing of G, we can find the separator in time O(n). Proof (of the lemma and of the theorem) in lecture...

13 The Algorithm Input: Plane triangulation G, spanning tree T G Output: Edge e E \ E(T ); C(e) is a separator with n ext (C(e)), n int (C(e)) 2 3 n. e = xy E \ E(T ) arbitrary, with direction. Run Clockwise-DFS(x, y, e) to determine n int (C(e)) and n Ext (C(e)). IF n ext (C(e)) > 2 3 n THEN Update y := x, x := y (e changes direction) IF n int (C(e)) > 2 3 n WHILE n int (C(e)) > 2 3 n DO z C(e) Int(C(e)) such that {z, x, y} is a face. Alternately run Clockwise-DFS(x, z, xz) and Anticlockwise-DFS(y, z, yz). IF Clockwise-DFS terminates first THEN n int (C(zx)) n int (C(zy)); Update e := zy. ELSE Update e := zx. Output e.

14 Algorithm Clockwise-DFS: Counting n Int and n Ext Clockwise-DFS(x,y,e) Input: plane triangulation G, spanning tree T G, cyclic lists L G v and L T v of the G-edges and T -edges incident to v V, target vertex x, start vertex y, edge e incident to y v current := y, e current := e. WHILE v current x DO e current := T -edge incident to v current coming first after e current in the clockwise direction. v current := the other endpoint of e current, Remarks The walk produced by Clockwise-DFS tends to bend in the clockwise direction. For Anticlockwise-DFS: Replace clockwise with anticlockwise. To count the vertices visited along the way, maintain a counter, mark each visited vertex, and increase the counter whenever an unmarked vertex is visited.