16.3-5 Give an 
algorithm to find the longest montonically
increasing sequence in a sequence of n numbers.
Ask yourself what would you like to know about the first n-1 elements to tell you the answer for the entire sequence?
.
(seems obvious)
will extend!
(not as obvious - this is the idea!)
Let 
be the length of the longest sequence ending with the
ith character:
| sequence | 5 | 2 | 8 | 7 | 3 | 1 | 6 | 4 | |
|
| 1 | 1 | 2 | 2 | 2 | 1 | 3 | 3 |
To find the longest sequence - we know it ends somewhere, so
Length = 
The Principle of Optimality
To use dynamic programming, the problem must observe the principle of optimality, that whatever the initial state is, remaining decisions must be optimal with regard the state following from the first decision.
Combinatorial problems may have this property but may use too much memory/time to be efficient.
Example: The Traveling Salesman Problem
be
the cost of the optimal tour for i to 1 that
goes thru each of the other cities once
Here there can be any subset of 
instead of
any subinterval - hence exponential.
Still, with other ideas (some type of pruning or best-first search) it can be effective for combinatorial search.
When can you use Dynamic Programming?
Dynamic programming computes recurrences efficiently by storing partial results. Thus dynamic programming can only be efficient when there are not too many partial results to compute!
There are n! permutations of an n-element set - we cannot use dynamic
programming to store the best solution for each subpermutation.
There are 
subsets of an n-element set - we cannot use dynamic
programming to store the best solution for each.
However, there are only n(n-1)/2 continguous substrings of a string, each described by a starting and ending point, so we can use it for string problems.
There are only n(n-1)/2 possible subtrees of a binary search tree, each described by a maximum and minimum key, so we can use it for optimizing binary search trees.
Dynamic programming works best on objects which are linearly ordered and cannot be rearranged - characters in a string, matrices in a chain, points around the boundary of a polygon, the left-to-right order of leaves in a search tree.
Whenever your objects are ordered in a left-to-right way, you should smell dynamic programming!
Minimum Length Triangulation
A triangulation of a polygon is a set of non-intersecting diagonals which partitions the polygon into diagonals.
We seek to find the minimum length triangulation. For a convex polygon, or part thereof:
Evaluation proceeds as in the matrix multiplication example -
values of t,
each of which takes O(j-i) time if we evaluate the sections in order
of increasing size.
Dynamic Programming and High Density Bar Codes
Symbol Technology has developed a new design for bar codes, PDF-417 that has a capacity of several hundred bytes. What is the best way to encode text for this design?
Originally, Symbol used a greedy algorithm to encode a string, making local decisions only. We realized that for any prefix, you want an optimal encoding which might leave you in every possible mode.
the cost of encoding the ith character
and ending up in node j).
Our simple dynamic programming algorithm improved to capacity of PDF-417
by an average of 
!
Dynamic Programming and Morphing
Morphing is the problem of creating a smooth series of intermediate images given a starting and ending image.
The key problem is establishing a correspondence between features in the two images. You want to morph an eye to an eye, not an ear to an ear.
We can do this matching on a line-by-line basis:
This algorithm was incorported into a morphing system, with the following results:
