## gap frequencies [& e] A riddle from The Riddler where brute-force simulation does not pay:

For a given integer N, pick at random without replacement integers between 1 and N by prohibiting consecutive integers until all possible entries are exhausted. What is the frequency of selected integers as N grows to infinity?

A simple implementation of the random experiment is as follows

generope=function(N){
frei=rep(1,N)
hus=0
while (max(frei)==1){
i=sample(rep((1:N)[frei==1],2),1)
frei[i]=frei[i+1]=frei[i-1]=0
hus=hus+1}
return(hus)}


It is however quite slow and does not exploit the recursive feature of the sampling, namely that the first draw breaks the set {1,…,N} into two sets:

generipe=function(N){
if (N<2){ return((N>0))}else{
i=sample(1:N,1)
return(1+generipe(i-2)+generipe(N-i-1))}}


But even this faster solution takes a while to run for large values of N:

frqns=function(N){
space=0
for (t in 1:1e3) space=space+generipe(N)
return(space/(N*1e3))}


as for instance

>  microbenchmark(frqns(100),time=10)
Unit: nanoseconds
expr       min        lq         mean    median        uq       max
frqns(100) 178720117 185020903 212212355.77 188710872 205865816 471395620
time         4         8        26.13        32        37       102


Hence this limits the realisation of simulation to, say, N=10⁴. Thinking further about the recursive aspect of the process however leads to a recursion on the frequencies qN, as it is straightforward to prove that $q_N=\frac{1}{N}+\frac{2}{N^2}\,\sum_{i=1}^{N-2} iq_i$

with q1=1 and q2=1/2. This recursion can be further simplified into $q_N=\frac{1}{N^2}+\frac{2(N-2)}{N^2}\,q_{N-2}+\frac{(N-1)^2}{N^2}q_{N-1}\qquad(1)$

which allows for a much faster computation

s=seq(1,1e7) #s[n]=n*q[n]
for (n in 3:1e7) s[n]=(1+2*s[n-2]+(n-1)*s[n-1])/n


and a limiting value of 0.4323324… Since storing s does not matter, a sliding update works even better:

a=b=1
for (n in 3:1e8){ c=(1+2*a+(n-1)*b)/n;a=b;b=c}


still producing a final value of 0.4323324, which may mean we have reached some limit in the precision.

As I could not figure out a way to find the limit of the sequence (1) above, I put it on the maths forum of Stack Exchange and very quickly got the answer (obtained by a power series representation) that the limit is (rather amazingly!) $\dfrac{1 - e^{-2}}{2}$

which is 0.432332358.., hence very close to the numerical value obtained for n=3×10⁸. (Which does not change from n=10⁸, once again for precision reasons.) Now I wonder whether or not an easier derivation of the above is feasible, but we should learn about it in a few hours on The Riddler. [Update: The solution published by The Riddler is exactly that one, using a power series expansion to figure out the limit of the series, unfortunately. I was hoping for a de Montmort trick or sort of…]

### 3 Responses to “gap frequencies [& e]”

1. jonocarroll Says:

Awesome work getting the right answer. I added some optimisations to the initial code and my own version; brute-force can still go a long way towards the exact answer in a (semi)-reasonable amount of time.

• 