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.

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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

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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

d e

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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

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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

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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

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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|)

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Adjacency Matrix

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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.

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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

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Intuition

Starting from source node s, Spread a wavefront to visit other nodes

Example Imagine given a tree What if a graph?

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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.

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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]:

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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.

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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

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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| )

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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

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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

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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