Introduction to Graphs CS2110, Spring 2011 Cornell University A - - PowerPoint PPT Presentation

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Introduction to Graphs CS2110, Spring 2011 Cornell University A - - PowerPoint PPT Presentation

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Introduction to Graphs CS2110, Spring 2011 Cornell University A graph is a data structure for representing relationships . Each graph is a set of nodes connected by edges . Synonym Graph Hostile Slick Icy Chilly Direct Nifty Cool Abrupt

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Introduction to Graphs

CS2110, Spring 2011 Cornell University

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A graph is a data structure for representing relationships.

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Each graph is a set of nodes connected by edges.

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Nifty Cool Sharp Chilly Composed Abrupt Hostile Direct Slick Icy

Synonym Graph

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Goals for Today

  • Learn the formalisms behind graphs.
  • Learn different representations for graphs.
  • Learn about paths and cycles in graphs.
  • See three ways of exploring a graph.
  • Explore applications of graphs to real-world

problems.

  • Explore algorithms for drawing graphs.
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Formalisms

  • A (directed) graph is a pair G = (V, E) where
  • V are the vertices (nodes) of the graph.
  • E are the edges (arcs) of the graph.
  • Each edge is a pair (u, v) of the start and end

(or source and sink) of the edge.

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CAT SAT RAT RAN MAN MAT CAN

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Directed and Undirected Graphs

  • A graph is directed if its edges specify which is

the start and end node.

  • Encodes asymmetric relationship.
  • A graph is undirected if the edges don't

distinguish between the start and end nodes.

  • Encodes symmetric relationship.
  • An undirected graph is a special case of a

directed graph (just add edges both ways).

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How Big is a Graph G = (V, E)?

  • Two measures:
  • Number of vertices: |V| (often denoted n)
  • Number of edges: |E| (often denoted m)
  • |E| can be at most O(|V|2)
  • A graph is called sparse if it has few edges. A

graph with many edges is called dense.

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

A B D F

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Navigating a Graph

A B D C E F

A C F

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A path from v0 to vn is a list of edges (v0, v1), (v1, v2), …, (vn-1, vn).

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The length of a path is the number of edges it contains.

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

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A node v is reachable from node u if there is a path from u to v.

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

B D B

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Navigating a Graph

A B D C E F

B D B D B

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Navigating a Graph

A B D C E F

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Navigating a Graph

A B D C E F

A B D B D F

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A cycle in a graph is a set of edges (v0, v1), (v1, v2), …, (vn, v0) that starts and ends at the same node.

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A simple path is a path that does not contain a cycle. A simple cycle is a cycle that does not contain a smaller cycle

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Properties of Nodes

A B D C E F

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The indegree of a node is the number of edges entering that node. The outdegree of a node is the number of edges leaving that node. In an undirected graph, these are the same and are called the degree of the node.

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Summary of Terminology

  • A path is a series of edges connecting two nodes.
  • The length of a path is the number of edges in the path.
  • A node v is reachable from u if there is a path from u to v.
  • A cycle is a path from a node to itself.
  • A simple path is a path without a cycle.
  • A simple cycle is a cycle that does not contain a nested

cycle.

  • The indegree and outdegree of a node are the number of

edges entering/leaving it.

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Representing Graphs

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Adjacency Matrices

  • n x n grid of boolean values.
  • Element Aij is 1 if edge from i

to j, 0 else.

  • Memory usage is O(n2)
  • Can check if an edge exists

in O(1).

  • Can find all edges entering
  • r leaving a node in O(n).

A B D C E F

1 1 1 A B C D E F A B C D E F 1 1 1 1 1 1

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Adjacency Lists

A B D C E F

A B C D E F B C D E F F B C E

  • List of edges leaving

each node.

  • Memory usage is

O(m+n)

  • Find edges leaving a

node in O(d+ (u))

  • Check if edge exists

in O(d+ (u))

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Graph Algorithms

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Representing Prerequisites

Graph Path Cycle Simple Path Simple Cycle Path Length Degree Reachability

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A directed acyclic graph (DAG) is a directed graph with no cycles.

