Computational Geometry

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The complexity of tangent words

The complexity of tangent words

์ด ๋…ผ๋ฌธ์€ ๋งค๋„๋Ÿฌ์šด ๊ณก์„ ๊ณผ ๊ทธ์— ๋Œ€ํ•œ ์ ‘์„  ๋‹จ์–ด์˜ ์–ธ์–ด์  ์ ‘๊ทผ๋ฒ•์„ ์ฒด๊ณ„์ ์œผ๋กœ ๋ถ„์„ํ•œ๋‹ค. ์ฃผ์š” ๋‚ด์šฉ์€ ๋‹ค์Œ๊ณผ ๊ฐ™๋‹ค: 1. ๋งค๋„๋Ÿฌ์šด ๊ณก์„ ์˜ ์ •์˜์™€ ํŠน์„ฑ : ๋งค๋„๋Ÿฌ์šด ๊ณก์„ ์€ ์‹ค์ˆ˜ ์ง์„  ์œ„์˜ ์ปดํŒฉํŠธ ๊ตฌ๊ฐ„ I์—์„œ ์ •์˜๋˜๋ฉฐ, ๋ชจ๋“  t โˆˆ I ์— ๋Œ€ํ•ด ||ฮณโ€ฒ(t)|| > 0 ์ด๋ผ๋Š” ๊ทœ์น™์„ฑ์„ ๋งŒ์กฑํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๊ณก์„ ์€ ์›ํ•˜๋Š” ๋งŒํผ ์žฌ๋งค๊ฐœ๋ณ€์ˆ˜ํ™”๋  ์ˆ˜ ์žˆ์œผ๋ฉฐ, ์ผ๋ฐ˜์ ์œผ๋กœ ๊ฐ t ์— ๋Œ€ํ•ด ||ฮณโ€ฒ(t)|| 1 ์ด๋‹ค. ๋งค๋„๋Ÿฌ์šด ๊ณก์„ ์˜ ๊ทผ์‚ฌ ๋ฐฉ๋ฒ•์œผ๋กœ ํ‰๋ฉด์— ๊ฒฉ์ž๋ง์„ ๊ทธ๋ ค์„œ ๊ณก์„ ์ด ๋งŒ๋‚˜๋Š” ์‚ฌ๊ฐํ˜•์˜ ์ˆœ์„œ๋ฅผ ์‚ดํŽด๋ณธ๋‹ค. ์ด ์ ˆ๋‹จ ์ˆœ์„œ๋Š” ์ƒ, ํ•˜, ์ขŒ, ์šฐ ๋ฐฉํ–ฅ์œผ

Formal Languages Computational Geometry Computer Science Discrete Mathematics
On the Non-Termination of Rupperts Algorithm

