Graph Theory, DFS & BFS

Kelly Choi 08-07-2006

What is a graph?

• A set of vertices and edges – Directed/Undirected – Weighted/Unweighted – Cyclic/Acyclic

vertex edge

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Representation of Graphs

• Adjacency Matrix – A V x V array, with matrix[i][j] storing whether there is an edge between the i

th

vertex and the j

th

vertex • Adjacency Linked List – One linked list per vertex, each storing directly reachable vertices • Edge List 08-07-2006 3

Representation of Graphs

Adjacency Matrix Adjacency Linked List Edge List Memory Storage

O(V 2 ) O(V+E) O(V+E)

Check whether (u,v) is an edge

O(1) O(deg(u)) O(deg(u))

Find all adjacent vertices of a vertex u

O(V) O(deg(u)) O(deg(u)) deg(u): the number of edges connecting vertex u 08-07-2006 4

Graph Searching

• Why do we do graph searching? What do we search for?

• What information can we find from graph searching?

• How do we search the graph? Do we need to visit all vertices? In what order?

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Depth-First Search (DFS)

• Strategy: Go as far as you can (if you have not visit there), otherwise, go back and try another way 08-07-2006 6

Implementation

DFS (vertex u) { mark u as visited for each vertex v directly reachable from u if v is unvisited DFS (v) } • Initially all vertices are marked as

unvisited

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Topological Sort

• Topological order: A numbering of the vertices of a directed acyclic graph such that every edge from a vertex numbered i to a vertex numbered j satisfies i

Example: Teacher’s Problem

• Emily wants to distribute candies to N students one by one, with a rule that if student A is teased by B, A can receive candy before B. • Given lists of students teased by each students, find a possible sequence to give the candies 08-07-2006 9

Shortest Paths?

• Given vertices s and t, if we try to find a path from s to t by DFS, the first path found may not be the shortest one.

• How can I find a shortest path?

• Let’s first review what we have done 08-07-2006 10

An Idea of What We Did

1. Start by visiting a certain vertex 2. Vertices fall into 3 categories: – Unvisited – Visited – Visited & Dead (All reachable vertices from these vertices are visited) 08-07-2006 11

An Idea of What We Did

3. Find unvisited vertices by expanding your path from visited (but not dead) vertices.

• In DFS, we choose the most recently visited vertex to expand.

• Are there any other strategies?

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Breadth-First Search (BFS)

• Instead of going as far as possible, BFS tries to search all paths.

• BFS makes use of a queue to store visited (but not dead) vertices, expanding the path from the earliest visited vertices.

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Simulation of BFS

• Queue:

1 4 3 5 2 6 4 1 2 3 5 6

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Implementation

while queue Q not empty dequeue the first vertex u from Q for each vertex v directly reachable from u if v is unvisited enqueue v to Q mark v as visited • Initially all vertices except the start vertex are marked as unvisited and the queue contains the start vertex only 08-07-2006 15

There is more…

• Other Graph Searching Algorithms: – Bidirectional search (BDS) – Iterative deepening search (IDS) 08-07-2006 16

Advantages

• Guarantee shortest paths for unweighted graphs • Use queue instead of recursive functions – Avoiding stack overflow 08-07-2006 17

Flood Fill

• An algorithm that determines the area connected to a given node in a multi-dimensional array • Start BFS/DFS from the given node, counting the total number of nodes visited • Example: Squareland (HKOI 2006) 08-07-2006 18

Graph Modeling

• Conversion of a problem into a graph problem • Essential in solving most graph problems 08-07-2006 19

Basics of graph modeling

• Identify the vertices and the edges • Identify the objective of the problem • State the objective in graph terms • Implementation: – construct the graph from the input instance – run the suitable graph algorithms on the graph – convert the output to the required format

(cx, 2004)

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Other Topics in Graph Theory

• Cut Vertices & Cut Edges • Euler Path/Circuit & Hamilton Path/Circuit • Planarity 08-07-2006 21

Cut Vertices & Cut Edges

• What is a cut vertex?

– A cut vertex is a type of graph vertex, the removal of which causes an increase in the number of connected components • What is a cut edge?

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Euler Path & Hamilton Path

• An Euler path is a path in a graph which visits each edge exactly once • A Hamilton path is a path in an undirected graph which visits each vertex exactly once.

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Planarity

• A planar graph is a graph that can be drawn so that no edges intersect • K 5 and K 3,3 are non-planar graphs 08-07-2006 24