A Survey paper on Planar drawings in the minimum number of layers

This survey paper was prepared as it was required for my undergraduate level computational geometry course

Transcript

1. A Survey paper on Planar Drawings in the minimum number of Layers Mizan Rahman Sadi Anwar Chowdhury Muhammad Rashed Alam
2. Definition • A layered drawing of a graph G is a planar straight-line drawing of G,where each vertex of G is placed on a set of horizontal lines called layers. • Such a drawing of G is also called a k-layer drawing of G when the number of layers in the drawing is k.
3. common variants of layered drawing proper upright short 4-layer planar not proper, not upright, short 3-layer planar not proper, upright, not short, 4-layer planar.
4. Planar graph with minimum number of layers an upward embeddable graph
5. Previous Results in Layered Drawings • Cornelsen et al. [CSW04] has given necessary and sufficient conditions for recognizing graphs which are a 2-layer planar and 3-layer planar. • Heath and Rosenberg [HR92] have shown that recognizing graphs with proper and planar drawing on layers is NP-complete. • Sugiyama et al. developed a simple method for drawing layered networks in 1979. • Dujmovic et al. [DFH+01] show that it can be solved in polynomial time when number of layers k is bounded by a constant
6. Previous Results in Layered Drawings(contd.) • Kaufmann [FK97] have recognized graphs that have proper and planar drawings on three layers • Vida, Michael & Matthew employ bounded pathwidth techniques On the Parameterized Complexity of Layered Graph Drawing?
7. Previous Results in Layered Drawings(contd.) • Sohaee et al prove that the minimum number of layers for layered upward embedding of a st-graph is one more than the length of longest path from source node s to sink node t. • genetic algorithms (GAs) with the problem of drawing of level planar graph or hierarchical planar graph, and explore the potential use of GAs to solve this particular problem.
8. Previous Results in Layered Drawings(contd.) • Jawaherul Alam et al gave a linear-time algorithm to obtain an upward drawing of a given tree T on the minimum number of layers. • They also gave a linear-time algorithm to check whether a given biconnected graph G admits an upright drawing on three layers and in case it admits one, our algorithm obtains such an upright drawing of G on three layers.
9. Algorithms • Sugiyama framework for layered drawings, which in itself is the most followed approach in this field. • algorithms for 2-layer drawings of planar graphs • algorithm for 3-layer proper drawing of biconnected graph
10. Sugiyama framework • Four phases. 1. Remove all cycles from the input graph 2. graph is layered with objective to minimize the number of layers or the number of vertices within a layer [DETT99]. 3. the vertices within each layer are permuted to reduce crossings among edges, typically using a layer-by-layer sweep algorithm [STT81] 4. assigns coordinates to the vertices [BJL01].
11. Sugiyama Method
12. Sugiyama framework • disadvantage : – layer assignments are not changed during the crossing minimization process in the second phase. • The worst-case running-time O(|V ||E| log |E|) requiring O(|V ||E|) memory. • Mutzel96 showed an way able to keep the number of dummy vertices and edges linear in the size of the graph without increasing the number of crossings. • They reduce the worst-case time complexity to O((|V | + |E|) log |E|) requiring O(|V | + |E|) space.
13. Two-layer Drawings • Trees: – A tree T has a planar two layer drawing if and only if it contains a spine i.e., a path S such that T- S is collection of paths. • Biconnected Graphs – A biconnected graph G has a planar two layer drawing if and only if G is outer planar • Bipartite Graphs – A connected bipartite graph G is two layer (proper) drawable if and only if G is a caterpillar.
14. Three-layer Drawings • Kaufmann [FK97] et al have recognized graphs that have proper planar 3-layer drawings • Findings: – If a graph G admits a 3-layer proper drawing , then G isbipartite. – If a graph G admits a 3-layer drawing, then there is no separator vertex v in G such that it has at least three neighbors with non-caterpillar components.
15. Problems yet to be solved: • The problem of determining whether a graph is k-layer planar for any given value of k is yet to be solved and till now, the results in [CSW04] has been the best known result for unrestricted layered drawings of general planar graphs. • To develop efficient algorithms to obtain a planar straight-line drawings of a given tree on the minimum number of layers.
16. Problems yet to be solved: • To provide necessary and sufficient conditions for upright drawings of biconnected graphs on a fixed number of layers greater than three. • To obtain necessary and sufficient conditions for upright drawings of general graphs. • To address the layered drawability problems of general graphs for both the unrestricted case and various restricted cases including upright drawings.
17. Reference • [DFH+01] V. Dujmovic, M. R. Fellows et al On the parameterized complexity of layered graph drawing. • [CSW04] S. Cornelsen, T. Schank, D. Wagner. Drawing Graphs on Two and Three Lines. • [DETT99] R.Tamassia and I. G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs • [FK97] M. Kaufmanne et al , Nice drawings for planar bipartite graphs • Upright drawings of graphs on three Layers M. Jawaherul Alam, Md. Mashfiqui Rabbi, Md. Saidur Rahman And Md. Rezaul Karim
18. Thank you 11/23/2011