A Parameterized Algorithm for Upward Planarity Testing of - PowerPoint PPT Presentation

A Parameterized Algorithm for Upward Planarity Testing of Biconnected Graphs Master’s thesis presentation Hubert Chan University of Waterloo May 6, 2003 – p. 1/48

Graph drawing • goal: visualization of graph structures • vertices represented by points, edges by curves • want drawings to satisfy certain criteria Master’s thesis presentation – May 6, 2003 – p. 2/48

Straight line drawing Master’s thesis presentation – May 6, 2003 – p. 3/48

Straight line drawing Master’s thesis presentation – May 6, 2003 – p. 3/48

Planarity Master’s thesis presentation – May 6, 2003 – p. 4/48

Planarity Master’s thesis presentation – May 6, 2003 – p. 4/48

Planarity Master’s thesis presentation – May 6, 2003 – p. 4/48

Upward planarity Master’s thesis presentation – May 6, 2003 – p. 5/48

Upward planarity Master’s thesis presentation – May 6, 2003 – p. 5/48

Our goal • Find an efficient solution to upward planarity testing. • Testing for planarity is linear time (Hopcroft and Tarjan 1974) • Testing for upward (crossings allowed) is linear time (e.g. Cormen et al. 2001, Brassard and Bratley 1996) Master’s thesis presentation – May 6, 2003 – p. 6/48

Our goal • Find an efficient solution to upward planarity testing. • Testing for planarity is linear time (Hopcroft and Tarjan 1974) • Testing for upward (crossings allowed) is linear time (e.g. Cormen et al. 2001, Brassard and Bratley 1996) • Unfortunately, upward planarity testing is NP-complete (Garg and Tamassia 2001). Master’s thesis presentation – May 6, 2003 – p. 6/48

Related work Class Complexity Reference st -graph O ( n ) Di Battista and Tamassia 1988 bipartite O ( n ) Di Battista, Liu, and Rival 1990 O ( n + r 2 ) triconnected Bertolazzi et al. 1994 O ( n 2 ) outerplanar Papakostas 1995 single source Bertolazzi et al. 1998 O ( n ) Master’s thesis presentation – May 6, 2003 – p. 7/48

Parameterized complexity • developed by Downey and Fellows • limit combinatorial explosion to some aspect of the problem Master’s thesis presentation – May 6, 2003 – p. 8/48

Parameterized complexity • developed by Downey and Fellows • limit combinatorial explosion to some aspect of the problem • e.g. VERTEX COVER Master’s thesis presentation – May 6, 2003 – p. 8/48

Parameterized complexity • developed by Downey and Fellows • limit combinatorial explosion to some aspect of the problem • e.g. VERTEX COVER • NP-complete Master’s thesis presentation – May 6, 2003 – p. 8/48

Parameterized complexity • developed by Downey and Fellows • limit combinatorial explosion to some aspect of the problem • e.g. VERTEX COVER • NP-complete � k k 2 � • to find a vertex cover of size k : O kn + 4 3 (Balasubramanian et al. 1998) Master’s thesis presentation – May 6, 2003 – p. 8/48

Related work in parameterized complexity • Zhou 2001 — treewidth/pathwidth and graph drawing • Dujmovi´ c et al. 2001 — layered drawings Master’s thesis presentation – May 6, 2003 – p. 9/48

Modified goal Develop a parameterized algorithm for upward planarity testing. our parameter: the number of triconnected components. Master’s thesis presentation – May 6, 2003 – p. 10/48

k -connectivity Definition. A graph is k -connected if there are at least k vertex-disjoint paths between any two vertices. Master’s thesis presentation – May 6, 2003 – p. 11/48

k -connectivity Definition. A graph is k -connected if there are at least k vertex-disjoint paths between any two vertices. Master’s thesis presentation – May 6, 2003 – p. 11/48

k -connectivity Definition. A graph is k -connected if there are at least k vertex-disjoint paths between any two vertices. 2-connected = biconnected 3-connected = triconnected Master’s thesis presentation – May 6, 2003 – p. 11/48

k -connected components Definition. A k -connected component is a maximal k -connected subgraph Master’s thesis presentation – May 6, 2003 – p. 12/48

Preliminary definitions — Embeddings • T wo different planar drawings may have similar structure • An embedding is a description of this structure Definition. The (planar) embedding associated with a drawing is the collection of clockwise orderings of the edges around each vertex. 1 5 5 1 2 4 4 2 3 3 Master’s thesis presentation – May 6, 2003 – p. 13/48

