Download presentation

Presentation is loading. Please wait.

Published byKrystal Gord Modified over 3 years ago

1
Chapter 9 (Layered drawings of digraphs) By: Waldo & Ludo uv

2
Chapter 9 (Graphically) 1.The hierarchical approach: Cyclic aCyclic Layered drawing of G DiGraph G Cycle Removal (9.4) Layer assignment Crossing reduction X coordinate assignment (9.1) (9.2) (9.3) Waldo Ludo Waldo Handled by :

3
Important requirements of layering: 1.The layered digraph should be compact. 2.The layering should be proper. 3.The number of dummy vertices should be small

4
The layering algorithm: 1.No labels are set 2.Assign labels (integer), such that s < v < t 3.Assign vertices to a layer, such that layer of t <= v <= s

5
selecting vertices to label 1.When choosing a vertex v all preceding vertices u (u,v) should be labeled and minimized. Minimization is accomplished by looking at the most significant labels. Example {6} < {3,7}; {1,2,9} < {2,3,9}; etc. For more on this definition see page 274 of the book

6
Phase one (assign labels) 3 1 4 2 13 12 10 85 11 9 76

7
selecting vertices to add to a layer 1.When choosing a vertex u all vertices v (u,v) should be placed in a layer lower than u.

8
Phase two (assign layers) 13 12 10 11 9 8 5 3 1 7 4 2 6 13 12 10 85 3 1 11 9 7 4 2 6

9
Input: Reduced digraph G=(V,E) and a positive integer W Output: Layering of G of width at most W Algorithm (Coffman-Graham layering) Initially, all vertices are unlabeled (trivial, as we’ve seen) For (i = 1 to |V|) perform a. Choose an unlabeled vertex v, such that {lbl(u) : (u,v) element of E } is minimized b. Lbl(v) = i K=1; L1=null; U=null. While U != V loop a. Choose u element of (V-U), such that every vertex in {v : (u,v) element of E} is in U, and lbl(u) is maximized b. If not |Lk| < W and for every edge (u,w), w is element of preceding levels then k++; add u to Lk c. Add u to U.

10
Phase two (adjusted) 13 12 10 11 9 5 8 3 1 7 4 2 6 13 12 10 85 3 1 11 9 7 4 2 6

11
Algorithm (Coffman-Graham layering) adjusted 1)Exercise.. (only for phase two – previous slide) A. Describe the adjusted algorithm B. Draw the iterational steps of the adjusted algorithm one by one.

12
Crossing Reduction Input: proper layered digraph Layer-by-Layer Sweep Two-Layer Crossing Problem

13
Each vertex in the two layers gets a unique x-coordinate, purely for ordering purposes: Two-Layer Crossing Problem

15
uv Crossing Numbers

16
uv

17
pur q pqur p0211 q5063 u6906 r2320

18
pur q pqur p0211 q5063 u6906 r2320

19
pur q pqur p0211 q5063 u6906 r2320

20
pur q pqur p0211 q5063 u6906 r2320

21
pur q pqur p0211 q5063 u6906 r2320

22
Algorithms for minimizing Adjacent Exchange Similar to Bubble-sort Split Similar to Quick-sort Barycenter Method Median Method Quadratic time Linear time

23
Adjacent-Exchange uv

24
uv

25
uv

26
uv

27
Split ap

28
ap

29
pb

30
pb

31
p

32
apb

33
u 431625 Barycenter Method 7 If same barycenter: seperate arbitrarily by small amount

34
u 431625 Median Method 7 X-coordinate of u is the median of its neighbours If no neighbours, then med(u) = 0 Special case, if med(u) = med(v)...odd degree left, even right Median ???

35
431625 Not always optimal 7 Barycenter 4316257 Median 89 10

36
Bends occur at dummy vertices Objective is to: – Reduce angles of bends (minimal width) – Keep ordering of crossing reduction step Horizontal Coordinate Assignment

37
4 3 1 6 2 5 0123456X 2 3

38
Cycle Removal 1 23 654 789 1 2 3 4 5 6 7 8 9 Vertex sequence for G : Dashed edges are the leftward edges Leftward edges form feedback set R Reversing R makes G acyclic

39
Cycle Removal Problem: – Minimizing leftward edges / feedback set R How? – Greedy Cycle Removal Algorithm

40
Cycle Removal 1 23 654 789 Iterate: prepend sinks to S r and remove them from G

41
Cycle Removal 1 23 654 78 Iterate: prepend sinks to S r and remove them from G

42
Cycle Removal 1 23 654 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G

43
Cycle Removal 23 654 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G

44
Cycle Removal 3 654 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G

45
Cycle Removal 654 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G

46
Cycle Removal 54 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G

47
Cycle Removal 54 78 Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G Choose vertex u such the outdegree(u) – indegree(u) is max, append to S l and remove from G

48
Cycle Removal Iterate: prepend sinks to S r and remove them from G Iterate: append sources to S l and remove them from G Choose vertex u such the outdegree(u) – indegree(u) is max, append to S l and remove from G Concatenate S l and S r to obtain S

49
Cycle Removal 1 2 3 6 4 5 7 8 9 1 23 654 789

50
1 2 3 6 4 5 7 8 9 1 23 654 789

51
Chapter 9 (in a nutshell) The hierarchical approach: Layer assignment Crossing reduction Horizontal coordinate assignment

52
Exercise 1)Prove Theorem 9.1 (Exercise 3 in book)

Similar presentations

OK

BOX PLOTS (BOX AND WHISKERS). Boxplot A graph of a set of data obtained by drawing a horizontal line from the minimum to maximum values with quartiles.

BOX PLOTS (BOX AND WHISKERS). Boxplot A graph of a set of data obtained by drawing a horizontal line from the minimum to maximum values with quartiles.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on coordinate geometry for class 9th notes Ppt on tata group of companies Ppt on indian festivals in hindi Ppt on tinder Ppt on articles of association for a church Ppt on image sensor Ppt on low level language in computer Ppt on holographic technology videos Ppt on amelia earhart biography Ppt on money and credit class 10 economics