Timeline for Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Current License: CC BY-SA 4.0
13 events
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| Nov 17, 2020 at 8:15 | history | edited | sebastian | CC BY-SA 4.0 |
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| Nov 17, 2020 at 8:04 | comment | added | sebastian | Ah, that makes sense. Yes, I misunderstood you, I'll change it. | |
| Nov 16, 2020 at 20:27 | comment | added | Penelope Benenati | Hence there is a factor $2$ between the number of pairs we were analyzing, which should not be a big deal anyway (I think it's my fault because I used the notation $(x,y)$ instead of $\{x,y\}$).(2) You also think that the same point $x$ alone can form a pair by being used twice, i.e. $(x,x)$. However I was talking about each pair as formed by two distinct points $x,y\in S$, which can possibly be coincident (for instance we can have $x=0.3$, $y=0.3$ and $(x,y)$ is a pair of points taken from $S$ -- "without replacement"). Just some little misunderstanding,I should have been clearer. | |
| Nov 16, 2020 at 20:26 | comment | added | Penelope Benenati | Thank you for your edit and your comment! About the expression $i^2+(n-i)^2$, you defined $i$ as the the number of points smaller than $p$. You wrote that the set $S_{<p}$ contains all pairs of points smaller than $p$, and its cardinality is equal to $i^2$. If $i=1$, the number such pairs is $0$. If $i=2$ the number of such pairs is $1$. Hence, I am understanding now that (1) when you talk about pairs, you mean ordered pairs, while I was meaning unordered pairs. | |
| Nov 16, 2020 at 17:42 | comment | added | sebastian | Thanks for your comments. I have added an edit for the case where the points are distributed equidistantly, but I didn't get so far yet. I'll let you know if I get anywhere (also wrt your conjecture about monotonicity). Considering the line "Thus $S_p$ contains $i^2 + (n - i)^2$ pairs": My idea was that the size of $S_{<p}$ is $i^2$ and the size of $S_{>p}$ is $(n - i)^2$. Together they make up $S_p$. Let me know if you still think there is a mistake, I don't see it yet. | |
| Nov 16, 2020 at 17:36 | history | edited | sebastian | CC BY-SA 4.0 |
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| Nov 16, 2020 at 13:51 | comment | added | Penelope Benenati | I feel confident that $m(n)$ is monotonically decreasing in $n$. @araomis, do you have any idea related to your expression about how to prove this property? | |
| Nov 16, 2020 at 0:12 | comment | added | Penelope Benenati | Thank you very much for your answer. The approach is clear and looks correct (perhaps there is a typo when you wrote "Thus $S_p$ contains exactly $i^2 + (n - i)^2$ pairs" if I am not wrong, anyway nothing really important). The difficult part is maximizing the last expression over all $s_i$ for $1\le i\le n$. Numerical simulations suggests that we obtain $0.40$ for $n=4$, $0.31$ for $n=8$, less than $0.29$ for $n=16$. Furthermore, the distance between two consecutive points seems to be equal up to $1/(n-1)$ up to a (small) constant factor, and we always have $s_1=0$ and $s_n=1$. | |
| Nov 15, 2020 at 11:15 | history | undeleted | sebastian | ||
| Nov 14, 2020 at 13:24 | history | deleted | sebastian | via Vote | |
| Nov 11, 2020 at 6:59 | history | edited | sebastian | CC BY-SA 4.0 |
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| Nov 10, 2020 at 22:42 | review | First posts | |||
| Nov 11, 2020 at 0:03 | |||||
| Nov 10, 2020 at 22:40 | history | answered | sebastian | CC BY-SA 4.0 |