while trying to implement a parallel QR decomposition, I was faced with a type of "sequences of integer series" which exceeds my mathematical understanding. So I'm asking You for help.
For a lack of a better name, I've dubbed the problem the "Turte Race", since it somewhat losely reminds me of Zeno's paradoxes. Imagine a race track with:
- $N \in \mathbb{N}$ lanes and
- a (maximum) track length of $M \in \mathbb{N}$.
In each lane there's a turtle participating in the "race". The race is "turn-based", i.e.:
- the time $t \in \mathbb{Z}_{\geq 0}$ takes discrete steps.
The progress of $p$ of the turtle in lane $n \in \{-1, 0, 1, \ldots, N-1\}$ at time $t$ is discrete as well and is denoted as:
$$p_{n,t} \in \{0, 1, \ldots, M-n-1\}$$
For simplicity's sake, there is a pretend turtle that already finished the race with a progress of $M$ at $n=\left(-1\right)$, i.e.:
$$p_{\left(-1\right),t} = M$$
All non-pretend turtles start with a progress of zero:
$$p_{n,0}=0 \text{ for } n \geq 0$$
During each turn of the race, each turtle is advancing by the current floored half of it's distance to its left neighbor:
$$p_{n,t+1} = p_{n,t} + \lfloor \frac{p_{n-1,t} - p_{n,t}}{2} \rfloor \text{ for } n \geq 0, t \geq 0$$
The "race" is finished at time $T$, when the race comes to a halt i.e.:
$$T = T_{M,N} = \min_{t} \{ t \in \mathbb{Z}_{\geq 0} : p_{n,t} = (M-n-1) \ \forall \ n \in \{0,1,\ldots,N-1\}\ \}$$
Since this all quite abstract, I've created a little HTML/JavaScript file where You can play around with different lane counts, track lengths and time steps (works best in Firefox). If You're familiar with JavaScript, You can download the page's source and play around with it as well.
What I am looking for:
A formula for $T_{M,N}$ or an good upper bound of $T_{M,N}$ that can be computed as fast as possible, ideally in $\mathcal{O}\left(\log{M}+\log{N}\right)$ time with $\mathcal{O}\left(M_{max} + N_{max}\right)$ memory.What I could find out about the problem so far:
- For a race with a single lane $N=1$, the race duration $T$ is given by: $$T_{M,1} = \lceil\log_2{M}\rceil \leq T_{M,N}$$
- For $M=N$, the series: $$T_{1,1},T_{2,2},T_{3,3}\ldots = 0, 1, 3, 4, 6, 8, 10, 11, \ldots$$ seems to correspond to the follwing series in OEIS.
- At least for $N \leq M \leq 1000$, the following is an upper bound: $$T_{M,N} \leq \min\{\lceil\log_2{M}\rceil n, \left(\frac{\lceil\log_2{M}\rceil^2}{3} + 2n\right), \left(2m - \lceil\log_2{M}\rceil - max\{M-N,0\}\right)\ \}$$ Sadly I have no insights/proofs as to wether or not that is the case for larger $M$.
This is a tough nut to crack... at least for me. Any insights You can give me about this problem would be greatly appreciated.
