Split Packing: An Algorithm for Packing Circles with up to Critical - - PowerPoint PPT Presentation

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Split Packing: An Algorithm for Packing Circles with up to Critical Density Sebastian Morr 2016-06-09 Sebastian Morr Split Packing 2016-06-09 1 / 57 Sebastian Morr Split Packing 2016-06-09 2 / 57 Sebastian Morr Split Packing

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Split Packing: An Algorithm for Packing Circles with up to Critical Density

Sebastian Morr 2016-06-09

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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Packing squares in a square

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SLIDE 12

Packing squares in a square

The critical density for packing squares is 1/2 [Moon & Moser, 1967]

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SLIDE 13

Outline

1

Packing circles in a square

2

Other container types

3

Other object types

4

Future work

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SLIDE 14

What about circles?

Critical density for packing circles into a square

What is the largest a so that any set of circles with a combined area of a can be packed into the unit square?

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What about circles?

Critical density for packing circles into a square

What is the largest a so that any set of circles with a combined area of a can be packed into the unit square?

→ Now: Constructive proof!

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Observation: Splitting in half is easy

1

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Observation: Splitting in half is easy

1 2 1 2

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Observation: Splitting in half is easy

1 2 1 4 1 4

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Observation: Splitting in half is easy

1 2 1 4 1 8 1 8

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Observation: Splitting in half is easy

1 2 1 4 1 8 1 16 1 16

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Observation: Splitting in half is easy

1 2 1 4 1 16 1 16

?

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

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Greedy splitting

A B

Split property:

All elements of larger group ≥ groups’ difference.

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Greedy splitting

A B

Split property:

All elements of larger group ≥ groups’ difference.

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Greedy splitting

A B

Split property:

All elements of larger group ≥ groups’ difference.

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An (a, b)-hat

b b b a

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Packing hats in a hat

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Analysis

Two perspectives:

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SLIDE 77

Analysis

Two perspectives:

Deciding packability

A tight sufficient density condition: Every instance with up to critical density d can be packed!

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Analysis

Two perspectives:

Deciding packability

A tight sufficient density condition: Every instance with up to critical density d can be packed!

Minimizing the container’s size

A constant-factor approximation algorithm: The ratio between the approximated and the optimal container area is at most 1/d.

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Analysis

Two perspectives:

Deciding packability

A tight sufficient density condition: Every instance with up to critical density d can be packed!

Minimizing the container’s size

A constant-factor approximation algorithm: The ratio between the approximated and the optimal container area is at most 1/d. Runtime:

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Analysis

Two perspectives:

Deciding packability

A tight sufficient density condition: Every instance with up to critical density d can be packed!

Minimizing the container’s size

A constant-factor approximation algorithm: The ratio between the approximated and the optimal container area is at most 1/d. Runtime: At most O(n2) numeric operations

◮ Worst-case greedy split: n + (n − 1) + (n − 2) + · · · + 1 operations Sebastian Morr Split Packing 2016-06-09 25 / 57

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SLIDE 81

Analysis

Two perspectives:

Deciding packability

A tight sufficient density condition: Every instance with up to critical density d can be packed!

Minimizing the container’s size

A constant-factor approximation algorithm: The ratio between the approximated and the optimal container area is at most 1/d. Runtime: At most O(n2) numeric operations

◮ Worst-case greedy split: n + (n − 1) + (n − 2) + · · · + 1 operations

Exactly 3n − 2 geometric constructions

◮ Full binary recursion tree with n leaf nodes Sebastian Morr Split Packing 2016-06-09 25 / 57

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Circles in a square

Critical density: π 3 + 2 √ 2 ≈ 53.90% Approximation factor: 3 + 2 √ 2 π ≈ 1.8552

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Circles in a square: Examples

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SLIDE 84

Outline

1

Packing circles in a square

2

Other container types

3

Other object types

4

Future work

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Splitting for asymmetric triangles

