1-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L ¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA Maarten L¨ offler Erin Chambers Michael Goodrich David Eppstein ?
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1-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
DRAWING GRAPHS IN THE PLANEWITH A PRESCRIBED OUTER FACE
AND POLYNOMIAL AREA
Maarten Loffler
Erin Chambers
Michael Goodrich
David Eppstein
?
2-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edges!
2-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edges!
2-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edges!
2-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edges!
2-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edges!
[Wagner, 1936] [Fary, 1948] [Stein, 1951]
3-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon.
3-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgesinside a given convex polygon. [Tutte, 1960]
4-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid.
4-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid.
4-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid.
4-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid.
4-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid.
4-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
FACTEvery planar graph can be drawn with straight edgeson a polynomial grid. [Schnyder, 1990]
[de Fraysseix & Pach & Pollack, 1990]
5-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
5-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
5-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
5-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
5-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
?
5-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
?
ANSWERObviously not...
5-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
QUESTIONCan every planar graph be drawn with straight edgesinside a given convex polygon on a polynomial grid?
?
ANSWERObviously not... ...butbutbut!
6-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-10 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-11 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-12 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-13 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-14 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-15 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
6-16 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
7-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon on a polynomial grid?
7-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon on a polynomial grid?
7-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon on a polynomial grid?
ANSWERNope.
7-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
TWICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon of bounded resolution on a polynomial grid?
7-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
TWICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon of bounded resolution on a polynomial grid?
7-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
TWICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is similar to a given convexpolygon of bounded resolution on a polynomial grid?
ANSWERStill no.
7-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
THRICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is almost similar to a givenconvex polygon of bounded resolution on apolynomial grid?
7-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
THRICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is almost similar to a givenconvex polygon of bounded resolution on apolynomial grid?
ANSWERYes!
7-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
THRICE REVISED QUESTIONCan every planar graph be drawn with straight edgesinside a polygon that is almost similar to a givenconvex polygon of bounded resolution on apolynomial grid?
ANSWERYes! [Us, Here, Now]
8-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
DEFINITIONThe resolution of a drawing is the ratio between thediameter and the shortest distance between twonon-incident features of the drawing.
8-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
DEFINITIONThe resolution of a drawing is the ratio between thediameter and the shortest distance between twonon-incident features of the drawing.
8-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
DEFINITIONThe resolution of a drawing is the ratio between thediameter and the shortest distance between twonon-incident features of the drawing.
8-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMAA drawing with polynomial resolution can betransformed into a drawing on a polynomial grid.
DEFINITIONThe resolution of a drawing is the ratio between thediameter and the shortest distance between twonon-incident features of the drawing.
9-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
THEOREMGiven a planar graph G with n vertices and a convexpolygon P with resolution d, there exists a drawing ofG inside P with resolution Ω(d/n3).
9-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
THEOREMGiven a planar graph G with n vertices and a convexpolygon P with resolution d, there exists a drawing ofG inside P with resolution Ω(d/n3).
PROOFJust use divide & conquer...
10-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (geometric split lemma)Every convex polygon with resolution d can be splitthrough any pair of edges into smaller polygons withresolutions d1 and d2 such that d1 + d2 ≥ d.
10-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (geometric split lemma)Every convex polygon with resolution d can be splitthrough any pair of edges into smaller polygons withresolutions d1 and d2 such that d1 + d2 ≥ d.
10-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (geometric split lemma)Every convex polygon with resolution d can be splitthrough any pair of edges into smaller polygons withresolutions d1 and d2 such that d1 + d2 ≥ d.
10-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (geometric split lemma)Every convex polygon with resolution d can be splitthrough any pair of edges into smaller polygons withresolutions d1 and d2 such that d1 + d2 ≥ d.
10-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (geometric split lemma)Every convex polygon with resolution d can be splitthrough any pair of edges into smaller polygons withresolutions d1 and d2 such that d1 + d2 ≥ d.
