TITLE: THE CELESTIAL NAVIGATOR'S CHALLENGE
PROBLEM STATEMENT
In the mystical world of Celestia, there exists a vast network of stars connected by ancient portals, forming a tree-like structure known as the "Stellar Web." Each star is a node, and each portal is an edge. The nodes possess unique magical properties, and the key to unlocking these properties lies
in the geometric alignment of the stars.
You are given a map of the Stellar Web, where each node has a coordinate in a 3D Euclidean space. Your task is to find the maximum magical potential that can be achieved by aligning the stars correctly through a series of portal traversals. The magical potential at any node is determined by the sum
of the distances to all other nodes, weighted by a celestial constant ( k ). However, the portals can only be traversed in a specific direction, and the traversal cost is proportional to the Euclidean distance between the stars.
Additionally, the nodes and edges form a tree structure, and you must use advanced mathematical concepts such as Euler geometry, Euler's characteristic formula, and complex numbers to solve this problem. You will also need to apply greedy algorithms, depth-first search (DFS), tree dynamic
programming (DP), and queue-based approaches.
INPUT FORMAT
The first line contains two integers ( n ) and ( k ), where ( n ) is the number of stars (nodes) and ( k ) is the celestial constant.
The next ( n ) lines contain three integers each ( x, y, z ), representing the coordinates of the star in 3D space.
The next ( n-1 ) lines contain two integers ( u ) and ( v ), representing a directed edge from star ( u ) to star ( v ).
OUTPUT FORMAT
Output a single integer representing the maximum magical potential.
CONSTRAINTS
* ( 2 \leq n \leq 10^5 )
* ( 1 \leq k \leq 10^9 )
* ( -10^9 \leq x, y, z \leq 10^9 )
* ( 1 \leq u, v \leq n )
SAMPLE INPUT
SAMPLE OUTPUT
EXPLANATION
The sample output is a placeholder, as the actual calculation would require implementing the solution.
NOTES
1. Utilize the properties of Euclidean geometry to calculate distances and determine the alignment of stars.
2. Apply greedy algorithms to select the most beneficial portal to traverse at each step, considering the traversal cost and the potential increase in magical potential.
3. Implement a depth-first search (DFS) to explore the tree structure of the Stellar Web and to calculate the magical potential dynamically.
4. Use tree-based dynamic programming to efficiently compute the sum of distances for each subtree.
5. Employ a queue to manage the traversal of the tree, ensuring that each node is visited in an optimal order.
6. Leverage Euler's characteristic formula and complex numbers to handle geometric transformations and calculations.
This problem is designed to challenge competitors in various domains, including advanced mathematics, algorithm design, and data structure implementation. It requires a deep understanding of Euclidean geometry, Euler's formulas, dynamic programming, and graph theory. Good luck, and may the celestial
forces guide you in your quest for the maximum magical potential!
