山东大学 Design and Analysis of Algorithm 2019年 期末考试

Summary of Course

如题，后天上午期末考试，因课表冲突不得不突击复习，以下内容包括教材信息以及重点分析 (本科学过图论算法，简单的概念和证明以及本人熟悉的部分一带而过)
刚刚考完，附上回忆版试题

Textbook

《Introduction to Algorithm (Third Edition)》by Thomas H. Cormen,Charles E. Leiserson, Ronald L. Rivest and Clifford Stein
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Selected Solutions for《Introduction to Algorithm》
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Key Points

Chapter 22 Basic Graph Algorithm

1. Graph Basics
   Graph (Edge&Vertex)
   Undirected Graph&Directed Graph
   Adjacency
   Repeated edge, Self-loop & Simple graph
   Degree, Isolated vertex
   Cycle, Sub-graph,
   Connected (connected component, strongly connected, strongly connected component)
   Special Graph (complete, bipartite, forest, tree, sparse, dense)
    
   Theorem 1.1
   Given an undirected graph $G=(V,E)$, ${\sum }_{v\in V}d(v)=2|E|$
    
   Copollary 1.2
   Given an undirected graph $G=(V,E)$, the number of vertices with odd degrees is even.
    
2. Representations of graphs
   
   For an undirected graph $G=(V,E)$，

• Adjacency-list:
  
  Advantage:
     When the graph is sparse, uses only O($|V|+|E|$) memory.
  Disvantage:
     No quicker way to determine if a given edge $(u,v)$ is present in the graph than to search for v in the adjacency list $Adj[u]$. O($|V|$)

• Adjacency matrix
  
  Advantage:
     Easily or quickly to determine if an edge is in the graph or not. O(1)
  Disvantage:
     Uses more memory to store a graph. O($|V|$²)

3. Elementary graph algorithms

• BFS(Breadth-First Search)
  

• DFS(Deep-First Search)
  

• White Path Theorem
  

• Topological Sort
  

• Strongly connected components
  

Chapter 23 Minimum spanning tree

• Kruskal algorithm
• Prim algorithm

Chapter 24 Single-Source Shortest Path

• Bellman-Ford algorithm
• Dijkstra Algorithm
• Proof of Dijkstra algorithm

Chapter 15 Dynamic programming

• Shortest path in DAGs
• Knapsack problem
• Bellman-Ford algorithm

Chapter 26 Maximum Flow

• Capacity-scaling algorithm
• Bipartite matching

试题（回忆版）

1.证明BFS生成的是最短路径树
2.证明一个图上的两颗最小生成树的边集的递增序列相同
3.设计O(VE)算法判断有向图是否有负圈，若没有，输出每个顶点的最短可达距离
(顶点v的最短可达距离：对于任意定点u∈V，dis=min{δ(u,v)}) 并证明算法正确性
4.设计DP算法，对于树T=(V,E)，去掉最少的点，使得没有边剩下
5.证明在ford-fulkerson算法求s→t最大流时，在残留网络求增广路径时，对于任意u，v∈V，s到u的最短路径长度都不比上一次短，v到t也是
6.形式化描述最大独立集和最小顶点覆盖问题，并证明其等价