improving the layout of splits networks 06/06/2005 philippe gambette & daniel huson caution!...
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Improving the layout of splits
networkshttp://philippe.gambette.free.fr/Tuebingen/
indexENG.htm06/06/2005
Philippe Gambette & Daniel Huson
Caution! Some parts of this presentation have become Outdated ! due to later results
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Outline
Phylogenetic networks and splits graphs
Drawing planar phylogenetic networks
A strategy to open the boxes of small graphs
Another strategy to open the boxes
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Splits graphs
For a tree: every edge splits the tree into 2 parts :
{x6,x1,x2}{x3,x4,x5}
S =
x1
x2
x3
x4x5
x6
Partition of the set of taxa
A splits graph codes for a set of splits.
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Splits graphs
Compatible splits:
{x6,x1,x2}{x3,x4,x5}
S =
x1x2
x3
x4x5
x6
all the splits are pairwise compatible tree
{x1,x2}{x3,x4,x5,x6}
S’ =
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Splits graphs
Incompatible splits:
{x6,x1}{x2,x3,x4,x5}
S =
x1
x2
x3
x4 x5
x6
a pair of incompatible splits box
{x1,x2}{x3,x4,x5,x6}
S’ =box
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{x2,x3,x4,x5}
Splits graphs
Circular split:
The split is circular
All the splits are circular outer-planar graph
{x6,x1}S =
x1
x2
x3
x4 x5
x6box
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Drawing planar splits graph: equal angle algorithm
=
Splits graph are associated with their taxa circle:the taxa are displayed regularly around the circle.
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« Opening boxes »
Improving the layout of the graphs: opening boxes.
The weight of the edges is fixed
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« Opening boxes » from the taxa circle
Advantages :- we don’t have to consider all the edges, just the splits:O(k) operations instead of O(n+k²).- we have a criteria for the graph to remain planar:keep the circular order of the taxa.
Disadvantage :- there is not a lot of space aroundthe taxa circle.- the criteria of keeping the circularorder is not necessary.
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« Opening boxes » by moving the taxa
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« Opening boxes » by moving the taxaStore a best position. Do the following loop n times:For each taxon, try to move it :
if it’s better : save it, try to move another taxon.if it’s better than the best, store as best.
else : save it with a probability p=20%.
nice results for small graphs
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
Considering only the split itself, changing a0:
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
Considering only the split itself, changing a0:
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
Considering collisions in the graph.
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
Identifying a defender and a striker:
4 extreme nodes
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
Identifying a defender and a striker:
4 extreme nodes
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« Opening boxes » once the graph is drawn
The graph must remain planar:Identify critical angles for the split angle.
E’’ is the new striker!
new angle
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« Opening boxes » once the graph is drawn
Danger area for our criteria:on its border, the striker arrives exactly on the the defender’s line.
Equation of the border of the area:
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« Opening boxes » once the graph is drawn
Danger area for our criteria, depending on the angle of the defender:
Those cases rarely happen.
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« Opening boxes » once the graph is drawn
An example:
Those cases rarely happen.
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Algorithm
Do the following loop n times:
If the total area of the boxes is not improved, break.
For each split:
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Results
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Evolution of the total area of the boxes
Vig Penny
Bad Opt Boxes
Hard
Chainletters
Mammals
Rubber
Primates
Algae
Bees
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Results
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-0,050,000,050,100,150,200,250,300,350,400,450,500,550,600,650,700,75
Improvement of the total area compared with the best area
Vig Penny
Bad Opt Boxes
Hard
Chainletters
Mammals
Rubber
Primates
Algae
Bees
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Results
Before the optimization
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Results
After 1 loop (10 secs on a 2.6GHz Pentium)
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Results
After 2 loops
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Results
After 3 loops
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Results
After 4 loops
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Results
After 5 loops
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Results
After 6 loops
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Results
After 7 loops
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Results
After 8 loops
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Results
After 9 loops
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Results
After 10 loops
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What about the names of the algorithms ???
Algorithm 1 : taxa, circular, before the layout… optimized angle algorithm.
Algorithm 2 : collisions, danger...
Both algorithms : box-opening