? Michael Goodrich Maarten L offler 1-1 ERIN CHAMBERS DAVID - PowerPoint PPT Presentation

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid? 7-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid? 7-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid? ANSWER Nope. 7-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid? 7-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid? 7-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Still no. 7-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid? 7-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Yes! 7-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Yes! [Us, Here, Now] 7-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing. 8-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing. 8-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing. 8-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing. LEMMA A drawing with polynomial resolution can be transformed into a drawing on a polynomial grid. 8-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THEOREM Given a planar graph G with n vertices and a convex polygon P with resolution d , there exists a drawing of G inside P with resolution Ω( d/n 3 ) . 9-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THEOREM Given a planar graph G with n vertices and a convex polygon P with resolution d , there exists a drawing of G inside P with resolution Ω( d/n 3 ) . PROOF Just use divide & conquer... 9-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER 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 split through any pair of edges into smaller polygons with resolutions d 1 and d 2 such that d 1 + d 2 ≥ d . 10-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER 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 split through any pair of edges into smaller polygons with resolutions d 1 and d 2 such that d 1 + d 2 ≥ d . 10-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER 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 split through any pair of edges into smaller polygons with resolutions d 1 and d 2 such that d 1 + d 2 ≥ d . 10-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER 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 split through any pair of edges into smaller polygons with resolutions d 1 and d 2 such that d 1 + d 2 ≥ d . 10-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER 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 split through any pair of edges into smaller polygons with resolutions d 1 and d 2 such that d 1 + d 2 ≥ d . 10-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. 11-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. SLIGHT PROBLEM The vertices along the cut need not be drawn in the same place by the subproblems. 11-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. SLIGHT PROBLEM The vertices along the cut need not be drawn in the same place by the subproblems. QUESTION What ever shall we do? 11-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip of faces. 12-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip of faces. [Duncan & Goodrich & Kobourov, 2009] 12-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip of faces. [Duncan & Goodrich & Kobourov, 2009] 12-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip of faces. [Duncan & Goodrich & Kobourov, 2009] 12-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip of faces. [Duncan & Goodrich & Kobourov, 2009] 12-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. 13-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. OBSERVATION Well, that didn’t work. 13-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. OBSERVATION Well, that didn’t work. JUST A THOUGHT Maybe it would have worked better if we had left some space for that strip of faces... 13-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. 14-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. EXCLAMATION Success! 14-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA