Visualising the Tutte Polynomial Computation Bennett Thompson, David J. Pearce Victoria University...

Visualising the Tutte Polynomial Computation

Bennett Thompson, David J. Pearce

Victoria University of Wellington,New Zealand

Gary HaggardBucknell, USA

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 bridgeT(G) = yT(G-e), if e is a loopT(G) = T(G-e) + T(G/e), otherwise

G = G–e = G/e =

COMP205 Software Design and Engineering

Tutte Computation Tree

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

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

COMP205 Software Design and Engineering

Optimising the Computation

• Caching previously seen graphs:

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Performance Data

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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?

COMP205 Software Design and Engineering

Visualising the Computation Tree

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

Minsdeg

Vorder

Minsdeg

Vorder

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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 !

COMP205 Software Design and Engineering

Graph Layout Algorithms?

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

advantages…