Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank - PowerPoint PPT Presentation

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank Ruskey University of Victoria & California Institute of Technology June 12, 2013 CanaDAM, St. John’s

Introduction Minimum Area Expansion Omitting the Empty Set Summary Outline Introduction Venn diagrams Minimum Area Definition Small examples Expansion Sufficient conditions One by eight expansion Omitting the Empty Set Summary What we know What we don’t know

Introduction Minimum Area Expansion Omitting the Empty Set Summary Types of curves Venn diagrams as circles

Introduction Minimum Area Expansion Omitting the Empty Set Summary Types of curves Venn diagrams as ovals

Introduction Minimum Area Expansion Omitting the Empty Set Summary Types of curves Venn diagrams as triangles

Introduction Minimum Area Expansion Omitting the Empty Set Summary Types of curves $n=2$ $n = 3$ $n = 4$ $n = 5$ $n = 6$ Venn diagrams with minimum intersections

Introduction Minimum Area Expansion Omitting the Empty Set Summary Types of curves A A A B C A B B B C C C A A B A C B A B B C C C polyomino Venn diagrams

Introduction Minimum Area Expansion Omitting the Empty Set Summary Venn diagram definition • A Venn diagram • is a collection of simple closed curves C = C 1 , C 2 , . . . , C n drawn on the plane

Introduction Minimum Area Expansion Omitting the Empty Set Summary Venn diagram definition • A Venn diagram • is a collection of simple closed curves C = C 1 , C 2 , . . . , C n drawn on the plane • such that each of the 2 n sets X 1 ∩ X 2 ∩ . . . ∩ X n is a nonempty and connected region where X i is either the bounded interior or unbounded exterior of C i .

Introduction Minimum Area Expansion Omitting the Empty Set Summary Venn diagram definition • A Venn diagram • is a collection of simple closed curves C = C 1 , C 2 , . . . , C n drawn on the plane • such that each of the 2 n sets X 1 ∩ X 2 ∩ . . . ∩ X n is a nonempty and connected region where X i is either the bounded interior or unbounded exterior of C i . We call the second requirement of the statement the region property.

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum area What is a minimum area polyomino Venn diagram? • A diagram

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum area What is a minimum area polyomino Venn diagram? • A diagram • with polyominoes,

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum area What is a minimum area polyomino Venn diagram? • A diagram • with polyominoes, • where each set � � interior ( P i ) ∩ exterior ( P i ) i ∈ I i / ∈ I together with a base region of unit squares, is a single unit square,

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum area What is a minimum area polyomino Venn diagram? • A diagram • with polyominoes, • where each set � � interior ( P i ) ∩ exterior ( P i ) i ∈ I i / ∈ I together with a base region of unit squares, is a single unit square, • for all I ⊆ [ n ] .

A A A A B B B A B Introduction Minimum Area Expansion Omitting the Empty Set Summary Base regions that are rectangles • A ( r , c ) -polyVenn is a minimum area polyomino Venn diagram confined to a 2 r × 2 c base rectangle.

A A A A B B B A B Introduction Minimum Area Expansion Omitting the Empty Set Summary Base regions that are rectangles • A ( r , c ) -polyVenn is a minimum area polyomino Venn diagram confined to a 2 r × 2 c base rectangle. • The number of polyominoes is n = r + c .

A A A A B B B A B Introduction Minimum Area Expansion Omitting the Empty Set Summary Base regions that are rectangles • A ( r , c ) -polyVenn is a minimum area polyomino Venn diagram confined to a 2 r × 2 c base rectangle. • The number of polyominoes is n = r + c . • The number of unit squares on the grid is 2 n .

Introduction Minimum Area Expansion Omitting the Empty Set Summary Base regions that are rectangles • A ( r , c ) -polyVenn is a minimum area polyomino Venn diagram confined to a 2 r × 2 c base rectangle. • The number of polyominoes is n = r + c . • The number of unit squares on the grid is 2 n . Example little polyVenns: A A A A B B B A B

Introduction Minimum Area Expansion Omitting the Empty Set Summary Are there any more single row polyVenns?

Introduction Minimum Area Expansion Omitting the Empty Set Summary Are there any more single row polyVenns? Place the leftmost polyomino A A A A

Introduction Minimum Area Expansion Omitting the Empty Set Summary Are there any more single row polyVenns? Only one choice for the second polyomino A A A B A B B B

Introduction Minimum Area Expansion Omitting the Empty Set Summary Are there any more single row polyVenns? No place for the third A A A B A B B B

Introduction Minimum Area Expansion Omitting the Empty Set Summary The known polyVenns 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 · · · 2 0 F F F × × × × × × · · · 2 1 F F ? ? ? ? ? ? · · · 2 2 ? ? ? ? ? ? ? · · · 2 3 ? ? ? ? ? ? · · · 2 4 ? ? ? ? ? · · · 2 5 ? ? ? ? · · · 2 6 ? ? ? · · · 2 7 ? ? · · · 2 8 ? · · · . ... . .

