CSE 780 Algorithms Advanced Algorithms Graph Algorithms Representations BFS

CSE 780 Algorithms Objectives At the end of this lecture, students should be able to: a. represent the graph b. write and analyze BFS algorithm.

CSE 780 Algorithms Intro. Graph representations are very useful in a variety of disciplines. Examples of applications: Network analysis Project Planning Shortest route planning Compilers: Data-Dependency graphs Natural Language Processing

CSE 780 Algorithms What Is A Graph Graph G = (V, E) V: set of nodes (vertices) E: set of edges Example: V ={ a, b, c, d, e, f } E ={(a, b), (a, d), (a, e), (b, c), (b, d), (b, e), (c, e), (e,f)} ba f c de

CSE 780 Algorithms More Terminologies Directed / undirected graph Weighted / unweighted graph Special graphs Complete graph, planar (plane) graph, tree Size of graph: |V|, |E| |E| = O ( ) |V| 2 Degree of a node indegree / outdegree Node u and v are connected If there is a path from E connecting u and v

CSE 780 Algorithms Properties of A Tree G = (V, E) is undirected graph, then is a tree has no cycle, and connected has |V|-1 edges, and connected

CSE 780 Algorithms Representations of Graphs Adjacency lists An array of |V| lists Each vertex u has a list, recording its neighbors I.e., all v’s such that (u, v)  E

CSE 780 Algorithms Adjacency Lists For vertex v  V, its list has size: outdegree(v) decide whether (v, u)  E or not in time O(outdegree(v)) Size of data structure (space complexity): O(|V| + |E|)

CSE 780 Algorithms Adjacency Matrix

CSE 780 Algorithms Adjacency Matrix Size of data structure: O ( |V|  |V|) Time to determine If (v, u)  E : O(1) Though larger, it is simpler compared to adjacency list.

CSE 780 Algorithms Traverse of Graph Given graph G = (V, E) Given a source node s  V Visit all nodes in V reachable from s If G is a tree … Need an efficient strategy to achieve that BFS: Breadth-first search

CSE 780 Algorithms Intuition Starting from source node s, Spread a wavefront to visit other nodes Example Imagine given a tree What if a graph?

CSE 780 Algorithms Intuition cont.  (u, v): Distance (smallest # edges) from node u to v in G A node: white: unvisited grey: discovered but not explored black: finished (explored) Goal: Start from source s, first visit those nodes of distance 1 from s, then 2, and so on.

CSE 780 Algorithms More Formally BFS Input: Graph G=(V, E) and source node s  V Output:  (s, u) for every u  V  (s,u) =  if u is unreachable from s Assume adjacency list representation for G Use FIFO queue Q to manage the traversal of vertices Keep track of the color of a vertex u using color[u]:

CSE 780 Algorithms BFS (cont.) white – not been discovered gray – has been put on Q black – its successors have been added to Q (explored) Keep track of the distance in d[u] Use л[u] to store the predecessor of u.

CSE 780 Algorithms Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms Example

CSE 780 Algorithms Analysis of Algorithm Initialise (Step 1 to 4): |V|-1 vertices are intialized to white in O(V). Queue Operations: Each enqueue and dequeu uses O(1) time, hence all queue operations take O(V) Scanning Adjacency Lists (step 12): The sum of the lists’ length is O(E). The time to scan the lists is O(E) Time complexity: O ( |V| + |E| )

CSE 780 Algorithms Breadth-First Tree v = л(u) if u is first discovered while exploring v The set of edges (л(u), u) for all u  V form a tree The breadth-first tree with root s All nodes of the same level of tree has same d[] value Nodes are explored level by level

CSE 780 Algorithms Print a shortest path from s to v Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms Correctness of Algorithm A node not reachable from s will not be visited A node reachable from s will be visited d(u) computed is correct: (i.e., =  (s, u) ) Proof by induction Consider the breadth-first tree generated by algorithm: Any node u from k’th level: d[u] = k Claim: all nodes v with  (s, v) = k are in level k and all nodes from level k has  (s, v) = k Observation: if edge (u,v) exists, Then  (s, v) ≤  (s, u) + 1

CSE 780 Algorithms Summary for BFS Works for both directed and undirected graph Starting from source node s, visits remaining nodes of graph from small distance to large distance Return distance between s to any reachable node in time O(|V| + |E|) Last updated: 13/02/2009