The first two steps of the development of a dynamic programming algorithm for a problem are

step 1 describe the structure of the subproblem

step 2 find a recurrence for the optimal value of the subproblem in terms of smaller subproblems

Perform just these two steps for this problem listed below. Do not write (pseudo) code - just the subproblem and recurrence structure.

You work for the OR state highway agency and must place warning signs along a certain road. Along the way there are n locations at which you may place a sign, at mileposts m ₁ < m₂ < < m_n, where each m_i is measured from the starting point m₁ = 0. The only places you are allowed to place a sign is at one of the given mileposts. In addition, you must place one at locations m₁ and m_n.

Your requirement is to place one every 100 miles, but this may not be possible (depending on the spacing of the mileposts). If you place two consecutive signs x miles apart, the penalty for that placement is (100x)². You want to arrange a placement so as to minimize the total penalty - that is, the sum, over all locations, of the penalties. Perform the two steps above to start the process of determining the minimum possible penalty.