Introduction Minimum Area Expansion Omitting the Empty Set Summary A (1 , 3)-polyVenn (symmetric) A A A A D C D C C C D D B B B B A B A B A B A B C C D D D C D C A B C D

Introduction Minimum Area Expansion Omitting the Empty Set Summary A (1 , 4)-polyVenn C B C B C A BC A B C A B C A C A B C A C C B C C A C A C C D D D E D D D E D E D E E E E B B A B A A B B B C A B A B A B A A D D D D E D E D E E D E D E E E E A B C D E

Introduction Minimum Area Expansion Omitting the Empty Set Summary The known polyVenns 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 · · · 2 0 F F F × × × × × × · · · 2 1 F F F F ? ? ? ? · · · 2 2 ? ? ? ? ? ? ? · · · 2 3 ? ? ? ? ? ? · · · 2 4 ? ? ? ? ? · · · 2 5 ? ? ? ? · · · 2 6 ? ? ? · · · 2 7 ? ? · · · 2 8 ? · · · . ... . .

Introduction Minimum Area Expansion Omitting the Empty Set Summary We can expand when... Suppose we have a 2 r × 2 c grid that holds an existing polyVenn. Then we can expand this polyVenn to create one on a 2 r + r ′ × 2 c + c ′ grid if the following is true:

Introduction Minimum Area Expansion Omitting the Empty Set Summary We can expand when... Suppose we have a 2 r × 2 c grid that holds an existing polyVenn. Then we can expand this polyVenn to create one on a 2 r + r ′ × 2 c + c ′ grid if the following is true: • A non-empty set of mini-systems exist that meet the required region property on small bounding rectangles of dimensions 2 r ′ × 2 c ′ .

Introduction Minimum Area Expansion Omitting the Empty Set Summary We can expand when... Suppose we have a 2 r × 2 c grid that holds an existing polyVenn. Then we can expand this polyVenn to create one on a 2 r + r ′ × 2 c + c ′ grid if the following is true: • A non-empty set of mini-systems exist that meet the required region property on small bounding rectangles of dimensions 2 r ′ × 2 c ′ . • There is a way to connect elements of these mini-systems on the large grid to create connected polyominoes.

Introduction Minimum Area Expansion Omitting the Empty Set Summary How it’s done The first polyomino is created by expanding the existing polyVenn. In this case each unit square is expanded into a 2 × 4 mini-grid.

Introduction Minimum Area Expansion Omitting the Empty Set Summary How it’s done Then find a system that meets the region requirement on the mini-grid.

Introduction Minimum Area Expansion Omitting the Empty Set Summary Then lay these out

Introduction Minimum Area Expansion Omitting the Empty Set Summary Another known polyVenn 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 · · · 2 0 F F F × × × × × × · · · 2 1 F F F F ? ? ? ? · · · 2 2 E ? ? ? ? ? ? · · · 2 3 ? ? ? ? ? ? · · · 2 4 ? ? ? ? ? · · · 2 5 ? ? ? ? · · · 2 6 ? ? ? · · · 2 7 ? ? · · · 2 8 ? · · · . ... . .

Introduction Minimum Area Expansion Omitting the Empty Set Summary One by eight expansion Theorem If there is a ( 2 , c ) -polyVenn, then there is a ( 2 , c + 3 ) -polyVenn.

Introduction Minimum Area Expansion Omitting the Empty Set Summary Find the systems ( ❑❑ ❏❏ , ▼ ▲ ▼ ▲ , ❑ ❏❏ ❑ ) , ( ❏❏ ❑❑ , ▼ ▲ ▼ ▲ , ❑ ❏❏ ❑ ) , ( ❏ ▼ ❑ ▲ , ❑ ❏ ❑ ❏ , ▼ ▲ ▼ ▲ ) , ( ▼ ❑ ▲ ❏ , ❏ ❑ ❏ ❑ , ▼ ▲ ▼ ▲ ) , ( ❏ ❑ ❏ ❑ , ▼ ❑ ▲ ❏ , ▼ ▲ ▼ ▲ ) , ( ❑ ❏ ❑ ❏ , ❏ ▼ ❑ ▲ , ▼ ▲ ▼ ▲ ) , ( ▼ ▲ ▼ ▲ , ❏❏ ❑❑ , ❏ ❑❑ ❏ ) , ( ▼ ▲ ▼ ▲ , ❑❑ ❏❏ , ❏ ❑❑ ❏ ) .

Introduction Minimum Area Expansion Omitting the Empty Set Summary Connect the mini-grids