#### CSCI 3104 - Algorithms - Spring 2010

### Problem Set #6

#### Due: 2pm, Apr 23rd, 2010

1. Text problem 6.1.
2. Text problem 6.4.

3. Text problem 6.8.

4. Text problem 6.22.

5.
A classic ice-breaking exercise is for a group of *n* people to
form a circle and then arbitrarily join hands with one another. This forms
a "human knot" since the players' arms are most likely intertwined. The
goal is then to unwind the knot to form a circle of players with no arms
crossed.

We now adapt this game to a more general and more abstract setting where
the physical constraints of the problem are gone. Suppose we represent the
initial knot with a
2-regular graph inscribed in a circle (ie, we have a graph with *n*
vertices with exactly two edges touching each vertex).
Initially, some edges may cross other edges and this is undesirable.
This is the "knot" we wish to unwind.

A "move" involves
moving any vertex to a new position on the circle, keeping its edges intact.
Our goal is to make the fewest possible moves such that we obtain an
*n*-sided polygon with no edge-crossings remaining.

For example, here is a knot on 4 vertices inscribed in a circle, but two
edges cross each other. By moving vertex 4
down to the position between 2 and 3, a graph without edge-crossings emerges.
This was achieved in a
single move, which is clearly optimal in this case.

When *n* is larger, things may not be quite as clear. Below we see a
knot on 6 vertices.
We might consider moving vertex 4 between 5 and 6, then
vertex 5 between 1 and 2, and finally vertex 6 between 3 and 4; this wins
in 3 moves.

But clearly we can solve the same knot in only two moves:

Give an efficient algorithm to solve this problem. You do not need
to code your solution. Although this is the chapter on Dynamic
Programming, you don't actually need DP here; but use one of our DP
algorithms as a subroutine to solve this problem. Your solution should
use O(n^{2}) time.