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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
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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Packing squares in a square
The critical density for packing squares is 1/2 [Moon & Moser, 1967]
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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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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 acan be packed into the unit square?
→ Now: Constructive proof!
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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 acan 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
12
12
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Observation: Splitting in half is easy
12
14
14
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Observation: Splitting in half is easy
12
14
18
18
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Observation: Splitting in half is easy
12
14
18
116
116
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Observation: Splitting in half is easy
12
14
116
116
?
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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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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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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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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
bb
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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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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AnalysisTwo 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 atmost 1/d.
Runtime:
At most O(n2) numeric operationsI Worst-case greedy split: n + (n − 1) + (n − 2) + · · ·+ 1 operations
Exactly 3n − 2 geometric constructionsI Full binary recursion tree with n leaf nodes
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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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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)3π < 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 2.0576Sebastian Morr Split Packing 2016-06-09 32 / 57
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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
4h ≈ 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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Outline
1 Packing circles in a square
2 Other container types
3 Other object types
4 Future work
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Other shapes in a square?
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Other shapes in a square?
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Other shapes in a square?
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Rubies!
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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 anonline 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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Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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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![Page 124: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/124.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squares
I circles into non-acute triangles, thick isosceles triangles, and longrectangles
. . . 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 125: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/125.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 126: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/126.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 127: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/127.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 128: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/128.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 129: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/129.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 130: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/130.jpg)
Contributions1 Algorithms for packing. . .
I circles, squares, and octagons into squaresI 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!
Sebastian Morr Split Packing 2016-06-09 56 / 57
![Page 131: Split Packing: An Algorithm for Packing Circles with up to ... · Packing squares in a square The critical density for packing squares is 1=2 [Moon & Moser, 1967] Sebastian Morr Split](https://reader033.vdocuments.us/reader033/viewer/2022052718/5f057bf67e708231d41330ff/html5/thumbnails/131.jpg)
Bonus slide: Applications
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