Consider the following process: we place $n$ points labelled $1...n$ uniformly at random on the interval $[0,1]$. At each time step, two points $i, j$ are selected uniformly at random and $i$ updates its position to be a point chosen uniformly at random in the interval between the positions of $i$ and $j$ (so the interval $[p(i),p(j)]$ if $p(i) < p(j)$ or $[p(j),p(i)]$ otherwise, where $p(x)$ denotes the position of the point labelled $x$).

  1. What is the expected time until all points are within distance $\varepsilon$ of each other for some fixed $\varepsilon > 0$?

  2. What is the expected time until all points are either to the left or right of $\frac{1}{2}$?

Asymptotic bounds are also very interesting to me.

  • $\begingroup$ You mean to the left or right of $\frac12$? $\endgroup$ – joriki Dec 19 '19 at 1:09
  • $\begingroup$ @joriki Yes, good catch :) $\endgroup$ – michael Dec 19 '19 at 23:48
  • $\begingroup$ Will we always have $n$ points? I don't know if I understood correctly, what you do is to uniformly choose a point k between $i$ and $j$ (assuming $i <j$) and replace $i$ with $k$. I think the question can be better written, you consider $X_{1}, X_{2},\ldots, X_{n} \sim\mathrm{Unif}[0,1]$ i.i.d, then you take a random pair $(i, j)$, then you choose a $U\sim \mathrm{unif} (X_{i},X_{j})$ and you do $X_{i} = U$. $\endgroup$ – Diego Fonseca Dec 20 '19 at 2:00
  • $\begingroup$ @DiegoFonseca I've tried rewriting the question to make it clearer, I think your interpretation is a little different from what I intended. $\endgroup$ – michael Dec 20 '19 at 16:41
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    $\begingroup$ @mathworker21 Yes, only $i$ updates its position, as the question states. (As in, two points are chosen one after the other, and only the first one will update its position. If the same point is chosen twice nothing happens.) $\endgroup$ – michael Dec 21 '19 at 0:06

Update: I'll prove that the expected time in question 1 (for fixed $\varepsilon$) is $\Theta(n \log n)$, and that the expected time in question 2 is $\geq \Omega(n \log n)$; at the moment I don't have a good upper bound for question 2.

Upper bound:

Define $S = \sum_{1 \leq a, b \leq n} (p(a) - p(b))^2$, and let $S_t$ be the value of $S$ at time $t$. Now, at a fixed time $t$, let $p$ denote the positions at time $t$, and $p'$ denote the positions at time $t+1$. Then $p'$ is determined from $p$ by independently picking $(i, j)$ uniformly from $\{1, \dots, n\}^2$ and $\lambda \sim \mathrm{Unif}[0, 1]$, and setting $p'(a) = p(a)$ for all $a \neq i$, and $p'(i) = \lambda p(i) + (1-\lambda) p(j)$. Then we have \begin{align*} S_t - S_{t+1} &= 2 \sum_{a \neq i} \left((p(i) - p(a))^2 - (p'(i) - p(a))^2\right) \\ &= 2(1-\lambda^2)(p(i) - p(j))^2 + 2 \sum_{a \neq i, j} \left((p(i) - p(a))^2 - (p'(i) - p(a))^2\right) \\ &= 2(1-\lambda^2)(p(i) - p(j))^2 + 2 \sum_{a \neq i, j} (p(i) - p'(i))(p(i) + p'(i) - 2p(a)) \\ &= 2\left(1-\lambda^2 + (n-2)\lambda(1-\lambda)\right)(p(i) - p(j))^2 \\ &\qquad+ 2 (1-\lambda) \sum_{a \neq i, j} (p(i) - p(j))(p(i) + p(j) - 2p(a)) \end{align*} There is a slight technicality here in that it appears we have assumed $i \neq j$, but in fact these expressions are still equal in the case $i = j$: the LHS is $0$ since then $S_{t+1} = S_t$, and the RHS is $0$ since $p(i) = p(j)$. Now, upon taking expectations conditional on $S_t$, the second term vanishes because it's antisymmetric in $i, j$ (i.e. it is negated when we swap $i, j$), so we have \begin{align*} S_t - \mathbb{E}[S_{t+1} | S_t] &= \mathbb{E}\bigg[ 2\left(1-\lambda^2 + (n-2)\lambda(1-\lambda)\right)(p(i) - p(j))^2 \,\bigg|\, S_t \bigg] \\ &= 2\,\mathbb{E}\big[1-\lambda^2 + (n-2)\lambda(1-\lambda)\,\big|\, S_t\big] \mathbb{E}[(p(i) - p(j))^2 \,|\, S_t] \\ &= 2\left(\frac{n+2}{6}\right)\left(\frac{S_t}{n^2}\right) \end{align*} (where the second equality follows by independence), hence $$\mathbb{E}[S_{t+1}|S_t] = \left(1 - \frac{n+2}{3n^2}\right)S_t$$ which means, in particular, that $\mathbb{E}[S_{t+1}] = \left(1 - \frac{n+2}{3n^2}\right)\mathbb{E}[S_t]$.

