Adam, that's very cool! Another not-so-long-tail interesting case is i^(-3/2). I can't think of an elegant enumeration. "not-so-long-tail" <=> integral converges. 4-house case: quite a steep dropoff (needs to be unbunched): 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 27 total visits ~ 1 month cycle. --- BTW, I know nothing of "Jim Propp's de-randomization tricks". They sound quite interesting. Any references/links? At 12:27 PM 3/17/2018, Adam P. Goucher wrote:

For the n^-2 case, a beautiful aperiodic solution is to visit house gcd(a, b) at time t, where T(a) is the largest triangular number below t, and b = t - T(a).

Best wishes,

Adam P. Goucher

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Sent: Saturday, March 17, 2018 at 4:53 PM From: "Michael Kleber" <michael.kleber@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Deterministic scheduling of Zipf's Law visits ?

This sounds like a problem for Jim Propp's de-randomization tricks -- converting a random method that has the right asymptotic distribution to a deterministic method that approximates its behavior. But I don't know what the random method with Zipfian distribution would be.

--Michael

On Sat, Mar 17, 2018 at 11:07 AM, Henry Baker <hbaker1@pipeline.com> wrote:

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Suppose that I'm a Fedex driver and I have N houses to visit, but only exactly 1 house every day (i.e., this is NOT a route planning problem). Since I'm an unimaginative Fedex driver, it's simply easier to go to the houses in the same order every cycle of days, whether or not I have something to deliver.

I'd like to come up with a *simple* house visit scheduling algorithm with the following properties:

* For each N, the output is a *fixed* *cyclic* schedule; i.e., after some # of days the entire schedule repeats; we'd like the cycle length to be relatively short and will accept some amount of deviation from Zipf's Law in return

* The % of visits to house #i in each cycle is very roughly proportional to 1/i

* Ideally, the deliveries to house #i should be spread out as much as possible -- i.e., not all bunched together on successive days

* The schedule for N+1 houses is simply computed from the schedule for N houses (alternatively, the schedule for N is computed from an a priori fixed number of schedules from smaller N's)

Here's an example of what NOT to do for Zipf's Law:

Suppose we have 4 houses, with visits scheduled like so:

1 1 2 1 2 3 1 2 3 4

Thus, we have 10 total visits each cycle, but the ratios are 40%, 30%, 20%, 10% -- i.e., this is a linear "tail", rather than a Zipf's Law tail.

Is there such an algorithm which approximates a Zipf's Law distribution?

What about other power laws/long-tail distributions -- e.g., i^(-2) ? Other common distributions? -- Forewarned is worth an octopus in the bush.