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SnowWalkers

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Dennis Shasha

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Omniheurist Course

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Computer Science

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*
Description
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Grid City is a small planned city laid out at a completely regular N by M grid
with streets going north-south and east-west.
There is one building per city block.
Grid occasionally
suffers major snow storms.
The Grid City Snow Clearing Department (GridClear)
wishes to make it possible to reach each
city block but the effort to move the snow is so great that
they resolve to make it possible to go from any block to any other
through paved paths.
This means that the path itself may snake through the city
and may even loop through
an intersection, but the plows won't double-back on a paved road
for fear of running over the many pedestrians.

The head of GridClear consults you to help plan the path.
You are told the path must start
somewhere on the outside boundary of the grid, but you
can choose where,
because Grid City has many garages available.
The goal is to ensure that for any two blocks that neighbor
one another north-south or east-west, a resident will
need to travel over only a few streets (or through a few buildings)
along the cleared path.
The "score" of a path is the worst case, i.e. the largest number of
such streets/buildings.
Crossing a plowed intersection costs nothing, but it is impossible
to cross an unplowed intersection or street.

You may assume that each building block has an entrance at every corner
and that walking through a building to any other corner (even
to the diagonally opposite one) costs
the same as walking one city block along a plowed street.
You accept this simplication, because there is no ice inside the buildings.

In what follows, you may find it helpful
to look at the very nice
applet of Arefin Huq
with whom I have worked
on this puzzle.

Players will be told M and N (the dimensions
of the grid), how many plows they have (up to and including 4),
and be given an upper bound (worst case)
of the score on the whole grid.
They are to obtain that score in the shortest time cost possible,
where the time cost is the maximum number of streets plowed by any
single plow.

Here are some examples along with solutions in the
human-solvable
version of this puzzle

## Architecture

The architecture group will extend Arefin Huq's
applet
to display the board,
reorganize the game to allow TCP connections, keep track of the plow
that takes the longest route.