Visualising the Tutte Polynomial Computation Bennett Thompson, David J. Pearce Victoria University...
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Transcript of 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:
COMP205 Software Design and Engineering
Performance Data
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?
COMP205 Software Design and Engineering
Visualising the Computation Tree
• Tree may have > 100K nodes– How can we visualise it?
Minsdeg
Vorder
Minsdeg
Vorder
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 !
COMP205 Software Design and Engineering
Graph Layout Algorithms?
• Simple layout algorithm used– Better ones exist that minimise crossings– But, simple approach has some
advantages…