3.32 Optimal district design in
terms of city blocks
In this problem we shall examine the question of optimal district
design for cases in
which the dimensions of a district can assume only integer values
due to the grid
structure of streets.
Suppose now that we wish to design a district so as to minimize E[D], subject to the constraint that the area of the district is greater than or equal to some value A (i.e., mn A). (It is not reasonable to set an equality constraint since m and n can take only integer values.) For instance, if A = 44, we wish to find n and m such that E[D] is minimized and the district contains at least 44 city blocks.
Hint: Q is symmetric in n and m.
STEP 1: Set i = [A], where [x] = smallest integer greater than or equal to x.
STEP 2: Set j = i - 1 if i(i - 1) A; otherwise, set j = i.
STEP 3: Set k = i + 1, = j - 1.
STEP 4: If kt A, set i = k, j = 1; return to Step 3.
Otherwise, stop; the
optimal district dimensions are n = i,