Traveling salesman part two

 Consider my 2D map of city points arbitrarily connected.

I lower the resolution of my map, always computing local density by region. 

If I consider partitoned regions by density, then my sales cost of visiting sparse regions is high. My total cost for the region is the frequency of visit times cost.

Cost is the number of steps to get through the region. That is, I can find a sales stop in log2(density) steps. I can say this because the region is equal density and I can systematically take lefts and rights.

So total cost in a complete search of all routes is -iLog2(i) where I nave some small number is i, each being the proportion of cities in that equal density region.  How many of the i densities do I have?  Consider the case I have 256 cities on my map and they are a perfect Gaussian fuzz ball centered on my map.  My selector is a tree of rank seven. I can pick 1 of 256 at random and select on of the cities at the proper rate of 2. This is the 2D case.  

  If the sales stop map is oval but gaussian then my regions have another dimension.  I have two colors. Then I remove skew, then kurtosis until the moments are less than the expected chaos of the streets. At that point I can do monte carlo simulations of typical paths. If I minimize total cost I am maximizing entropy and I should be able to transform a map with M moments into the best gaussian approximation.  But my total searches goes up exponentially.

So we pick a small number of density quants. The salesman takes his quantized map and enter a region that is under served. He processes some sales until he reaches an exit. Wash rinse and repeat. He will find the shortest map with an error because the number sales stop per region is not a multiple of the number of regional visits.  He will have to take small detours in each region at he end of each sales trip.

We can bound that round off error, i is elated to the number of moment we cancelled on the city nap.

Let us add competing sales people. Everything the same, the same number of colors, each reagion manages the coloring for a fixed number pre regional density.  On entering a region the salesman faces a trading pit. He can stay and service sales calls, earning paying interest, or he can deposit and earn interest on  exit.  In this case, all the shortest paths become optimally congested.   We are using the standard S/L for congestion pricing.  This is a good model of the economy and the sandbox. It is what our new Nobels do, it is the completion of agglomeration theory. It is Markov theory, it is path thinning in AI decision trees. and it is standard logistics in supply chains.