The.Algorithm.Design.Manual.Springer-Verlag.1998

Longest Increasing Sequence
Next: Minimum Weight Triangulation Up: Dynamic Programming Previous: Approximate String
Matching
Longest Increasing Sequence
Hopefully, a pattern is emerging. Every dynamic programming solution has three components:
1. Formulate the answer as a recurrence relation or recursive algorithm.
2. Show that the number of different values of your recurrence is bounded by a (hopefully small)
polynomial.
3. Specify an order of evaluation for the recurrence so you always have the partial results you need
available when you need them.
To see how this is done, let's see how we would develop an algorithm to find the longest monotonically
increasing sequence in a sequence of n numbers. This problem arises in pattern matching on
permutations, as described in Section . We distinguish an increasing sequence from a run, in that the
selected elements need not be neighbors of each other. The selected elements must be in sorted order
from left to right. For example, consider the sequence
The longest increasing subsequence of has length 3 and is either (2,3,4) or (2,3,6). The longest increasing
run is of length 2, either (2,8) or (1,6).
Finding the longest increasing run in a numerical sequence is straightforward, indeed you should be able
to devise a linear-time algorithm fairly easily. However, finding the longest increasing subsequence is
considerably trickier. How can we identify which scattered elements to skip? To apply dynamic
programming, we need to construct a recurrence computing the length of the longest sequence. To find
the right recurrence, ask what information about the first n-1 elements of S, coupled with the last element
, would enable you to find the answer for the entire sequence?
● The length of the longest increasing sequence in seems a useful thing to know. In
fact, this will be the longest increasing sequence in S, unless extends it or another increasing
sequence of the same length. Unfortunately, knowing just this length is not enough. Suppose I
told you that the longest increasing sequence in was of length 5 and that .
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