| As in Chapter 1, the mine can be thought of as a rooted tree, with one node per cave, the root being node \(1\), and each tunnel corresponding to an edge. | |
| If \(K = 0\), the answer is \(C_1\). Otherwise, we can think of Minerva's route as consisting of any set of \(K\) edge-disjoint paths in the tree (that is, with no edge shared between multiple paths), with at least one of them including node \(1\). A path including node \(1\) will implicitly correspond to both the first and last portions of her route: we can imagine her starting at node \(1\) and driving to one of the path's endpoints before drilling her first tunnel, and then eventually drilling her final tunnel to the path's other endpoint and driving back to node \(1\). The remaining \(K-1\) paths more directly correspond to the intermediate portions of her route, with her repeatedly drilling to a path's endpoint and then driving along that complete path before drilling another tunnel to the next path. The total value of such a set of paths is equal to the sum of \(C\) values across all nodes which are part of at least one path, and we want to find a set of paths which maximizes this total. | |
| We'll take a dynamic programming (DP) approach. Let \(F(i, j, k)\) be the maximum total value of paths if only node \(i\)'s subtree is considered, such that: | |
| - There are \(j\) paths contained entirely within that subtree | |
| - There's a path with an endpoint at node \(i\) (potentially connecting to \(i\)'s parent) available to be extended further if and only if \(k = 1\) (defined for \(0 \le k \le 1\)) | |
| We'll additionally enforce that, if \(i = 1\), at least one of the paths must include node \(i\). Our final answer will be the maximum of \(C_1\) and of \(F(1, j, 0)\) (for \(1 \le j \le K\)). | |
| Computing \(F(i, j, k)\) will require processing \(i\) and its children in a secondary level of DP. Let \(G(i, f, j, k, c)\) be the maximum total value of paths if only node \(i\) and its first \(f\) children are considered, such that: | |
| - There are \(j\) paths contained entirely within those nodes | |
| - There's a path with an endpoint at node \(i\) (potentially connecting to \(i\)'s parent) available to be extended further if and only if \(k = 1\) (defined for \(0 \le k \le 1\)) | |
| - There's at least one path connecting \(i\) to one of those \(f\) children if and only if \(c = 1\) (defined for \(0 \le c \le 1\)) | |
| We'll recurse through the tree, computing these values. Once we've computed the \(F\) values for all children of a node \(i\), we'll be able to compute node \(i\)'s \(G\) values. After starting with the base case of \(G(i, 0, 0, 0, 0) = 0\), we can compute \(G(f, ...)\) for each \(f\)th child of node \(i\) by combining \(G(f-1, ...)\) with that child's \(F\) values, while considering the options of either having or not having a path connecting node \(i\) and that child. If node \(i\) has \(Z_i\) children, we can then transform \(G(i, Z_i, ...)\) into \(F(i, ...)\). Further details around the DP transitions involved can be found in the official solution's source code. | |
| Overall, it takes a total of \(O(NK^2)\) time to compute all \(O(NK)\) relevant DP values. | |
| [See David Harmeyer's solution video here](https://youtu.be/V1p4ZX-uDM8). | |