Basic Graph Algorithms Stanford University-Books Pdf

Basic Graph Algorithms Stanford University
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Adjacency Matrix and Adjacency List,Special Graphs. Depth First and Breadth First Search,Topological Sort. Eulerian Circuit,Minimum Spanning Tree MST,Strongly Connected Components SCC. An abstract way of representing connectivity using nodes also. called vertices and edges,We will label the nodes from 1 to n. m edges connect some pairs of nodes, Edges can be either one directional directed or bidirectional.
Nodes and edges can have some auxiliary information. Why Study Graphs, Lots of problems formulated and solved in terms of graphs. Shortest path problems,Network flow problems,Matching problems. 2 SAT problem,Graph coloring problem,Traveling Salesman Problem TSP still unsolved. and many more,Adjacency Matrix and Adjacency List,Special Graphs. Depth First and Breadth First Search,Topological Sort.
Eulerian Circuit,Minimum Spanning Tree MST,Strongly Connected Components SCC. Adjacency Matrix and Adjacency List 5,Storing Graphs. Need to store both the set of nodes V and the set of edges E. Nodes can be stored in an array,Edges must be stored in some other way. Want to support operations such as, Retrieving all edges incident to a particular node. Testing if given two nodes are directly connected, Use either adjacency matrix or adjacency list to store the.
Adjacency Matrix and Adjacency List 6,Adjacency Matrix. An easy way to store connectivity information, Checking if two nodes are directly connected O 1 time. Make an n n matrix A,aij 1 if there is an edge from i to j. aij 0 otherwise,Uses n2 memory,Only use when n is less than a few thousands. and when the graph is dense,Adjacency Matrix and Adjacency List 7.
Adjacency List,Each node has a list of outgoing edges from it. Easy to iterate over edges incident to a certain node. The lists have variable lengths,Space usage n m,Adjacency Matrix and Adjacency List 8. Implementing Adjacency List,Solution 1 Using linked lists. Too much memory time overhead,Using dynamic allocated memory or pointers is bad. Solution 2 Using an array of vectors,Easier to code no bad memory issues.
But very slow,Solution 3 Using arrays,Assuming the total number of edges is known. Very fast and memory efficient,Adjacency Matrix and Adjacency List 9. Implementation Using Arrays,Adjacency Matrix and Adjacency List 10. Implementation Using Arrays,Have two arrays E of size m and LE of size n. E contains the edges, LE contains the starting pointers of the edge lists.
Initialize LE i 1 for all i,LE i 0 is also fine if the arrays are 1 indexed. Inserting a new edge from u to v with ID k,E k nextID LE u. Adjacency Matrix and Adjacency List 11,Implementation Using Arrays. Iterating over all edges starting at u,for ID LE u ID 1 ID E ID nextID. E ID is an edge starting from u,Once built it s hard to modify the edges.
The graph better be static,But adding more edges is easy. Adjacency Matrix and Adjacency List 12,Adjacency Matrix and Adjacency List. Special Graphs,Depth First and Breadth First Search. Topological Sort,Eulerian Circuit,Minimum Spanning Tree MST. Strongly Connected Components SCC,Special Graphs 13.
A connected acyclic graph,Most important type of special graphs. Many problems are easier to solve on trees,Alternate equivalent definitions. A connected graph with n 1 edges,An acyclic graph with n 1 edges. There is exactly one path between every pair of nodes. An acyclic graph but adding any edge results in a cycle. A connected graph but removing any edge disconnects it. Special Graphs 14,Other Special Graphs, Directed Acyclic Graph DAG the name says what it is. Equivalent to a partial ordering of nodes, Bipartite Graph Nodes can be separated into two groups S.
and T such that edges exist between S and T only no edges.

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