Algorithms for Graph Visualization

Transcription

1 Algorithms for Graph Visualization Summer Semester 2016 Lecture #4 Divide-and-Conquer Algorithms: Trees and Series-Parallel Graphs (based on slides from Martin Nöllenburg and Robert Görke, KIT) 1

2 Uses of Divide & Conquer Well suited for inductively/recursively defined Graph Classes Rooted Binary Trees: 2 r root Subtree T (v) v left child left subtree T l (v) right child right subtree Tr (v) 1. draw the left subtree 2. draw the right subtree 3. combine together + draw root Terminology depth(v): distance from the root traversal preorder inorder postorder

3 Overview balanced drawings of binary trees radial drawings of trees Grid Size O(nh) O(nh) compact drawings of trees O(n log n) upward drawings of series parallel graphs exp. 3

4 Algorithm of Reingold and Tilford ( 81) Trivial: If T = {v}, draw v (e.g., as a small disk). Divide: Run Alg. recursively on the left and right subtrees. Conquer: shift the partial drawings to up to 2 units apart place the root r one above and centrally btw. children. or 3! Grid Drawing? 2 4-7

5 Algorithm of Reingold and Tilford ( 81) Implementation in 2 Phases: 1. postorder (bottom-up): contours and x-offsets gather the predecessors 2 2. preorder (top-down): calculate absolute coordinates Contour: linked list of vertices (-coordinates) 4-9

6 Algorithm of Reingold and Tilford ( 81) Phase 1: 1. compute T l (v) und T r (v) 2. trace the right contour of T l (v) and left of T r (v) 3. Find d v = min. horiz. distance between v l und v r 4. x-offset(v l ) = d v /2, x-offset(v r ) = d v /2 5. Build left contour of T v from: v v, v l left contour of T l (v), left contour of any low hanging part of T r (v) 6. Symmetrically for right contour. 5-3 v r

7 Algorithm of Reingold and Tilford ( 81) Phase 1: 1. compute T l (v) und T r (v) 2. trace the right contour of T l (v) and left of T r (v) 3. Find d v = min. horiz. distance between v l und v r 4. x-offset(v l ) = d v /2, x-offset(v r ) = d v /2 5. Build left contour of T v from: v v, v l left contour of T l (v), left contour of any low hanging part of T r (v) 6. Symmetrically for right contour Runtime? v (1 + min{h l(v), h r (v)}) = n + v min{... } n + n v r

8 Algorithm of Reingold und Tilford ( 81) Phase 2: Set y-coordinate y(v) = depth(v) for each vertex v. Set x(w) := 0 for the root w, then in preorder for v V : x(v l ) := x(v) + x-offset(v l ) and x(v r ) := x(v) + x-offset(v r ). Runtime? O(n) 6

9 Summary for Balanced Drawings of Binary Trees Theorem [Reingold & Tilford 81] For a binary tree with n vertices, in O(n) time we can produce a drawing Γ such that: Min. width (but without the grid):by LP! [Supowit & Reingold 83] 7 Min. width and on the grid: NP-hard! Easily generalizes to arbitrary trees! Γ is layered, i.e., y depth, Γ is planar, straightline, and strictly downward, Γ matches the embedding (i.e., right children on the right), all vertices: horiz. & vert. distances 1, and on the grid, the area is O(n 2 ), example? parent always centered above children.

10 Example of width variation Output of the Algorithm: Optimal Drawing: 8

11 2. Radial Drawings of Trees balanced drawings of binary trees radial drawings of trees Grid Size O(nh) O(nh) compact drawings of trees O(n log n) upward drawings of series parallel graphs exp. 9

12 Example: Radial Tree Layouts 10

13 An Algorithm for Radial Layout?

14 Restricting to Smaller Sectors α v α min τ ρ i ρ i+1 v 12 α max cos τ = ρ i ρ i+1 { α min = α v arccos ρ i ρ i+1 α max = α v + arccos ρ i ρ i+1

15 Pseudocode for radial tree layout RadialTreeLayout(tree T, root r T, radii ρ 1 < < ρ k ) begin postorder(r) preorder(r, 0, 0, 2π) return (d v, α v ) v V (T ) {vertex pos./ polar coord.} postorder(vertex v) n v 1 foreach child w von v do postorder(w) n v n v + n w 13 size of the subtree T (v) preorder(vertex v, t, α min, α max ) d v ρ t { output } α v (α min + α max )/2 if t > 0 then α min max{α min, α v arccos ρ t ρ t+1 } α max min{α max, α v +arccos ρ t ρ t+1 } left α min foreach child w von v do right left + n w n v 1 (α max α min ) preorder(w, t + 1, left, right) left right Runtime? O(n). Correctness?

16 Overview balanced drawings of binary trees radial drawings of trees Grid Size O(nh) O(nh) compact drawings of trees O(n log n) upward drawings of series parallel graphs exp. 14

17 hv-drawings Definition. An hv-drawing of a binary tree is a straight line drawing, so that for each vertex v: each child of v is either directly right or directly below v. the smallest axis-parallel rectangle enclosing the subtrees of the children of v are disjoint. 15 horizontal combination vertical combination

18 Algorithm RightHeavyHVTreeDraw Recursively construct drawings of the left and right subtrees from the root. Place the larger subtree on the right using the horizontal combination, and the smaller on the left Size of a subtree := number of vertices at least 2 at least 2 at least 2 16 Obs. The drawing has width n, height log 2 n.

19 Overview balanced drawings of binary trees radial drawings of trees Grid Size O(nh) O(nh) compact drawings of trees O(n log n) upward drawings of series parallel graphs exp. 17

20 Series Parallel Graphs simple series parallel graph t Induction: combining two series parallel graphs G 1, G 2... t 1 s t 2 G 1 G 2 s 1 s series or parallel. 18 s 1 t 1 = s 2 t 2 G 1 G 2 G 1 s 1 = s 2 G 2 t 1 = t 2

21 Decomposition Tree for SP-graphs P S P S S Q P S Q Q Q Q S S Q 19 Generalization: SPQR-Tree Q Q Q Q

22 SP-Graphs: applications Flow Charts PERT-Diagrams (Program Evaluation and Review Technique) Provides: 20 Linear time algorithms for NP-complete problems (e.g., Maximum Independent Set)

23 Grid Size Theorem [Bertolazzi et al. 92] There is a family (G n ) n N of embedded SP-graphs where G n has 2n vertices and every upward planar drawing of G n requires Ω(4 n ) area. t n+1 Proof: t n+1 2 t 0 t n tn G n G n s n 1 21 G 0 s 0 s n s n+1 G n+1 s n+1 1 Π sn a(g n+1 ) a(π) + a( 1 ) + a( 2 ) 2 a(π) 4 a(g n )