What is dynamic programming?
“Programming”: has no particular connection to computer programming. It actually comes from
the term "mathematical programming" (e.g. linear programming), a synonym for optimization.
Here program means a plan of some events. For example, in a conference, the final schedule of the
events is sometimes called a program.
Motivating example: finding shortest path
To find the shortest path from start to goal, we have to go to either A, B, or C. If we are greedy, we
should go to the closest city, B. But this may not be the shortest path overall. Suppose that we
have already found the shortest paths from A to goal, B to goal, and C to goal, we could calculate
the shortest path from start to goal by:
SP(start, goal) = min {Dist(start, x) + SP(x, goal)}, x = A, B, C
We can use the same idea to get SP(x, goal), except that we are now closer to our goal than we
originally were.
Good idea. But does it always work?
Not necessarily. Depending on how the smaller sub-problems can be solved and used.
Example 1
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,
6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, …
F(n) = F(n-1) + F(n-2), F(0) = F(1) = 1.
Analytically: F(n) ~ 1.618^n / 1.382
Pseudo-code for computing the n-th Fibonacci number by recursion:
function fib(n)
if (n == 0 or n == 1)
return 1;
else
return fib(n-1) + fib(n-2)
If we call fib(5), then:
fib(5)
= fib(4) + fib(3)
= (fib(3)+fib(2)) + (fib(2)+fib(1))
= (fib(2)+fib(1)) + (fib(1)+fib(0)) + ((fib(1)+fib(0)) + 1)
