On the edge-length ratio of 2 -trees V aclav Bla zej, Ji r - - PowerPoint PPT Presentation

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On the edge-length ratio of 2 -trees V aclav Bla zej, Ji r - - PowerPoint PPT Presentation

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On the edge-length ratio of 2 -trees V aclav Bla zej, Ji r Fiala, and Giuseppe Liotta blazeva1@fit.cvut.cz March 15, 2020 EuroCG 2020 Research partially supported by MIUR, the Italian Ministry of Education, University and

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On the edge-length ratio of 2-trees

V´ aclav Blaˇ zej, Jiˇ r´ ı Fiala, and Giuseppe Liotta

blazeva1@fit.cvut.cz

March 15, 2020

EuroCG 2020

Research partially supported by MIUR, the Italian Ministry of Education, University and Research, under Grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data. This work has been supported by OP VVV (OP RDE) No.: CZ.02.1.01/0.0/0.0/16 019/0000765. The work of J. Fiala was supported by the grant 19-17314J of the GA ˇ CR.

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Graph drawings

We want straight-line planar drawings

vertices: {a, b, c, d, e, f} edges:

  • {a, b}, {b, c}, . . .
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Graph drawings

We want straight-line planar drawings

straight-line drawing = edges are line segments

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Graph drawings

We want straight-line planar drawings

Planar drawing = crossings are forbidden

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Graph drawings

Position changes matter to us because we care about edge lengths.

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Graph drawings

Position changes matter to us because we care about edge lengths.

shortest edge longest edge

edge-length ratio = |longest edge| |shortest edge|

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Edge-length – Previous work

  • If given edge-lengths, then NP-complete [Eades, Wormald]

2 1 3 1 2 3 3 3 2 1 2 1 NP-complete

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Edge-length – Previous work

  • If given edge-lengths, then NP-complete [Eades, Wormald]
  • if all edge-lengths are equal, then NP-complete [Cabello,

Demaine, Rote]

1 1 NP-complete 1 1 1 1

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Edge-length – Previous work

  • If given edge-lengths, then NP-complete [Eades, Wormald]
  • if all edge-lengths are equal, then NP-complete [Cabello,

Demaine, Rote]

  • if degree-4 trees on integer grid with all edge-lengths equal,

then NP-complete [Bhatt and Cosmadakis]

1 1 NP-complete 1 1 1 1

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Edge-length – Previous work

  • Hoffmann, Van Kreveld, Kusters, Rote proposed relaxation:

edge-length ratio

shortest edge longest edge Figure: edge-length ratio is between the longest and the shortest edge

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Edge-length – Previous work

  • Hoffmann, Van Kreveld, Kusters, Rote proposed relaxation:

edge-length ratio

  • minimizing edge-length ratio is hard for general graphs [Chen,

Jiang, Kanj, Xia, Zhang]

shortest edge longest edge Figure: edge-length ratio is between the longest and the shortest edge

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Edge-length – Previous work

  • Hoffmann, Van Kreveld, Kusters, Rote proposed relaxation:

edge-length ratio

  • minimizing edge-length ratio is hard for general graphs [Chen,

Jiang, Kanj, Xia, Zhang]

  • outerplanar graph have bounded edge-length ratio 2 [Lazard,

Lenhart, Liotta]

Figure: edge-length ratio 2 for outerplanar graphs

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Edge-length – Previous work

  • Hoffmann, Van Kreveld, Kusters, Rote proposed relaxation:

edge-length ratio

  • minimizing edge-length ratio is hard for general graphs [Chen,

Jiang, Kanj, Xia, Zhang]

  • outerplanar graph have bounded edge-length ratio 2 [Lazard,

Lenhart, Liotta]

+

series parallel

Can series-parallel graphs be drawn with constant edge-length ratio?

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Edge-length ratio

Can series-parallel graphs be drawn with constant edge-length ratio? NO – they have unbounded edge-length ratio! Because subclass of series-parallel graphs called 2-trees have unbounded edge-length ratio. 2-trees definition:

Figure: 2-trees are defined constructively

Edge is a 2-tree; adding a vertex connected to two neighboring vertices to a 2-tree is still a 2-tree.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Result 1: 2-trees have unbounded edge-length ratio

Outline of the proof:

  • Start with a big 2-tree,
  • consider its (fixed) drawing,
  • shrinking area chain of triangles,
  • shrinking perimeter chain,
  • small perimeter

= ⇒ short edges = ⇒ small ratio.

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Local edge-length ratio shortest edge longest edge neighborhood

  • f each vertex

consider locally locally

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Local edge-length ratio shortest edge longest edge neighborhood

  • f each vertex

consider locally locally

local edge-length ratio = max |A| |B| where edges A and B are incident.

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Result 2: 2-trees have bounded local edge-length ratio

Outline of the proof:

  • Find graph layers (BFS),
  • decompose it into parts,
  • draw each part separately

and guarantee its children can be drawn.

Figure: graph to be decomposed

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Result 2: 2-trees have bounded local edge-length ratio

Outline of the proof:

  • Find graph layers (BFS),
  • decompose it into parts,
  • draw each part separately

and guarantee its children can be drawn.

Figure: graph decomposition into parts

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Result 2: 2-trees have bounded local edge-length ratio

Outline of the proof:

  • Find graph layers (BFS),
  • decompose it into parts,
  • draw each part separately

and guarantee its children can be drawn.

Figure: drawing a graph part

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Results and open problems

1.R: 2-trees have unbounded edge-length ratio ratio = Ω(log(graph size)).

  • Close the gap between edge-length ratio lower (logarithmic)

and upper bound (linear) of 2-trees.

2.R: Local edge-length ratio of 2-trees is upper bound by 4.

  • Is 4 tight local edge-length ratio for 2-trees?
  • Investigate interplay of edge-length ratio with other

parameters, such as angular resolution, to make the graph drawings readable.

Thanks for watching!