We can now use this to get an upper bound for question 1. Note we have $\mathbb{E}[S_0] \leq n^2$ and $\mathbb{E}[S_{t+1}] \leq e^{-1/3n} \mathbb{E}[S_t]$, so $\mathbb{E}[S_t] \leq n^2 e^{-t/3n}$, and thus the probability that there are two points which are at least $\varepsilon$ apart at time $t$ is at most $(n^2 / \varepsilon^2) e^{-t/3n}$ (by Markov's inequality, since if this holds we must have $S_t \geq \varepsilon^2$). Letting $T_1$ be the time at which all points are within distance $\varepsilon$ of each other, this means $\mathbb{P}[T_1 > t] \leq (n^2 / \varepsilon^2) e^{-t/3n}$, hence $$\mathbb{E}[T_1] = \sum_{t=0}^\infty \mathbb{P}[T_1 > t] \leq \sum_{t=0}^\infty \min \{(n^2 / \varepsilon^2) e^{-t/3n}, 1\},$$ which we can approximate as $$6n \log(n / \varepsilon) + \int_{6n\log(n/\varepsilon)}^\infty (n^2 / \varepsilon^2) e^{-t/3n} \,dt = 6n\log(n / \varepsilon) + 3n.$$ Therefore $\mathbb{E}[T_1] \leq O(n \log(n/\varepsilon))$.

Lower bounds:

I should have thought of this before -- we can actually get some basic coupon-collector-type lower bounds for both questions. I'm not really optimizing for good constants below.

Lemma: Suppose we have two disjoint sets $A, B \subset \{1, \dots, n\}$ with $|A|, |B| \geq k$. At each time step we choose a uniformly random element of $\{1, \dots, n\}$. Then the expected time until either all elements of $A$ have been chosen at least once or all elements of $B$ have been chosen at least once is at least $(n/2) \log k$.

Proof: This is essentially the same as the proof for the coupon collector's problem found here. Let $a_1, \dots, a_k$ be distinct elements of $A$, and $b_1, \dots, b_k$ distinct elements of $B$, and say that the pair $(a_i, b_i)$ is hit if one of $a_i, b_i$ is chosen. Note that our condition -- that all elements of $A$ are chosen or all elements of $B$ are chosen -- is satisfied only if all pairs $(a_1, b_1), \dots, (a_k, b_k)$ are hit. Let $t_r$ be the time until the $r$-th pair is hit after $r-1$ pairs are hit. After $r-1$ pairs are hit, the probability of hitting a new pair is $\frac{2(k-r+1)}{n}$, hence $t_r$ has geometric distribution with expectation $\frac{n}{2(k-r+1)}$, and the expected time until all pairs are hit is thus at least $$\sum_{r=1}^k \mathbb{E}[t_r] = \frac{n}{2} \sum_{r=1}^k \frac{1}{k-r+1} = \frac{n}{2} \sum_{r=1}^k \frac{1}{r} \geq \frac{n}{2} \log k.$$

Note that for question 1, at time $T_1$ all points lie in some interval $I$ of length $\leq \varepsilon$, so all points initially outside of $I$ have moved, i.e. all points initially outside of $I$ have been chosen as the $i$-value at some time-step. Defining $I^- = [0, (1+\varepsilon)/2]$ and $I^+ = [(1-\varepsilon)/2, 1]$, this interval necessarily satisfies either $I \subset I^-$ or $I \subset I^+$. Letting $A$ be the set of points outside of $I^-$ at $t = 0$, and $B$ be the set of points outside of $I^+$ at $t = 0$, this means that at time $T_1$, either every element of $A$ has been chosen as $i$ or every element of $B$ has been chosen as $i$. The complement of each of $I^-$ and $I^+$ is an interval of length $\frac{1 - \varepsilon}{2}$, hence since the points are i.i.d. uniform on $[0, 1]$ at $t = 0$, we have $|A|, |B| \geq \frac{1 - \varepsilon}{3} n$ with probability $1 - o(1)$. Conditioning on the arrangement of points at $t = 0$, by the Lemma the conditional expectation has $\mathbb{E}[T_1 \mid p|_{t = 0}] \geq \frac{1}{2} n \log (\frac{1-\varepsilon}{3}n)$ when $|A|, |B| \geq \frac{1 - \varepsilon}{3} n$, hence the unconditional expectation has $\mathbb{E}[T_1] \geq \frac{1}{2} (1 - o(1)) n \log(\frac{1-\varepsilon}{3}n) = \Omega(n \log n)$.

Similarly, for question 2, let $A$ be the set of points in $[0, 1/2)$ at $t = 0$, and $B$ be the set of points in $(1/2, 1]$ at $t = 0$. Then at time $T_2$ (when all points lie on one side of $1/2$), either all points in $A$ have moved, or all points in $B$ have moved. Since with probability $1 - o(1)$ we have $|A|, |B| \geq n/3$, the same argument as above gives $\mathbb{E}[T_2] \geq \frac{1}{2} (1 - o(1)) n \log(n/3) = \Omega(n \log n)$.

