Timeline for Sweep-segment bot: Will this random walk sweep the plane?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 8, 2017 at 10:57 | vote | accept | Joseph O'Rourke | ||
| May 8, 2017 at 10:57 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed. Additional minor edits.
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| Dec 11, 2010 at 2:38 | history | edited | Joseph O'Rourke | CC BY-SA 2.5 |
Addendum, thanks, editorializing.
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| Dec 11, 2010 at 0:12 | answer | added | fedja | timeline score: 9 | |
| Dec 11, 2010 at 0:02 | comment | added | Joseph O'Rourke | @George: You mean there is not much commercial promise here to undercut Roomba? :-) | |
| Dec 10, 2010 at 23:17 | answer | added | Anthony Quas | timeline score: 3 | |
| Dec 10, 2010 at 22:57 | comment | added | George Lowther | @Joseph: It will sweep the plane, but not very efficiently. How long would it take to sweep a circle of radius $R^2$ (asymptotically as $R\to\infty$)? I've no idea, but I expect it takes much longer than $O(R^2)$. | |
| Dec 10, 2010 at 20:32 | answer | added | Did | timeline score: 7 | |
| Dec 10, 2010 at 20:18 | answer | added | Hugh J | timeline score: 13 | |
| Dec 10, 2010 at 19:13 | comment | added | Joseph O'Rourke | @Didier: Yes, angles uniform and independent. Good point about the discretized version being a biased walk. Perhaps if the angle is one of $-\pi/2,0,\pi/2$ ... I will investigate reinforced random walks, a new term to me. Thanks for your interest! | |
| Dec 10, 2010 at 19:07 | comment | added | Did | When $\theta_i$ is $+\pi/2$ or $-\pi/2$ or $\pi$ ($-\pi$ being superfluous), the successive positions of the midpoint do not perform a simple random walk in the classical sense. To wit, the legal steps at time $n+1$ depend on the step performed at time $n$. For instance, after a $(+1,0)$ step, the next step can be $(+1,0)$, $(+1/2,+1/2)$ and $(+1/2,-1/2)$ only. More generally, the set of all possible steps ever has size $8$ but only $3$ of them are possible after each given one. This looks more like an odd kind of reinforced random walk to me (and these are notoriously difficult to analyze). | |
| Dec 10, 2010 at 18:47 | comment | added | Did | Sweet problem! Are the random angles $\theta_i$ independent and uniform on the interval $(-\pi,+\pi)$? | |
| Dec 10, 2010 at 13:11 | history | asked | Joseph O'Rourke | CC BY-SA 2.5 |