Lecture 14: Graph Algorithms

Lecture 14:Graph Algorithms Shang-Hua Teng

A B C D E F G H I J K L M N O P 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 B C D E F G H I 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)

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 matrixrepresents 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: 1 a d 2 4 b c 3

Adjacency Matrix Representation • Undirected graph  matrix A is symmetric • Storage requirements: O(V2) • 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 1 2 4 3

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 “www.google.com” • 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

A B C D 0 E F G H I J K L M N O P

A B C D 0 1 E F G H 1 1 I J K L M N O P

A B C D 0 2 1 E F G H 1 1 I J K L 2 M N O P

A B C D 0 3 2 1 E F G H 1 1 3 3 I J K L 2 M N O P 3 3

A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 M N O P 3 3

A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 5 M N O P 3 3 5

Breadth-First Search BFS(G, s) • For each vertex u in V – {s}, • color[u] = white; d[u] = infty; p[u] = NIL • color[s] = GRAY; d[s] = 0; p[s] = NIL; Q = {} • ENQUEUE(Q,s) // Q is a FIFO queue • while (Q not empty) • u = DEQUEUE(Q) • for each v  Adj[u] • if color[v] = WHITE • then color[v] = GREY • d[v] = d[u] + 1; p[v] = u • ENQUEUE(Q, v); • color[u] = BLACK; What does v->d represent? What does v->p represent?

Breadth-first Search Algorithm • Algorithm BFS(s): • initialize container L to contain start vertex s. • i 0 • while Li is not empty do • create container Li+1 to initially be empty • for each vertex v in Li do • for each edge e incident on v do • if edge e is unexplored then • let w be the other endpoint of e • if vertex w is unexplored then • label e as a discovery edge • insert w into Li+1 • else • label e as a cross edge • 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 leasti 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 r s t u         v w x y

Example r s t u  0       v w x y Q: s

Example r s t u 1 0    1   v w x y Q: w r

Example r s t u 1 0 2   1 2  v w x y Q: r t x

Example r s t u 1 0 2  2 1 2  v w x y Q: t x v

Example r s t u 1 0 2 3 2 1 2  v w x y Q: x v u

Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: v u y

Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: u y

Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: y

Example r s t u 1 0 2 3 2 1 2 3 v w x y 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

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.

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.

Lecture 14: Graph Algorithms