Algorithm Design and Analysis 3/18/2008
Lecture 24: More Dynamic Programming
Instructor: Mike Kowalczyk
1 Path Finding
Suppose that we have a terrain represented by an nby ngrid of positive numbers, each of which
represents the cost for Jeff’s robot to travel through that square. Jeff’s robot can choose to start
at any square on the bottom of the grid, and can move forward, diagonal left, or diagonal right.
The grid doesn’t “wrap around”, and the goal is for the robot to reach the top of the grid with the
lowest possible cost of movement.
We could try this by brute force, but how many possibilities would we need to consider? Brute
force gets us somewhere between n2n−1and n3n−1possibilities; way too time consuming. Greedy
algorithms probably won’t apply becuase there might be a “wall” of numbers to get through, and
something that doesn’t look too good at first may turn out to be a better global choice. Is this
problem in P? Can we use dynamic programing to find a polynomial time algorithm that outputs
such a path? For example, given the grid below, the progam should output something like this:
Total cost 108
Path is (3,3)->(2,3)->(1,2)->(0,1)
1 3 77 17
1003 5000 4 1
0 1007 1080 2
1 2 60000 99
1
