A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 16.1 Dynamic Programming 533dynamic control systems, which got its start before what we think of as computerprogramming. The act of storing precomputed values in a table for later reuse isreferred to as “programming” in that field.Dynamic programming is a powerful alternative to the standard principle ofdivide and conquer. In divide and conquer, a problem is split into subproblems,the subproblems are solved (independently), and then recombined into a solutionfor the problem being solved. Dynamic programming is appropriate whenever thesubproblems to be solved are overlapping in some way and if we can find a suitableway of doing the necessary bookkeeping. Dynamic programming algorithms areusually not implemented by simply using a table to store subproblems for recursivecalls (i.e., going backwards as is done by Fibrt). Instead, such algorithms typicallyimplemented by building the table of subproblems from the bottom up. Thus,Fibi better represents the most common form of dynamic programming than doesFibrt, even though it doesn’t need the actual table.16.1.1 The Knapsack ProblemWe will next consider a problem that appears with many variations in a varietyof industrial settings. Many businesses need to package items with the greatestefficiency. One way to describe this basic idea is in terms of packing items intoa knapsack, and so we will refer to this as the Knapsack Problem. First definea particular form of the problem, and then discuss an algorithm for it based ondynamic programming. We will see other versions of the problem in the exercisesand in Chapter 17.Assume that we have a knapsack with a certain amount of space that we willdefine using integer value K. We also have n items each with a certain size suchthat that item i has integer size k i . The problem is to find a subset of the n itemswhose sizes exactly sum to K, if one exists. For example, if our knapsack hascapacity K = 5 and the two items are of size k 1 = 2 and k 2 = 4, then no suchsubset exists. But if we have add a third item of size k 3 = 1, then we can fill theknapsack exactly with the first and third items. We can define the problem moreformally as: Find S ⊂ {1, 2, ..., n} such that∑k i = K.i∈SExample 16.1 Assume that we are given a knapsack of size K = 163and 10 items of sizes 4, 9, 15, 19, 27, 44, 54, 68, 73, 101. Can we find asubset of the items that exactly fills the knapsack? You should take a fewminutes and try to do this before reading on and looking at the answer.