AI第三次搞题这次应该较难，看看？
2024-10-20 20:17:47
发布于：浙江
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
5 2
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
1 2
2 3
3 4
4 5
Sample Output
50
Explanation
The sample output is a placeholder, as the actual calculation would require implementing the solution.
Notes
- Utilize the properties of Euclidean geometry to calculate distances and determine the alignment of stars.
- 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.
- Implement a depth-first search (DFS) to explore the tree structure of the Stellar Web and to calculate the magical potential dynamically.
- Use tree-based dynamic programming to efficiently compute the sum of distances for each subtree.
- Employ a queue to manage the traversal of the tree, ensuring that each node is visited in an optimal order.
- 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!
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翻译：
The Celestial Navigator's Challenge
问题陈述
在神秘的 Celestia 世界中，存在着一个巨大的星际网络，由古老的传送门连接，形成了一个被称为“恒星网”的树状结构。每个星星都是一个节点，每个门户都是一条边。这些节点具有独特的神奇特性，而解锁这些特性的关键在于星星的几何排列。您将获得一张 Stellar Web 的地图，其中每个节点在 3D 欧几里得空间中都有一个坐标。你的任务是通过一系列传送门穿越正确对齐星星来找到可以实现的最大魔法潜力。任何节点的神奇势能都由到所有其他节点的距离之和决定，由天体常数 （ ） 加权。然而，传送门只能在特定方向上穿越，并且穿越成本与恒星之间的欧几里得距离成正比。
此外，节点和边形成树状结构。
输入格式
第一行包含两个整数 （ ） 和 （ ），其中 （ ） 是恒星（节点）的数量，（ ） 是天体常数。接下来的 （ ） 行包含三个整数，每个整数 （ ），表示星星在 3D 空间中的坐标。
接下来的 （ ） 行包含两个整数 （ ） 和 （ ），表示从星形 （ ） 到星形 （ ） 的有向边。
输出格式
输出一个整数，表示最大魔法潜力。约束
.
输入样例5 2 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 1 2 2 3 3 4 4 5输出样例
50笔记
利用欧几里得几何的属性来计算距离并确定星星的对齐方式。
应用贪心算法，在每一步中选择最有利的传送门进行遍历，同时考虑到遍历成本和魔法潜力的潜在增加。
实施深度优先搜索 （DFS） 来探索 Stellar Web 的树结构并动态计算神奇的潜力。
使用基于树的动态规划来有效地计算每个子树的距离总和。
使用队列来管理树的遍历，确保以最佳顺序访问每个节点。
利用 Euler 的特征公式和复数来处理几何变换和计算。
该问题旨在挑战各个领域的竞争对手，包括高级数学、算法设计和数据结构实现。它需要对欧几里得几何、欧拉公式、动态规划和图论有深入的理解。祝你好运，愿天体力量指引你追求最大的魔法潜力！2024-10-20 来自 广东
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