Computational results:

I don't have an upper bound for question 2 right now, but I ran some simulations of the process $50$ times for each of $n = 10, 11, \dots, 250$. The estimates for $\mathbb{E}[T_2]/n$ are plotted below.

question 2

Based on this, $\mathbb{E}[T_2]/n$ appears to be linear in $\log n$, suggesting the possibility of a matching upper bound for $\mathbb{E}[T_2]$. Least squares gives a best-fit line of $\mathbb{E}[T_2]/n \approx 5.75 \log n - 7.19$.

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  • $\begingroup$ wait, this is a proof that $\mathbb{E}[T] \le O(n\log(n/\epsilon))$, right? If so, you should say that at the start of your answer. also, is $n\log(1/\epsilon)$ a trivial lower bound? something like that feels like a trivial lower bound. $\endgroup$ – mathworker21 Dec 21 '19 at 11:05
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    $\begingroup$ I came up with $O(n \log(1/\varepsilon))$ via the intuition that at time $T$ we'd probably have $S_t \approx n^2 \varepsilon^2$ rather than $S_t \approx \varepsilon^2$, which only takes $O(n \log(1/\varepsilon))$ steps. that said I haven't been able to prove it's a lower bound; maybe I'm missing something obvious. It seems like you would need to find a way to get a good lower bound on the probability that some points are at least $\varepsilon$ apart at time $t$ $\endgroup$ – user125932 Dec 21 '19 at 19:01
  • $\begingroup$ can you answer the first question in my comment above $\endgroup$ – mathworker21 Dec 22 '19 at 1:23
  • $\begingroup$ right, it is a proof that $\mathbb{E}[T] \leq O(n \log (n/\varepsilon))$ -- I had already edited the answer to say that when I replied $\endgroup$ – user125932 Dec 22 '19 at 1:47
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    $\begingroup$ thanks. I did some coding and it appears that at time $T$, $S$ is around $n^{3/2}$ (for fixed $\epsilon$ and large $n$). I found this rather surprising. Unfortunately, I'm not able to see what bound this gives for OP's question 1. Are you able to? (assuming the $n^{3/2}$ thing is true). $\endgroup$ – mathworker21 Dec 25 '19 at 8:46

This isn't an answer but I did some numerical simulations in order to at least check out what would happen.

Actual Data

For these, I ran $1000$ different random sequences for $n=2,3,\cdots 30$ and $\epsilon=1/2^k$ for $k=1,2,\cdots 6$. From this, I would conjecture that a terrible upper bound on this sequence is given by $(kn)^2$, where $k$ is related to $\epsilon$ by

$$k=\frac{\log \left(\frac{1}{\epsilon}\right)}{\log (2)}$$

Upper Bound

while an asymptotic bound might be given by $(kn)^{5/4}$

Asymptotic Bound

Note that in both of these graphs the black graph is the actual data while the red graph is the conjectured bound.

For your second question, I proceeded in much that same manner except I made $2000$ random sequences instead of $1000$. However, the results were much the same as before


where the black plot is the actual data while the red plot is $n^2$. Again, it would seem a crude upper bound is $n^2$.

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    $\begingroup$ The large-$n$ behavior of that last data set might be more apparent in a log-log plot. $\endgroup$ – Semiclassical Dec 20 '19 at 17:59
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    $\begingroup$ It'd be helpful to add axis labels to these plots, I think. $\endgroup$ – cdipaolo Dec 20 '19 at 20:57

Exact solution for $n=2$. Consider the process when the points are at $(a, b)$ with $0 < a < b < 1$ and a target of $\epsilon$. It is equivalent to a process with points $(0, 1)$ and a target of $\frac{\epsilon}{a-b}$. With this observation, let $f(t)$ be the expected number of steps when points are at $(0, 1)$ and the target is $\epsilon$. The next step is terminates the process with probability $\epsilon$. If it doesn't terminate, the remaining interval has a width $U(\epsilon, 1)$. So $f$ must satisfy the following recurrence

$$f(t) = 1 + \int_t^1 f(t/x) \mathrm{d}x$$ which is satisfied by $f(t) = 1 - \ln(t).$ The initial points are both $U(0, 1)$, so the expected number of steps with $n=2$ is given by

$$\int_0^1 (2- 2w)\left(1 - \ln\left(\frac{\epsilon}{d}\right)\right) \mathrm{d}w \\ = \frac{\epsilon^2-1}{2} + \log{\left(\frac{1}{\epsilon} \right)}$$ Which matches experiments. case n=2

For larger $n$ this gives the "trivial" lower bound of $O(-n \log{\epsilon})$ since for larger $n$, the smalest and largest points are close to $(0, 1)$ and they are moves on average every $2/n$ steps.

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  • $\begingroup$ a trivial bound of big O something? cmon... $\endgroup$ – mathworker21 Dec 27 '19 at 10:04

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