11-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
11-10 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
SLIGHT PROBLEMThe vertices along the cut need not be drawn in thesame place by the subproblems.
11-11 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
IDEASplit graph and polygon in equal fractions.
SLIGHT PROBLEMThe vertices along the cut need not be drawn in thesame place by the subproblems.
QUESTIONWhat ever shall we do?
12-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (combinatorial split lemma)Every embedded triangulated graph can be splitthrough any pair of edges by a simple connected stripof faces.
12-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (combinatorial split lemma)Every embedded triangulated graph can be splitthrough any pair of edges by a simple connected stripof faces. [Duncan & Goodrich & Kobourov, 2009]
12-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (combinatorial split lemma)Every embedded triangulated graph can be splitthrough any pair of edges by a simple connected stripof faces. [Duncan & Goodrich & Kobourov, 2009]
12-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (combinatorial split lemma)Every embedded triangulated graph can be splitthrough any pair of edges by a simple connected stripof faces. [Duncan & Goodrich & Kobourov, 2009]
12-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
LEMMA (combinatorial split lemma)Every embedded triangulated graph can be splitthrough any pair of edges by a simple connected stripof faces. [Duncan & Goodrich & Kobourov, 2009]
13-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
13-10 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
OBSERVATIONWell, that didn’t work.
13-11 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROPOSITIONLet’s try that again.
JUST A THOUGHTMaybe it would have worked better if we had leftsome space for that strip of faces...
OBSERVATIONWell, that didn’t work.
14-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-10 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
14-11 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
EXCLAMATIONSuccess!
14-12 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
PROVERBThird time lucky.
EXCLAMATIONSuccess!
Several details have been swept under the carpet here. The geometric split lemma was only applied partiallyon one side. It is possible that the combinatorial split lemma generates more than two subproblems. Also,we may need to include extra faces inside the strip to avoid chords on the boundary. Oh, and I failed tomention that the resolution of a polygon is not exactly what we need either – in the paper you will find thedefinition of the potential resolution of a polygon. In short, don’t try this at home. At least not until youread the paper. No people or animals have been hurt in the production of this presentation.
DISCLAIMER
15-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
TENACIOUS INTERROGATIONOk, so then how does it really work?
16-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-7 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
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ALGORITHMJust follow these easy steps:
16-8 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
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ALGORITHMJust follow these easy steps:
16-9 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-10 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-11 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-12 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-13 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-14 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
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ALGORITHMJust follow these easy steps:
16-15 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
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9 13
ALGORITHMJust follow these easy steps:
16-16 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-17 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-18 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-19 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-20 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-21 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-22 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-23 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-24 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
16-25 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
ALGORITHMJust follow these easy steps:
17-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
17-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
17-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
17-4 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
17-5 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
17-6 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
APPLICATIONGraphs embedded on a surface of genus g are oftendrawn in the plane inside a regular 4g-gon.
18-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
MAIN RESULT REPEATED ONCE MOREGiven a planar graph G with n vertices and a convexpolygon P with resolution d, there exists a drawing ofG inside P with resolution Ω(d/n3).
18-2 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
MAIN RESULT REPEATED ONCE MOREGiven a planar graph G with n vertices and a convexpolygon P with resolution d, there exists a drawing ofG inside P with resolution Ω(d/n3).
COROLLARYGraphs can be drawn inside a given polygon and on apolynomial grid, under some natural restrictions.
18-3 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
MAIN RESULT REPEATED ONCE MOREGiven a planar graph G with n vertices and a convexpolygon P with resolution d, there exists a drawing ofG inside P with resolution Ω(d/n3).
COROLLARYGraphs can be drawn inside a given polygon and on apolynomial grid, under some natural restrictions.
OPEN QUESTIONCan the bound of Ω(d/n3) be improved?
19-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN LOFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA
?
ONE MORE QUESTIONDo you have any questions?
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