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Circles in a right/obtuse triangle

b a c Condition: a2 + b2 ≤ c2 Critical density:

  • −(a − b − c)(a + b − c)(a − b + c)

(a + b + c)3 π < 53.91% Approximation factor: Larger than 1.8552

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Circles in a right/obtuse triangle: Examples

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Circles in a thick isosceles triangle

c b Condition: c √ 2 ≤ b ≤ c Critical density: 48.60% < (c − 2b + √ 4b2 − c2)2π 2c √ 4b2 − c2 < 53.91% Approximation factor: Between 1.8552 and 2.0576

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Circles in a thick isosceles triangle: Examples

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The problem with acute triangles

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The problem with acute triangles

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The problem with acute triangles

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Circles in a long rectangle

w h Condition: w ≥ 2 + 3 √ 2 4 h ≈ 1.5607h Critical density: πh 4w < 50.33% Approximation factor: 4w πh > 1.9870

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Circles in a long rectangle: Examples

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SLIDE 95

Outline

1

Packing circles in a square

2

Other container types

3

Other object types

4

Future work

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SLIDE 96

Other shapes in a square?

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SLIDE 97

Other shapes in a square?

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SLIDE 98

Other shapes in a square?

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SLIDE 99

Rubies!

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SLIDE 100

Rubies!

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Rubies in a square

Critical density: 8

  • 2(

√ 2 − 1) + 6 √ 2 − 15 ≈ 76.67% Approximation factor: ≈ 1.3043

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Rubies in a square: Examples

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Squares in a square

Critical density: 50% Approximation factor: 2

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Octagons in a square

Critical density: 8(5 √ 2 − 7) ≈ 56.85% Approximation factor: ≈ 1.7589

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Other shapes in an isosceles right triangle?

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Other shapes in an isosceles right triangle?

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Other shapes in an isosceles right triangle?

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Other shapes in an isosceles right triangle?

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“Sharp rubies” in an isosceles right triangle

Critical density: 4

  • 2(

√ 2 − 1) + 3 √ 2 − 7 ≈ 88.34% Approximation factor: ≈ 1.1320

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“Sharp rubies” in an isosceles right triangle: Examples

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Outline

1

Packing circles in a square

2

Other container types

3

Other object types

4

Future work

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Future work: More container types

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Future work: Acute triangles

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Future work: More object types

Ovals Rectangles General convex polygons? What do the critical instances look like?

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Future work: Maximum object size

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Future work: Online

Current best achievable density for packing squares into a square in an

  • nline setting: 2/5

[Brubach 2015]

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Future work: Circle/river packing

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Future work: 3D

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Future work: Covering

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Future work: Covering

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Future work: Covering

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Future work: Covering

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Contributions

1 Algorithms for packing. . . Sebastian Morr Split Packing 2016-06-09 56 / 57

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Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares Sebastian Morr Split Packing 2016-06-09 56 / 57

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Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

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Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

. . . with critical density!

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SLIDE 127

Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

. . . with critical density!

2 Constant-factor approximation algorithms for these problems Sebastian Morr Split Packing 2016-06-09 56 / 57

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SLIDE 128

Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

. . . with critical density!

2 Constant-factor approximation algorithms for these problems 3 Interactive visualization, at https://morr.cc/split-packing/ Sebastian Morr Split Packing 2016-06-09 56 / 57

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SLIDE 129

Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

. . . with critical density!

2 Constant-factor approximation algorithms for these problems 3 Interactive visualization, at https://morr.cc/split-packing/ 4 Promising future work Sebastian Morr Split Packing 2016-06-09 56 / 57

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SLIDE 130

Contributions

1 Algorithms for packing. . . ◮ circles, squares, and octagons into squares ◮ circles into non-acute triangles, thick isosceles triangles, and long

rectangles

. . . with critical density!

2 Constant-factor approximation algorithms for these problems 3 Interactive visualization, at https://morr.cc/split-packing/ 4 Promising future work

Thanks!

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Bonus slide: Applications

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