1. Visualising the Tutte Polynomial Computation Bennett Thompson, David J. Pearce Victoria University of Wellington, New Zealand Gary Haggard Bucknell, USA

2. COMP205 Software Design and Engineering 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 =G–e =G/e =

3. COMP205 Software Design and Engineering Tutte Computation Tree

4. COMP205 Software Design and Engineering Great, but why do we care? Many applications of Tutte polynomial –Physics, Biology and probably lots more … Knots –Tangled cords which can’t be unravelled –Problem: how do we know when two knots are same? –Tutte polynomial can be used to answer this

5. COMP205 Software Design and Engineering GREAT, but why do we care? Many applications of Tutte polynomial –Physics, Biology and probably lots more … For example –Tangled cords which can’t be unravelled –Double Helix of DNA actually forms a Knot -- N.R. Cozzarelli and A. Stasiak

6. COMP205 Software Design and Engineering Optimising the Computation Caching previously seen graphs:

7. COMP205 Software Design and Engineering Performance Data

8. COMP205 Software Design and Engineering 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. COMP205 Software Design and Engineering Visualising the Computation Tree Tree may have > 100K nodes –How can we visualise it?

10. Minsdeg

11. Vorder

12. Minsdeg

13. Vorder

15. COMP205 Software Design and Engineering 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 …, I’ll tell you later !

16. COMP205 Software Design and Engineering Graph Layout Algorithms? Simple layout algorithm used –Better ones exist that minimise crossings –But, simple approach has some advantages…