Consider the problem of storing n books on a bookshelf, where the length of each shelf is L. The order of the books is fixed and cannot be rearranged. Therefore, we can speak of a book b_i, where 1 i n, that has a thickness t_i and height h_i. Assume there is enough space to store all the books regardless of the book placement plan. Define the cost of a book placement plan to be the number of shelves that it uses.

Suppose all the books have the same height h (i.e. h = h_i = h_j for all i, j) and the vertical distances between two neighboring shelves are all greater than h, so any book fits on any shelf. A greedy algorithm to minimize the cost would fill the first shelf with as many books as we can until we get the smallest i such that b_i does not fit, and then repeat with subsequent shelves. Show that the greedy algorithm indeed minimizes the cost. Analyze the time complexity of the algorithm.