rectangle packing -...
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![Page 1: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/1.jpg)
![Page 2: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/2.jpg)
Rectangle Packing[Demaine & Demaine 2007]
⁄𝑛𝑛 3
𝑡𝑡 + 𝑛𝑛
𝑛𝑛3
![Page 3: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/3.jpg)
Edge-Matching Puzzles[1890s—]
![Page 4: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/4.jpg)
US$2,000,000prize
![Page 5: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/5.jpg)
Near-Solutions toEternity II
Louis Verhaardhttp://www.shortestpath.se/
eii/results.html
467 edge matches out of 480
($10,000)
![Page 6: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/6.jpg)
“Solution” to Eternity II
Louis Verhaard http://www.shortestpath.se/eii/results.html
![Page 7: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/7.jpg)
• Simulate rectangle packing
Edge-MatchingPuzzles[Demaine & Demaine 2007]
![Page 8: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/8.jpg)
Edge-Matching → Signed Edge-Matching[Demaine & Demaine 2007]
![Page 9: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/9.jpg)
Jigsaw Puzzles [1760s–]
http://www.jigsawplanet.com/
![Page 10: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/10.jpg)
Signed Edge Matching ↔ Jigsaw[Demaine & Demaine 2007]
![Page 11: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/11.jpg)
£1,000,000prize
PolyformPackingPuzzles
![Page 12: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/12.jpg)
PolyformPackingPuzzles
image by Ed Pegg Jr.solution by Alex Selby & Oliver Riordan, 2000
£1,000,000prize
![Page 13: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/13.jpg)
Signed Edge Matching ↔ Jigsaw→ Polyomino Packing [Demaine & Demaine 2007]
![Page 14: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/14.jpg)
Polyform Packing Puzzle →Edge-Matching Puzzle [Demaine & Demaine 2007]
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![Page 15: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/15.jpg)
Packing Squares into a Square isStrongly NP-complete• Generalization: Rectangular target
[Leung, Tam, Wong, Young, Chin 1990]
3𝐵𝐵 + 𝑡𝑡
𝐵𝐵 + 𝑎𝑎𝑖𝑖
𝐵𝐵 + 𝑡𝑡
𝐵𝐵 + 𝑡𝑡𝑛𝑛3
![Page 16: Rectangle Packing - courses.csail.mit.educourses.csail.mit.edu/6.892/spring19/lectures/L02_images.pdfJigsaw Puzzles [1760s–] Jigsaw generator: 爀䤀洀愀最攀㨀 栀琀琀瀀㨀](https://reader033.vdocuments.net/reader033/viewer/2022051805/5ff239358f980f00182b48c2/html5/thumbnails/16.jpg)
Packing Squares into a Square isStrongly NP-complete• Rectangle gadget: (infrastructure)
[Leung, Tam, Wong, Young, Chin 1990]
𝑥𝑥 𝑥𝑥 + 1 − 1
𝑥𝑥 + 1
𝑥𝑥
𝑥𝑥 + 1 𝑥𝑥 𝑥𝑥 + 1 − 1
𝑥𝑥