Embeddings Master’s thesis presentation – May 6, 2003 – p. 14/48

Equivalence of drawings Definition. T wo drawings are equivalent if they have the same embedding, and are strongly equivalent if the have the same embedding and the same outer face. Master’s thesis presentation – May 6, 2003 – p. 15/48

Outline • Transformations of drawings • Edge contraction • Joining subgraphs • Parameterized algorithm for biconnected graphs • Conclusion Master’s thesis presentation – May 6, 2003 – p. 16/48

Transformations of drawings • If a graph is upward planar, we can draw it so that a specified edge is drawn vertically • We can scale and translate drawings, preserving upward planarity Master’s thesis presentation – May 6, 2003 – p. 17/48

Edge contraction Master’s thesis presentation – May 6, 2003 – p. 18/48

Edge contraction Question: after contracting an edge ǫ , is the resulting embedding still upward planar? Master’s thesis presentation – May 6, 2003 – p. 19/48

Edge contraction • look at the edge ordering neighbours ... a α t β b ǫ c γ s δ d ... • consider all possibilities for the orientations of the neighbours Master’s thesis presentation – May 6, 2003 – p. 20/48

Is the contracted graph upward planar? ? ? • yes: (Hutton and Lubiw 1996) ? ? * • no: (corollary of Tamassia and Tollis 1986) • if and only if G ← ǫ is upward planar: remaining cases Master’s thesis presentation – May 6, 2003 – p. 21/48

Edge contraction Use characterization by Hutton and Lubiw: Theorem. Given φ , a planar drawing of a directed acyclic graph G , there is an upward planar drawing strongly equivalent to φ if and only if every vertex v is a sink on the outer face of φ v . v v Master’s thesis presentation – May 6, 2003 – p. 22/48

Proof outline In the contracted graph: • v is a sink ( ← only show this) • G/ǫ is acyclic • v is on the outer face only need to consider vertices that have s as a predecessor but not t t s Master’s thesis presentation – May 6, 2003 – p. 23/48

Proof outline In the contracted graph: • v is a sink ( ← only show this) • G/ǫ is acyclic • v is on the outer face only need to consider vertices that have s as a predecessor but not t t s Master’s thesis presentation – May 6, 2003 – p. 23/48

Proof outline In the contracted graph: • v is a sink ( ← only show this) • G/ǫ is acyclic • v is on the outer face only need to consider vertices that have s as a predecessor but not t t s Master’s thesis presentation – May 6, 2003 – p. 23/48

Proof outline In the contracted graph: • v is a sink ( ← only show this) • G/ǫ is acyclic • v is on the outer face only need to consider vertices that have s as a predecessor but not t t s Master’s thesis presentation – May 6, 2003 – p. 23/48

v is a sink • we can draw ǫ vertically ... α A a t β ǫ b C B c γ s d δ ... D Master’s thesis presentation – May 6, 2003 – p. 24/48

v is a sink • we can draw ǫ vertically • where can vertices be in relation to ǫ ? ... α A a t β ǫ b C B c γ s d δ ... D Master’s thesis presentation – May 6, 2003 – p. 24/48

Locations of vertices • predecessors of t must be in B or D • successors of s must be in A or C ... A α a t β ǫ b C B c γ s d δ ... D Master’s thesis presentation – May 6, 2003 – p. 25/48

v is a sink • if not: there is an outgoing edge ( v, v 1 ) • v was a sink in the original graph • v 1 must be a predecessor of t • v must be a predecessor of t Master’s thesis presentation – May 6, 2003 – p. 26/48

Where can v be drawn? • v is a predecessor of t — must be in B or D • v is a successor of s — must be in A or C ... A α a t β ǫ b C B c γ s d δ ... D Master’s thesis presentation – May 6, 2003 – p. 27/48

Joining subgraphs Contracting edges allows us to join two upward planar subgraphs • draw G 1 and G 2 G 1 G 2 v 2 v 1 Master’s thesis presentation – May 6, 2003 – p. 28/48

Joining subgraphs Contracting edges allows us to join two upward planar subgraphs • draw G 1 and G 2 • draw a curve connecting v 1 and v 2 G 1 G 2 v 2 v 1 Master’s thesis presentation – May 6, 2003 – p. 28/48

Joining subgraphs Contracting edges allows us to join two upward planar subgraphs • draw G 1 and G 2 • draw a curve connecting v 1 and v 2 • contract the edge ( v 1 , v 2 ) G 1 G 2 v Master’s thesis presentation – May 6, 2003 – p. 28/48

Joining subgraphs Goal: characterize when we can join two upward planar graphs to produce a new upward planar graph Master’s thesis presentation – May 6, 2003 – p. 29/48