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)

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 $G = (V, E)$G=(V,E), $\sum_{v\in V}d(v)=2\lvert E\rvert$∑v∈V​d(v)=2∣E∣
    
   Copollary 1.2
   Given an undirected graph $G = (V, E)$G=(V,E), the number of vertices with odd degrees is even.
    
2. Representations of graphs
   
   For an undirected graph $G = (V, E)$G=(V,E)，

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