Computer Science 340
Reasoning about Computation
Review Session 2
Friday, January 11, 2008
Problem 1
Recall the cow grazing problem from Homework 5. The cow is on a path that leads to
food, but the cow doesn’t know what direction the food lies in. The goal was to design an
online algorithm for the cow to traverse the path and find the food.
The deterministic algorithm we proposed in the solution was the following.
The cow
starts in an arbitrary direction for one unit of distance. If it does not find food, it comes
back to the starting position and tries the other direction going 2 units of distance. Each
time it comes back to the start position, it goes in the other direction for twice the distance
it tried before. We argued that the competitive ratio of this algorithm was 9. Several of you
got this, or some variant of this idea.
Next we asked you to design a randomized algorithm with better competitive ratio.
Several students proposed randomizing the previous deterministic algorithm by picking a
direction to traverse at random every time the cow returns to the origin. So, the cow goes
distance 2
i

1
in the
i
th iteration (and returns to the start if unsuccessful), but the direction
of travel is picked at random at every iteration.
Determine the competitive ratio of this
algorithm.
Solution:
The algorithm is not competitive. Suppose that the field is at position 1. Compute the
expected distance traveled by the algorithm in this case. The expected distance is
∞
. The
expression for expected distance turns out to be
1
2
(1) +
1
4
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 Fall '07
 CharikarandChazelle
 Graph Theory, Analysis of algorithms

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