federal government of CS-landia

The federal government of CS-landia has gotten word that Algotopia’s border walls are actually
virtual firewalls and will do nothing to contain the plague. Luckily, the city does have the ability to construct actual
walls on the border, just as they do in the interior of the city. The federal council is worried about a pandemic and
wants to be absolutely certain that the plague does not escape the city limits. The plague escaptes city limits if it can
spread beyond the city grid, that is if a square on the border of the grid gets infected and the wall separating the border
square from outside the city has not been built.
In addition, the federal council realizes that the head theorist’s cost function was simplistic. Infecting an originally
healthy square (i, j) now incurs an arbitrary cost Ii,j corresponding to the population of square. Additionally, each
wall costs a different amount to build. Let Wi,j,⇤ be the cost to build a wall on the side of square i, j indicated by ⇤,
which should be a direction in {“, !, , #}. The costs will be consistent, so both ways of indicating a particular wall
have the same cost. For example, Wi,j,” = Wi,j1,#.
Give an efficient algorithm to determine which walls to erect so as to minimize the total cost (the cost of building
walls plus the cost for infecting originally healthy squares), while ensuring that plague doesn’t escape the city. Prove
your algorithm is correct and state and analyze its runtime in terms of n. You may assume that the costs W and I are
bounded by a constant.
Figure 2: The grid on the left represents a possible instance of problem 1B (incomplete because it doesn’t contain
cost information), and the two grids on the right represent two possible solutions. The red represents intially infected
squares, the orange initially healthy squares that will get infected, and the white squares that begin and stay healthy.
The dark bolded lines represent walls that are constructed. Note that walls on the boundary of the grid do not initially
exist and must be constucted.

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