IS CS-2020S2-04

题目来源：Problem 4 日期：2024-08-11 题目主题：CS-图论-最小生成树

解题思路

这道题目考察了图论中的最小生成树（Minimum Spanning Tree, MST）的相关性质和算法实现。题目由多个小问组成，涵盖了 MST 的关键性质和相关算法的时间复杂度分析。以下是解题思路：

1. 第一问考察了图中环与 MST 的关系，要求证明某个最大边权的边不在 MST 中。这个问题可以通过反证法和 MST 的基本性质来证明。
2. 第二问要求证明特定子集间的最小边必然在 MST 中。可以通过切割定理（Cut Property）来证明这个性质。
3. 第三问要求实现一个 $O(|E|)$ 复杂度的算法，找到图中任意两个节点之间的一条路径。可以通过深度优先搜索（DFS）或广度优先搜索（BFS）来实现。
4. 第四问要求在给定 MST 的基础上，找到包含一条新边的图的 MST。这个问题可以通过 MST 更新算法来解决，算法的复杂度要求是 $O(|V|)$。
5. 第五问要求对第四问中的算法进行正确性证明，需要结合 MST 的性质进行分析。

Solution

Question 1

Let $C$ be an arbitrary cycle in $G$, and let $e$ be the edge in $C$ with the maximum weight. We need to prove that there exists a minimum spanning tree (MST) that does not contain $e$.

Proof

• Step 1: Assume $T$ is an MST that contains $e$.
• Step 2: Consider the cycle $C$ in $G$ that includes $e$. Since $T$ is a tree and contains all vertices of $G$, adding $e$ to $T$ will create a cycle.
• Step 3: In the cycle $C$, remove the edge $e$ (which has the maximum weight in $C$) to break the cycle, forming a new tree $T^{\prime }$. Since we removed the edge with the maximum weight, $T^{\prime }$ has a smaller or equal total weight compared to $T$.
• Step 4: Conclude that $T^{\prime }$ is also an MST, but it does not contain $e$.

Therefore, there exists a minimum spanning tree that does not contain the edge $e$.

Question 2

Consider an arbitrary vertex subset $V^{\prime }$ of $V$ ($V^{\prime }\ne V,V^{\prime }\ne \varnothing$). Let $e$ be the edge with the minimum weight among the edges $(u,v)\in E$ such that $u\in V^{\prime }$ and $v\in V-V^{\prime }$. We need to prove that there is a minimum spanning tree that contains $e$.

Proof

• Step 1: Apply the Cut Property.** According to the Cut Property, for any cut in the graph, the minimum weight edge crossing the cut must be in the MST.
• Step 2: Define the cut associated with $V^{\prime }$ as the set of edges $(u,v)$ where $u\in V^{\prime }$ and $v\in V-V^{\prime }$.
• Step 3: Since $e$ is the minimum weight edge across this cut, by the Cut Property, $e$ must be included in every MST of $G$.

Thus, there is a minimum spanning tree that contains the edge $e$.

Question 3

Describe an $O(|E|)$-time algorithm that finds an arbitrary path between two nodes $u,v\in V$ on graph $G=(V,E)$.

Algorithm

1. Initialization: Initialize a stack (or queue) and mark all vertices as unvisited.
2. Depth-First Search (DFS):
   • Push the starting vertex $u$ onto the stack and mark it as visited.
   • While the stack is not empty:
     • Pop a vertex $w$ from the stack.
     • If $w=v$, return the path from $u$ to $v$.
     • For each adjacent vertex $x$ of $w$ that is not visited, push $x$ onto the stack and mark it as visited.
3. Termination: If the search ends without finding $v$, return “No Path”.

This algorithm runs in $O(|E|)$ time because in the worst case, it visits each edge once.

Question 4

Assume that we are given a graph $G=(V,E)$ and its minimum spanning tree $T$. Let $G^{\prime }$ be the graph obtained by adding to $G$ a new edge $e=(u,v)\notin E$ $(u,v\in V)$ with weight $w>0$. Describe an $O(|V|)$-time algorithm that finds a minimum spanning tree of $G^{\prime }$.

Algorithm

1. Step 1: Add the new edge $e=(u,v)$ to the MST $T$. This will create a cycle in the tree.
2. Step 2: Find the maximum weight edge $f$ in this cycle.
3. Step 3: If $f=e$, then the original MST $T$ is still valid.
4. Step 4: If $f\ne e$, remove $f$ from the cycle. The resulting graph is a tree and is the new MST.

The time complexity is $O(|V|)$ because finding the cycle in a tree and identifying the maximum weight edge in the cycle can be done in linear time.

Question 5

Prove the correctness of the algorithm described in question (4).

Proof

• Step 1: Adding edge $e$ to the MST $T$ creates exactly one cycle, as $T$ was a tree.
• Step 2: In the cycle, removing the maximum weight edge ensures that the resulting graph is still a tree with a total weight less than or equal to the original MST.
• Step 3: If $f\ne e$, then $f$ must be removed to maintain the minimality of the spanning tree, as $e$ introduces a smaller weight.
• Step 4: This process guarantees that the resulting tree is the MST of $G^{\prime }$, thus proving the algorithm’s correctness.

知识点

最小生成树图论DFSBFS 贪心算法

解题技巧和信息

• 最小生成树的关键性质可以通过切割定理和环路定理进行理解。
• 对于图中的路径查找问题，DFS 和 BFS 都是常用的线性时间算法。
• 解决最小生成树更新问题时，可以通过引入新边后检查形成的环并删除最大边的方式实现最小生成树的维护。

重点词汇

• Minimum Spanning Tree (MST) 最小生成树
• Cycle 环
• Cut Property 切割定理
• Depth-First Search (DFS) 深度优先搜索
• Breadth-First Search (BFS) 广度优先搜索

参考资料

1. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press. Chap. 23: Minimum Spanning Trees.
2. Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed.). Addison-Wesley. Sections on Graph Algorithms.