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Returning to our discussion of longest increasing subsequences: the formula for L(j) also suggests an alternative, recursive algorithm Wouldn t that be even simpler Actually, recursion is a very bad idea: the resulting procedure would require exponential time! To see why, suppose that the dag contains edges (i, j) for all i < j that is, the given sequence of numbers a1 , a2 , , an is sorted In that case, the formula for subproblem L(j) becomes L(j) = 1 + max{L(1), L(2), , L(j 1)}
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The following gure unravels the recursion for L(5) Notice that the same subproblems get solved over and over again!
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For L(n) this tree has exponentially many nodes (can you bound it ), and so a recursive solution is disastrous Then why did recursion work so well with divide-and-conquer The key point is that in divide-and-conquer, a problem is expressed in terms of subproblems that are substantially smaller, say half the size For instance, mergesort sorts an array of size n by recursively sorting two subarrays of size n/2 Because of this sharp drop in problem size, the full recursion tree has only logarithmic depth and a polynomial number of nodes In contrast, in a typical dynamic programming formulation, a problem is reduced to subproblems that are only slightly smaller for instance, L(j) relies on L(j 1) Thus the full recursion tree generally has polynomial depth and an exponential number of nodes However, it turns out that most of these nodes are repeats, that there are not too many distinct subproblems among them Ef ciency is therefore obtained by explicitly enumerating the distinct subproblems and solving them in the right order
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The origin of the term dynamic programming has very little to do with writing code It was rst coined by Richard Bellman in the 1950s, a time when computer programming was an esoteric activity practiced by so few people as to not even merit a name Back then programming meant planning, and dynamic programming was conceived to optimally plan multistage processes The dag of Figure 62 can be thought of as describing the possible ways in which such a process can evolve: each node denotes a state, the leftmost node is the starting point, and the edges leaving a state represent possible actions, leading to different states in the next unit of time The etymology of linear programming, the subject of 7, is similar
63 Edit distance
When a spell checker encounters a possible misspelling, it looks in its dictionary for other words that are close by What is the appropriate notion of closeness in this case A natural measure of the distance between two strings is the extent to which they can be aligned, or matched up Technically, an alignment is simply a way of writing the strings one above the other For instance, here are two possible alignments of SNOWY and SUNNY: S S U N O W N N Cost: 3 Y Y S S U N O W N Cost: 5 N Y Y
The indicates a gap ; any number of these can be placed in either string The cost of an alignment is the number of columns in which the letters differ And the edit distance between two strings is the cost of their best possible alignment Do you see that there is no better alignment of SNOWY and SUNNY than the one shown here with a cost of 3 Edit distance is so named because it can also be thought of as the minimum number of edits insertions, deletions, and substitutions of characters needed to transform the rst string into the second For instance, the alignment shown on the left corresponds to three edits: insert U, substitute O N, and delete W In general, there are so many possible alignments between two strings that it would be terribly inef cient to search through all of them for the best one Instead we turn to dynamic programming A dynamic programming solution When solving a problem by dynamic programming, the most crucial question is, What are the subproblems As long as they are chosen so as to have the property (*) from page 159 it is an easy matter to write down the algorithm: iteratively solve one subproblem after the other, in order of increasing size Our goal is to nd the edit distance between two strings x[1 m] and y[1 n] What is a good subproblem Well, it should go part of the way toward solving the whole problem; so how about looking at the edit distance between some pre x of the rst string, x[1 i], and some pre x of the second, y[1 j] Call this subproblem E(i, j) (see Figure 63) Our nal objective, then, is to compute E(m, n) 161
Figure 63 The subproblem E(7, 5)