For local networks, it often is necessary to use rectilinear or
``checkerboard'' distances, instead of straight Euclidean lines.
distances in this metric are computed as shown in figure 2.
Suppose you wish to design a minimum cost spanning tree for a local network for 9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0), i(10,3). You are restricted to using rectilinear lines. Moreover, all ``phantom'' stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length.
Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day.
The table gives the time, in seconds, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of the water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.