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Examples of DAGs

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Examples of DAGs

4 2 6 1 3 5 7

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Examples of DAGs

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Traversing a DAG

Graph Path Cycle Simple Path Simple Cycle Path Length Degree Reachability

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Traversing a DAG

Graph Path Cycle Simple Path Simple Cycle Path Length Degree Reachability

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Traversing a DAG

Path Cycle Simple Path Simple Cycle Path Length Degree Reachability Graph

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Traversing a DAG

Path Cycle Simple Path Simple Cycle Path Length Degree Reachability Graph

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Traversing a DAG

Path Simple Path Simple Cycle Path Length Degree Reachability Cycle Graph

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Traversing a DAG

Path Simple Path Simple Cycle Path Length Degree Reachability Cycle Graph

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Traversing a DAG

Path Simple Path Simple Cycle Path Length Degree Reachability Cycle Graph

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Traversing a DAG

Path Simple Path Simple Cycle Path Length Degree Reachability Cycle Graph

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path

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Traversing a DAG

Simple Path Simple Cycle Path Length Degree Reachability Graph Cycle Path Graph Path Cycle Simple Path Simple Cycle Path Length Degree Reachability

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Topological Sort

  • Order the nodes of a DAG so no node is picked

before its parents.

  • Algorithm:
  • Find a node with no incoming edges (indegree 0)
  • Remove it from the graph.
  • Add it to the resulting ordering.
  • Not necessarily unique.
  • Question: When is it unique?
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Analyzing Topological Sort

  • Assumes at each step that the DAG has a node

with indegree zero. Is this always true?

  • Claim one: Every DAG has such a node.
  • Proof sketch: If this isn't true, then each node has at

least one incoming edge. Start at any node and keep following backwards across that edge. Eventually you will find the same node twice and have found a cycle.

  • Claim two: Removing such a node leaves the

DAG a DAG.

  • Proof sketch: If the resulting graph has a cycle, the old

graph had a cycle as well.

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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Traversing an Arbitrary Graph

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General Graph Search Algorithm

  • Maintain a collection C of nodes to visit.
  • Initialize C with some set of nodes.
  • While C is not empty:
  • Pick a node v out of C.
  • Follow all outgoing edges from v, adding each

unvisited node found this way to C.

  • Eventually explores all nodes reachable from

the starting set of nodes. (Why?)

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Depth-First Search

  • Specialization of the general search algorithm

where nodes to visit are put on a stack.

  • Explores down a path as far as possible, then

backs up.

  • Simple graph search algorithm useful for exploring

a complete graph.

  • Useful as a subroutine in many important graph

algorithms.

  • Runs in O(m + n) with adjacency lists, O(n2 ) with

adjacency matrix.

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Depth-first search

A B D E C F

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Depth-first search

A B D E C F Stack

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Depth-first search

A B D E C F Stack A

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Depth-first search

A B D E C F Stack

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Depth-first search

A B D E C F Stack B E

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Depth-first search

A B D E C F Stack B E

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Depth-first search

A B D E C F Stack B

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Depth-first search

A B D E C F Stack B D F C

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Depth-first search

A B D E C F Stack B D F C

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Depth-first search

A B D E C F Stack B D F

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Depth-first search

A B D E C F Stack B D F

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Depth-first search

A B D E C F Stack B D F

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Depth-first search

A B D E C F Stack B D

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Depth-first search

A B D E C F Stack B D

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Depth-first search

A B D E C F Stack B

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Depth-first search

A B D E C F Stack B

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Depth-first search

A B D E C F Stack

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Depth-first search

A B D E C F Stack

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Implementing DFS

DFS(Node v, Set<Node> visited) { if (v is in visited) return; Add v to visited; for (Node u connected to v) DFS(u, visited); }

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Graph Search Trees

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Mazes as Graphs

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Mazes as Graphs

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Mazes as Graphs

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Mazes as Graphs

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Creating a Maze with DFS

  • Create a grid graph of the appropriate size.
  • Starting at any node, run a depth-first search,

adding the arcs to the stack in random order.

