Question

Country XYZ has n important towns, which are connected by roads. There are only n − 1 roads and the road connectivity is such that every town is reachable from every other town by a unique sequence of roads. Two towns are said to be neighbors if there is a direct road connecting them. The king of country XYZ decided to station military troupes in the towns to counter possible attacks from enemies. Since the country lacks sufficient military budget, the king decided not to post a troupe in every town. Every town has the option of posting 0, 1 or 2 troupes such that any town with 0 troupes has exactly one neighboring town with 2 troupes. Design an algorithm to assign troupes to towns so that the total number of troupes assigned is minimized.

If you are giving a Dynamic programming algorithm, state and prove the recurrence relation and also give the bottom-up implementation.
For greedy algorithms, please provide a proof of correctness and pseudo-code.

Answer 1

To solve this problem, we can use a dynamic programming approach. The minimum number of troupes assigned can be calculated by considering two cases. We can start the dynamic programming algorithm from the root of the tree and use a bottom-up approach to compute the values of dp(v, p) for each vertex v in the tree.

Step-by-step explanation:

To solve this problem, we can use a dynamic programming approach. Let's define a function dp(v, p) that represents the minimum number of troupes assigned when considering the subtree rooted at vertex v, where p is the parent of v. We can calculate this value by considering two cases:

1. If v is assigned 0 troupes, then each child u of v must be assigned either 0 or 2 troupes. In this case, the minimum number of troupes assigned can be calculated as follows:
   dp(v, p) = sum(max(dp(u, v), dp(u, v) + 2) for each child u of v)
2. If v is assigned 1 troupe, then at least one child u of v must be assigned 2 troupes. In this case, the minimum number of troupes assigned can be calculated as follows:
   dp(v, p) = 1 + sum(max(dp(u, v), dp(u, v) + 2) for each child u of v)

We can start the dynamic programming algorithm from the root of the tree and use a bottom-up approach to compute the values of dp(v, p) for each vertex v in the tree. The minimum number of troupes assigned overall will be min(dp(root, null), dp(root, null) + 1), where root is the root of the tree.

The question involves designing an algorithm to optimize military troupe placements in a network of towns structured as a tree. The aim is to minimize the total number of troupes while ensuring that towns without troupes have a neighbor with two troupes. Suitable techniques include depth-first search and dynamic programming.

The question posed involves creating an algorithm to optimize the placement of military troupes across a network of towns in a manner that minimizes the total number of troupes required, with specific constraints. This problem fits within the field of graph theory and combinatorial optimization in computer science, particularly concerning tree structures, as the connectivity between the towns forms a tree if every town is reachable from every other without cycles. A suitable algorithm would likely involve a depth-first search (DFS) or dynamic programming technique to minimize troupe assignment while adhering to the given conditions.

An example algorithm might operate as follows:

Start with a root town and perform a DFS to visit each town.

At each visited town, determine if it has already been assigned troupe support from a neighbor (either directly or indirectly through the constraint).

If not, assign two troupes to the town's parent if the parent has not been assigned troupe support yet.

Backtrack and repeat the process for all towns.

The pseudo-code for this algorithm and a proof of its correctness would be based on the properties of tree structures, recursive optimization, and strategic assignment.