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

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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

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

Rectangle Packing[Demaine & Demaine 2007]

⁄𝑛𝑛 3

𝑡𝑡 + 𝑛𝑛

𝑛𝑛3

Presenter

Presentation Notes

Figure 3 of http://erikdemaine.org/papers/Jigsaw_GC/

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

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

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

Near-Solutions toEternity II

Louis Verhaardhttp://www.shortestpath.se/

eii/results.html

467 edge matches out of 480

($10,000)

Presenter

Presentation Notes

http://www.shortestpath.se/eii/results.html

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

“Solution” to Eternity II

Louis Verhaard http://www.shortestpath.se/eii/results.html

Presenter

Presentation Notes

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

• Simulate rectangle packing

Edge-MatchingPuzzles[Demaine & Demaine 2007]

Presenter

Presentation Notes

Figures 4 & 5 of http://erikdemaine.org/papers/Jigsaw_GC/

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

Edge-Matching → Signed Edge-Matching[Demaine & Demaine 2007]

Presenter

Presentation Notes

Figure 6 of http://erikdemaine.org/papers/Jigsaw_GC/

Jigsaw Puzzles [1760s–]

http://www.jigsawplanet.com/

Presenter

Presentation Notes

Jigsaw generator: http://www.jigsawplanet.com/ Image: http://courses.csail.mit.edu/6.890/fall14/poster_500.jpg

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

Signed Edge Matching ↔ Jigsaw[Demaine & Demaine 2007]

Presenter

Presentation Notes

Figure 7 of http://erikdemaine.org/papers/Jigsaw_GC/

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

£1,000,000prize

PolyformPackingPuzzles

Presenter

Presentation Notes

http://www.mathpuzzle.com/eternity.html

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

PolyformPackingPuzzles

image by Ed Pegg Jr.solution by Alex Selby & Oliver Riordan, 2000

£1,000,000prize

Presenter

Presentation Notes

http://www.mathpuzzle.com/eternity.html

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

Signed Edge Matching ↔ Jigsaw→ Polyomino Packing [Demaine & Demaine 2007]

Presenter

Presentation Notes

Figure 7 of http://erikdemaine.org/papers/Jigsaw_GC/

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

Polyform Packing Puzzle →Edge-Matching Puzzle [Demaine & Demaine 2007]

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

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

Presenter

Presentation Notes

Based on http://dx.doi.org/10.1016/0743-7315(90)90019-L

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

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

Presenter

Presentation Notes

Based on Figure 3 of http://dx.doi.org/10.1016/0743-7315(90)90019-L

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2017 Women’s Health Conference · This is on Mondays at noon \⠀㈀㨀㔀†ጀ 倀䴀琀愀氀欀尩. We give CME credit, and there is salad for lunch. Attendance is usually

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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

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