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

• 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

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

• An ordering of vertices v1, …, vn is canonical if for each index k, 3 ≤ k ≤ n:

1. Gk is 2-connected and internally triangulated

2. (v1, v2) is on Co(Gk)

3. 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 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 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 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 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

• General idea:– Insert vertices in canonical order in drawing

– Invariants when inserting vk:1. x(v1) = 0, y(v1) = 0, x(v2) = 2k–6, y(v2) = 0

2. x(w1) < … < x(wt) where Co(Gk–1) = w1, …, wt with w1 = v1, wt = v2

3. Each edge on Co(Gk–1) except (v1, v2) is drawn by a straight line having slope +1 or –1

The shift algorithm

• General idea:– Some

vertices need to be shifted when inserting vk

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 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 algorithm

• Updatingthe tree

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

• The shift method uses– The canonical ordering– A 2n–4 x n–2 grid, obviously– Linear time

• Next time: the realizer method

• Questions?

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.