  • The resulting DFS tree is a maze with one

solution.

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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.
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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.

Stack

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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.

Stack

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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.

A B C Stack

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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.

A B C Stack C B A

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Problems with DFS

  • Useful when trying to explore everything.
  • Not good at finding specific nodes.

A B C Stack C B A

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Breadth-First Search

  • Specialization of the general search algorithm

where nodes to visit are put into a queue.

  • Explores nodes one hop away, then two hops

away, etc.

  • Finds path with fewest edges from start node to

all other nodes.

  • Runs in O(m + n) with adjacency lists, O(n2 )

with adjacency matrix.

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Breadth-first search

A B D E C F

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Breadth-first search

A B D E C F Queue

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Breadth-first search

A B D E C F Queue A

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Breadth-first search

A B D E C F Queue

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Breadth-first search

A B D E C F Queue B E

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Breadth-first search

A B D E C F Queue B E

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Breadth-first search

A B D E C F Queue E

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Breadth-first search

A B D E C F Queue E C

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Breadth-first search

A B D E C F Queue E C

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Breadth-first search

A B D E C F Queue C

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Breadth-first search

A B D E C F Queue C D F

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Breadth-first search

A B D E C F Queue C D F

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Breadth-first search

A B D E C F Queue D F

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Breadth-first search

A B D E C F Queue D F

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Breadth-first search

A B D E C F Queue F

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Breadth-first search

A B D E C F Queue F

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Breadth-first search

A B D E C F Queue

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Breadth-first search

A B D E C F Queue

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Implementing BFS

BFS(Node v, Set<Node> visited) { Create a Queue<Node> of nodes to visit; Add v to the queue; while (The queue is not empty) { Dequeue a node from the queue, let it be u; if (u has been visited) continue; Add u to the visited set; for (Node w connected to u) Enqueue w in the queue; } }

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Classic Graph Algorithms

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Graph Coloring

  • Given a graph G, assign colors to the nodes so

that no edge has endpoints of the same color.

  • The chromatic number of a graph is the

fewest number of colors needed to color it.

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Graph Coloring is Hard.

  • Determining whether a graph can be colored

with k colors (for k > 2) is NP-complete.

  • It is not known whether this problem can be

solved in polynomial time.

  • Want $1,000,000? Find a polynomial-time

algorithm or prove that none exists.

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Matching

  • A matching in a graph is a subset of the edges

that don't share any endpoints.

  • Intuitively, pairing up nodes in the graph.
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Matching

  • A matching in a graph is a subset of the edges

that don't share any endpoints.

  • Intuitively, pairing up nodes in the graph.
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Applications of Matching

  • Unlike graph coloring, matching can be done

quickly.

  • Sample application: divvying up desserts.
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Divvying Up Desserts

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Divvying Up Desserts

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Divvying Up Desserts

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Divvying Up Desserts

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Divvying Up Desserts

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Divvying Up Desserts

3 7

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Divvying Up Desserts

3 7 4 6

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Divvying Up Desserts

3 7 4 6 2 8

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Divvying Up Desserts

3 7 4 6 2 8 5 5

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Divvying Up Desserts

3 6 8 5

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Drawing Graphs

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Nifty Cool Sharp Chilly Composed Abrupt Hostile Direct Slick Icy

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Nifty Cool Sharp Chilly Composed Abrupt Hostile Direct Slick Icy

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Idea: Treat the graph as a physical system that exerts forces on itself.

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This is called a force-directed layout algorithm.

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Summary

  • Graphs are a powerful abstraction for modeling relationships

and connectivity.

  • Adjacency lists and adjacency matrices are two common

representations of graphs.

  • Directed acyclic graphs can be visited via a topological sort.
  • Depth-first search is a simple graph exploration algorithm.
  • Breadth-first search searches a graph one layer at a time.
  • There are many classic algorithms on graphs:
  • Graph coloring tries to color nodes so no two nodes of the same color

are connected.

  • Matchings represent pairing up of graph elements.
  • Graph drawing seeks to render aesthetically-pleasing graphs.