Visualising the Tutte Polynomial Computation

1
Visualising the Tutte Polynomial Computation

• Bennett Thompson, David J. Pearce
• Victoria University of Wellington,
• New Zealand
• Gary Haggard
• Bucknell, USA

2
The Tutte Polynomial

• Delete/Contract Operations
• Tutte Definition
• T(G) 1, if G ?
• T(G) xT(G/e), if e is a bridge
• T(G) yT(G-e), if e is a loop
• T(G) T(G-e) T(G/e), otherwise

G
Ge
G/e
3
Tutte Computation Tree
4
Great, but why do we care?

• Many applications of Tutte polynomial
• Physics, Biology and probably lots more
• Knots
• Tangled cords which cant be unravelled
• Problem how do we know when two knots are same?
• Tutte polynomial can be used to answer this

5
GREAT, but why do we care?
-- N.R. Cozzarelli and A. Stasiak

• Many applications of Tutte polynomial
• Physics, Biology and probably lots more
• For example
• Tangled cords which cant be unravelled
• Double Helix of DNA actually forms a Knot

6
Optimising the Computation

• Caching previously seen graphs

7
Performance Data
8
Optimising the Computation

• Degrees of Freedom
• Can apply Tutte rules in any order
• Can choose any edge to delete/contract
• Our choices affect size of computation tree
• Edge Selection Heuristics
• Developed heuristics Minsdeg, Vorder
• But, why are they any good?

9
Visualising the Computation Tree

• Tree may have gt 100K nodes
• How can we visualise it?

10
Minsdeg
11
Vorder
12
Minsdeg
13
Vorder
14
(No Transcript)
15
To be continued

• Edge Selection Heuristics
• Q) How do we know why they work?
• A) Visualise them!
• Q) So, does it really help?
• A) Er , Ill tell you later !

16
Graph Layout Algorithms?

• Simple layout algorithm used
• Better ones exist that minimise crossings
• But, simple approach has some advantages