Lecture 14: Graph Algorithms Shang-Hua Teng

Undirected Graphs A graph G = (V, E) –V: vertices –E : edges, unordered pairs of vertices from V  V –(u,v) is same as (v,u) –Thus |E| <= |V| (|V|-1)/2 A ONM LKJ EFGH DCB I P

Undirected Graphs A graph G = (V, E) –V: vertices –E : edges, ordered pairs of vertices from V  V –(u,v) is different from (v,u) –Thus |E| <= |V| (|V|-1) A EFGH DCB I

Basic Graph Properties An undirected graph is connected if it has a path from every vertex to every other A directed graph is strongly connected if it has a path from every vertex to every other A weighted graph associates weights with either the edges or the vertices E.g., a road map: edges might be weighted w/ distance A multigraph allows multiple edges between the same vertices

Dense and Sparse Graphs We will typically express running times in terms of |E| and |V| (often dropping the |’s) –If |E|  |V| 2 the graph is dense –If |E|  |V| the graph is sparse Different data structures may be used to express sparse and dense graph

Graph in Applications Internet: web graphs –Each page is a vertex –Each edge represent a hyperlink –Directed GPS: highway maps –Each city is a vertex –Each freeway segment is an undirected edge –Undirected Graphs are ubiquitous in computer science

Representing Graphs Assume V = {1, 2, …, n} An adjacency matrix represents the graph as a n x n matrix A: –A[i, j] = 1 if edge (i, j)  E (or weight of edge) = 0 if edge (i, j)  E

Adjacency Matrix Example: a d bc A

Adjacency Matrix Representation Undirected graph  matrix A is symmetric Storage requirements: O(V 2 ) –A dense representation The adjacency matrix is a dense representation –Usually too much storage for large graphs –But can be very efficient for small graphs Most large interesting graphs are sparse –For example, trees and planar graphs (such as highway map) have |E| = O(|V|) –For this reason, we often needs more sparse representation

Adjacency List Representation Adjacency list: for each vertex v  V, store a list of vertices adjacent to v Example: –Adj[1] = {2}{3} –Adj[2] = {3} –Adj[3] = {} –Adj[4] = {3} Variation: can also keep a list of edges coming into vertex

Adjacency List Representation The degree of a vertex v in an undirected graph is equal to the number of incident edges For directed graphs, we can define in-degree, out-degree for each vertex For directed graphs, the total size of the adjacency lists is  out-degree(v) = |E| takes  (V + E) storage For undirected graphs, the total size of the adjacent lists is  degree(v) = 2 |E| also  (V + E) storage So: Adjacency lists take O(V+E) storage

Google Problem:Graph Searching How to search and explore in the Web Space? How to find all web-pages and all the hyperlinks? –Start from one vertex “ –Systematically follow hyperlinks from the discovered vertex/web page –Build a search tree along the exploration

General Graph Searching Input: a graph G = (V, E), directed or undirected Objective: methodically explore every vertex and every edge Build a tree on the graph –Pick a vertex as the root –Choose certain edges to produce a tree –Note: might also build a forest if graph is not connected

Breadth-first Search Objective: Traverse all vertices of a graph G that can be reached from a starting vertex, called root Method: –Search for all vertices that are directly reachable from the root (called level 1 vertices) –After mark all these vertices, visit all vertices that are directly reachable from any level 1 vertices (called level 2 vertices), and so on. –In general level k vertices are directly reachable from a level k – 1 vertices

An Example

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Breadth-First Search BFS(G, s) 1.For each vertex u in V – {s}, 2. color[u] = white; d[u] = infty;  [u] = NIL 3.color[s] = GRAY; d[s] = 0;  [s] = NIL; Q = {} 4.ENQUEUE(Q,s) // Q is a FIFO queue 5.while (Q not empty) 6. u = DEQUEUE(Q) 7. for each v  Adj[u] 8. if color[v] = WHITE 9. then color[v] = GREY 10. d[v] = d[u] + 1;  [v] = u 11. ENQUEUE(Q, v); 12. color[u] = BLACK;

Breadth-first Search Algorithm 1.Algorithm BFS(s): 1.initialize container L to contain start vertex s. 2.i  0 3.while L i is not empty do 1.create container L i+1 to initially be empty 2.for each vertex v in L i do 1.for each edge e incident on v do 1.if edge e is unexplored then 2. let w be the other endpoint of e 3. if vertex w is unexplored then 4. label e as a discovery edge 5.insert w into L i+1 6. else 7.label e as a cross edge 4.i  i + 1

Breadth-first Search Visited all vertices reachable from the root A spanning tree For any vertex at level i, the spanning tree path from s to i has i edges, and any other path from s to i has at least i edges (shortest path property) Edges not in the spanning tree are at most, 1 level apart (level by level property)

Breadth-First Search: the Color Scheme White vertices have not been discovered –All vertices start out white Grey vertices are discovered but not fully explored –They may be adjacent to white vertices Black vertices are discovered and fully explored –They are adjacent only to black and gray vertices Explore vertices by scanning adjacency list of grey vertices

Example         rstu vwxy

  0      rstu vwxy s Q:

Example 1  0 1     rstu vwxy w Q: r

Example 1    rstu vwxy r Q: tx

Example   rstu vwxy Q: txv

Example  rstu vwxy Q: xvu

Example rstu vwxy Q: vuy

Example rstu vwxy Q: uy

Example rstu vwxy Q: y

Example rstu vwxy Q: Ø

Properties of BFS BFS calculates the shortest-path distance to the source node –Shortest-path distance  (s,v) = minimum number of edges from s to v, or  if v not reachable from s –Prove using blackboard BFS builds breadth-first tree, in which paths to root represent shortest paths in G –Thus can use BFS to calculate shortest path from one vertex to another in O(V+E) time

39 Properties of BFS Proposition: Let G be an undirected graph on which a BFS traversal starting at vertex s has been performed. Then –The traversal visits all vertices in the connected component of s. –The discovery-edges form a spanning tree T, which we call the BFS tree, of the connected component of s –For each vertex v at level i, the path of the BFS tree T between s and v has i edges, and any other path of G between s and v has at least i edges. –If (u, v) is an edge that is not in the BFS tree, then the level numbers of u and v differ by at most one.

40 Properties of BFS Proposition: Let G be a graph with n vertices and m edges. A BFS traversal of G takes time O(n + m). Also, there exist O(n + m) time algorithms based on BFS for the following problems: –Testing whether G is connected. –Computing a spanning tree of G –Computing the connected components of G –Computing, for every vertex v of G, the minimum number of edges of any path between s and v.