On the Non-Termination of Rupperts Algorithm

๋ฃจํผํŠธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ์‚ผ๊ฐํ™” ๊ณผ์ •์—์„œ ๊ฐ๋„์— ๋Œ€ํ•œ ์—„๊ฒฉํ•œ ์ œํ•œ์„ ์„ค์ •ํ•จ์œผ๋กœ์จ, ํ‰๋ฉด ๊ทธ๋ž˜ํ”„๋ฅผ ์•ˆ์ •์ ์ธ ๊ตฌ์กฐ๋กœ ๋ณ€ํ™˜ํ•˜๋Š” ๋ฐ ์ค‘์ ์„ ๋‘๊ณ  ์žˆ๋‹ค. ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ํ•ต์‹ฌ์€ ๋ชจ๋“  ์‚ผ๊ฐํ˜•์ด ์ตœ์†Œ ๊ฐ๋„ ฮฑ๋ณด๋‹ค ์ž‘๊ฑฐ๋‚˜ ๊ฐ™์€ ๊ฐ๋„๋ฅผ ๊ฐ€์ง€๋„๋ก ํ•˜๋Š” ๊ฒƒ์ด๋‹ค. ์ดˆ๊ธฐ ์—ฐ๊ตฌ์—์„œ๋Š” 20.7ยฐ๊ฐ€ ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด ์ข…๋ฃŒ๋˜๋Š” ๊ฐ€์žฅ ์ž‘์€ ฮฑ ๊ฐ’์œผ๋กœ ์ œ์‹œ๋˜์—ˆ์œผ๋‚˜, ํ›„์† ์‹คํ—˜์„ ํ†ตํ•ด ์ด๋Š” ๋งค์šฐ ๋ณด์ˆ˜์ ์ธ ์ถ”์ •์ด๋ผ๋Š” ๊ฒƒ์ด ๋ฐํ˜€์กŒ๋‹ค. ํŒŒ๋ธŒ ์˜ˆ์ œ์˜ ๋ถ„์„: ์Šคํ‹ฐ๋ธ ํŒŒ๋ธŒ๋Š” ๋ฃจํผํŠธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์—์„œ ๋น„์ข…๊ฒฐ์„ฑ์ด ๋ฐœ์ƒํ•˜๋Š” ๊ฒฝ์šฐ๋ฅผ ๋ณด์—ฌ์ฃผ๋Š” ์˜ˆ์ œ๋ฅผ ์ œ์‹œํ–ˆ๋‹ค. ์ด ์˜ˆ์ œ์—์„œ๋Š” ๋‘ ๊ฐœ์˜ ์ธ์ ‘ํ•œ ์„ธ๊ทธ๋จผํŠธ๊ฐ€ 3

Computer Science Computational Geometry
A deterministic algorithm for fitting a step function to a weighted point-set

A deterministic algorithm for fitting a step function to a weighted point-set

: ๋ณธ ๋…ผ๋ฌธ์€ k๋‹จ๊ณ„ ํ•จ์ˆ˜๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ๊ฐ€์ค‘๋œ ์  ์ง‘ํ•ฉ์„ ๊ทผ์‚ฌํ•˜๋Š” ๋ฌธ์ œ์— ๋Œ€ํ•œ ์ƒˆ๋กœ์šด ์ ‘๊ทผ ๋ฐฉ์‹์„ ์ œ์•ˆํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์ด ๋ฌธ์ œ๋Š” ๋ฐ์ดํ„ฐ๋ฒ ์ด์Šค ์‘์šฉ ํ”„๋กœ๊ทธ๋žจ์—์„œ ํšจ์œจ์ ์ธ ๋ฐ์ดํ„ฐ ์ €์žฅ ๋ฐ ์ฟผ๋ฆฌ ์ฒ˜๋ฆฌ ์†๋„ ํ–ฅ์ƒ์„ ์œ„ํ•ด ์ค‘์š”ํ•œ ์—ญํ• ์„ ํ•ฉ๋‹ˆ๋‹ค. 1. k๋‹จ๊ณ„ ํ•จ์ˆ˜์™€ ์ค‘๋Ÿ‰์ ์˜ ์ •์˜ k๋‹จ๊ณ„ ํ•จ์ˆ˜๋Š” ์‹ค์ˆ˜ ์‹œํ€€์Šค๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ๊ฐ ๊ตฌ๊ฐ„์— ์ƒ์ˆ˜ ๊ฐ’์„ ๊ฐ€์ง€๋Š” ํ•จ์ˆ˜์ž…๋‹ˆ๋‹ค. ์ค‘๋Ÿ‰์ ์€ (x, y, w)๋กœ ํ‘œํ˜„๋˜๋ฉฐ, ์—ฌ๊ธฐ์„œ x์™€ y๋Š” ์ ์˜ ์ขŒํ‘œ์ด๊ณ , w๋Š” ํ•ด๋‹น ์ ์˜ ๋ฌด๊ฒŒ๋ฅผ ๋‚˜ํƒ€๋ƒ…๋‹ˆ๋‹ค. 2. ๊ทผ์‚ฌ ๋ฌธ์ œ ์ฃผ์–ด์ง„ ๊ฐ€์ค‘๋œ ์  ์ง‘ํ•ฉ P์— ๋Œ€ํ•ด k๋‹จ๊ณ„ ํ•จ์ˆ˜ f๋ฅผ ์ฐพ์•„์„œ P์™€

Computational Geometry Computer Science Data Structures

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