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Straight line drawings of planar graphs – part I. Roeland Luitwieler. Outline. Introduction The shift method The canonical ordering The shift algorithm Next time: the realizer method. Introduction. This presentation is about: Straight line grid drawings of planar graphs Minimized area

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Presentation Transcript
outline
Outline
  • Introduction
  • The shift method
    • The canonical ordering
    • The shift algorithm
  • Next time: the realizer method
introduction
Introduction
  • This presentation is about:
    • Straight line grid drawings of planar graphs
    • Minimized area
  • De Fraysseix, Pach & Pollack, 1988:
    • The shift method: 2n–4 x n–2 grid
  • Schnyder, 1990:
    • The realizer method: n–2 x n–2 grid
  • Both are quite different, but can be implemented as linear time algorithms
introduction1
Introduction
  • Assumptions (can be taken care of in linear time)
    • n ≥ 3 (trivial otherwise)
    • Graph is maximal planar (=triangulated)
    • Graph has a topological embedding
  • Terminology
    • Given an ordering on the vertices of graph G:
      • Gk is the induced subgraph of vertices v1, …, vk
    • Co(G) is the outer cycle of a 2-connected graph G
    • Given a cycle C with u and v on it, but (u,v) not on it:
      • The edge (u,v) is a chord of C
the canonical ordering
The canonical ordering
  • An ordering of vertices v1, …, vn is canonical if for each index k, 3 ≤ k ≤ n:
    • Gk is 2-connected and internally triangulated
    • (v1, v2) is on Co(Gk)
    • If k+1≤n, vk+1 is in the exterior face of Gk and its neighbours in Gk appear on Co(Gk) consecutively
the canonical ordering1
The canonical ordering
  • Lemma: every triangulated plane graph has a canonical ordering
  • Trivial for n=3, so assume n≥4
  • Proof by reverse induction
  • Base case (k=n): conditions hold since Gn = G
    • Co(Gn) consists of v1, v2 and vn
  • We now assume that vn, vn-1, ..., vk+1 for k+1≥4 have been appropriately chosen and that the conditions hold for k
  • To prove: the conditions hold for k–1
the canonical ordering2
The canonical ordering
  • If we can find a vertex (= vk) on Co(Gk) which is not an end of a chord, the conditions hold
    • Follows from Gk–1 = Gk – vk
  • Such a vertex always exists, because a chord always skips at least one vertex on the cycle
the canonical ordering3
The canonical ordering
  • A canonical ordering of a triangulated plane graph can be found in linear time
    • Use labels for vertices
      • Label -1 means not yet visited
      • Label 0 means visited once
      • Label i means visited more than once and there are i intervals of visited neighbours in the circular order around the vertex
    • Algorithm:
      • Let v1 and v2 be two consecutive vertices on Co(G) in counter clockwise order
      • Label all other vertices -1
      • Process v1
      • Process v2
      • for k = 3 to n do
        • Chose a vertex v with label 1
        • vk = v
        • Process vk
the canonical ordering4
The canonical ordering
  • (Sub-algorithm) Process v:
    • For all neighbours w of v not yet processed do
      • If label -1 then label(w) = 0
      • If label 0 then (w has one processed neighbour u)
        • If w next to u in circular order of neighbours of v then label(w) = 1
        • Otherwise label(w) = 2
      • Otherwise (label i)
        • If vertices next to w in circular order of neighbours of v both have been processed then label(w) = i–1
        • If neither has been processed yet then label(w) = i+1
  • Correctness
    • All processed vertices form Gk and the chosen vertex is vk+1
      • Verify the conditions hold now
    • Successful termination guaranteed by previous proof
the shift algorithm
The shift algorithm
  • General idea:
    • Insert vertices in canonical order in drawing
    • Invariants when inserting vk:
      • x(v1) = 0, y(v1) = 0, x(v2) = 2k–6, y(v2) = 0
      • x(w1) < … < x(wt) where Co(Gk–1) = w1, …, wt with w1 = v1, wt = v2
      • Each edge on Co(Gk–1) except (v1, v2) is drawn by a straight line having slope +1 or –1
the shift algorithm1
The shift algorithm
  • General idea:
    • Some vertices need to be shifted when inserting vk
the shift algorithm2
The shift algorithm
  • To obtain a linear time algorithm, x-offsets are used in stead of x-coordinates
  • We use a binary tree to represent shift dependencies
    • Each node shifts whenever its parent shifts
    • Store x-offset and y-coordinate in each node
  • Using the tree, the x-offsets can be accumulated in linear time
the shift algorithm3
The shift algorithm
  • Compute x-offsets and y-coordinates:
    • Compute values and

create the tree for G3

    • For k = 4 to n do
      • Let w1, …, wt = Co(Gk–1) with w1 = v1, wt = v2
      • Let wp, wp+1, …, wq be the neighbours of vk on Co(Gk–1)
      • Increase Δx(wp+1) and Δx(wq) by one
      • Compute Δx(vk) and y(vk)
      • Update the tree accordingly
the shift algorithm4
The shift algorithm
  • Updating

the tree

the shift algorithm5
The shift algorithm
  • Computing Δx(vk) and y(vk)
    • Δx(wp, wq) = Δx(wp+1) + ... + Δx(wq)
    • Δx(vk) = ½ {Δx(wp, wq) + y(wq) – y(wp)}
    • y(vk) = ½ {Δx(wp, wq) + y(wq) + y(wp)}
    • Other necessary updates:
      • Δx(wq) = Δx(wp, wq) – Δx(vk)
      • If p+1 <> q then Δx(wp+1) = Δx(wp+1) – Δx(vk)
conclusions
Conclusions
  • The shift method uses
    • The canonical ordering
    • A 2n–4 x n–2 grid, obviously
    • Linear time
  • Next time: the realizer method
  • Questions?
references
References
  • H. de Fraysseix, J. Pach and R. Pollack, How to draw a planar graph on a grid, Combinatorica 10 (1), 1990, pp. 41–51.
  • D. Harel and M. Sardas, An Algorithm for Straight-Line Drawing of Planar Graphs, Algorithmica 20, 1998, pp. 119–135.
  • T. Nishizeki and Md. S. Rahman, Planar Graph Drawing, World Scientific, Singapore, 2004, pp. 45–88.
  • W. Schnyder, Embedding planar graphs on the grid, in: Proceedings of the First ACM-SIAM Symposium on Discrete Algorithms, 1990, pp. 138–148.