CSE 326: Data StructuresLecture #20Graphs I.5

Alon Halevy

Spring Quarter 2001

Outline

• Some definitions• Topological Sort• Graph Data Structures• Graph Properties• Shortest Path Problem

Directed vs. Undirected Graphs

Han

Leia

Luke

Han

Leia

Luke

aka: di-graphs

• In directed graphs, edges have a specific direction:

• In undirected graphs, they don’t (edges are two-way):

• Vertices u and v are adjacent if (u, v) E

Weighted Graphs

20

30

35

60

Mukilteo

Edmonds

Seattle

Bremerton

Bainbridge

Kingston

Clinton

There may be more information in the graph as well.

Each edge has an associated weight or cost.

Paths

A path is a list of vertices {v1, v2, …, vn} such that (vi, vi+1) E for all 0 i < n.

Seattle

San FranciscoDallas

Chicago

Salt Lake City

p = {Seattle, Salt Lake City, Chicago, Dallas, San Francisco, Seattle}

Path Length and Cost

Path length: the number of edges in the path

Path cost: the sum of the costs of each edge

Seattle

San FranciscoDallas

Chicago

Salt Lake City

3.5

2 2

2.5

3

22.5

2.5

length(p) = 5 cost(p) = 11.5

Simple Paths and CyclesA simple path repeats no vertices (except that the first can be the last):

– p = {Seattle, Salt Lake City, San Francisco, Dallas}– p = {Seattle, Salt Lake City, Dallas, San Francisco, Seattle}

A cycle is a path that starts and ends at the same node:– p = {Seattle, Salt Lake City, Dallas, San Francisco, Seattle}

A simple cycle is a cycle that repeats no vertices except that the first vertex is also the last (in undirected graphs, no edge can be repeated)

ConnectivityUndirected graphs are connected if there is a path between

any two vertices

Directed graphs are strongly connected if there is a path from any one vertex to any other

Di-graphs are weakly connected if there is a path between any two vertices, ignoring direction

A complete graph has an edge between every pair of vertices

Graph Density

A sparse graph has O(|V|) edges

A dense graph has (|V|2) edges

Anything in between is either sparsish or densy depending on the context.

We use V and E as the parameters in our analysis.

Trees as Graphs

• Every tree is a graph with some restrictions:– the tree is directed

– there are no cycles (directed or undirected)

– there is a directed path from the root to every node

A

B

D E

C

F

HG

JI

Directed Acyclic Graphs (DAGs)

DAGs are directed graphs with no cycles.

main()

add()

access()

mult()

read()

Trees DAGs Graphs

Topological Sort

Given a graph, G = (V, E), output all the vertices in V such that no vertex is output before any other vertex with an edge to it.

check inairport

calltaxi

taxi toairport

reserveflight

packbags

takeflight

locategate

Topo-Sort Take One

Label each vertex’s in-degree (# of inbound edges)

While there are vertices remaining

Pick a vertex with in-degree of zero and output it

Reduce the in-degree of all vertices adjacent to it

Remove it from the list of vertices

runtime:

Topo-Sort Take Two

Label each vertex’s in-degree

Initialize a queue to contain all in-degree zero vertices

While there are vertices remaining in the queue

Pick a vertex v with in-degree of zero and output it

Reduce the in-degree of all vertices adjacent to v

Put any of these with new in-degree zero on the queue

Remove v from the queue

runtime:

Graph Representations

• List of vertices + list of edges

• 2-D matrix of vertices (marking edges in the cells)“adjacency matrix”

• List of vertices each with a list of adjacent vertices“adjacency list”

Han

Leia

Luke

Adjacency Matrix

A |V| x |V| array in which an element (u, v) is true if and only if there is an edge from u to v

Han

Leia

Luke

Han Luke Leia

Han

Luke

Leia

runtime: space requirements:

Adjacency List

A |V|-ary list (array) in which each entry stores a list (linked list) of all adjacent vertices

Han

Leia

LukeHan

Luke

Leia

runtime: space requirements:

Single Source, Shortest Path

Given a graph G = (V, E) and a vertex s V, find the shortest path from s to every vertex in V

Many variations:– weighted vs. unweighted– cyclic vs. acyclic– positive weights only vs. negative weights allowed– multiple weight types to optimize

The Trouble with Negative Weighted Cycles

A B

C D

E

2 10

1-5

2

What’s the shortest path from A to E?(or to B, C, or D, for that matter)

Unweighted Shortest Path Problem

Assume source vertex is C…

A

C

B

D

F H

G

E

Distance to: A B C D E F G H

Dijkstra

Legendary figure in computer science; now a professor at University of Texas.

Supports teaching introductory computer courses without computers (pencil and paper programming)

Supposedly wouldn’t (until recently) read his e-mail; so, his staff had to print out messages and put them in his box.

Dijkstra’s Algorithm for Single Source Shortest Path

• Classic algorithm for solving shortest path in weighted graphs without negative weights

• A greedy algorithm (irrevocably makes decisions without considering future consequences)

• Intuition:– shortest path from source vertex to itself is 0

– cost of going to adjacent nodes is at most edge weights

– cheapest of these must be shortest path to that node

– update paths for new node and continue picking cheapest path

Intuition in Action

A

C

B

D

F H

G

E

2 2 3

21

1

410

8

11

94

2

7

Dijkstra’s Pseudocode(actually, our pseudocode for Dijkstra’s algorithm)

Initialize the cost of each node to Initialize the cost of the source to 0

While there are unknown nodes left in the graphSelect the unknown node with the lowest cost: n

Mark n as known

For each node a which is adjacent to n

a’s cost = min(a’s old cost, n’s cost + cost of (n, a))

Dijkstra’s Algorithm in Action

A

C

B

D

F H

G

E

2 2 3

21

1

410

8

11

94

2

7

vertex known costABCDEFGH

THE KNOWNCLOUD

G

Next shortest path from inside the known cloud

P

Better pathto the same node

The Cloud Proof

But, if the path to G is the next shortest path, the path to P must be at least as long.

So, how can the path through P to G be shorter?

Source

Inside the Cloud (Proof)

Everything inside the cloud has the correct shortest path

Proof is by induction on the # of nodes in the cloud:– initial cloud is just the source with shortest path 0

– inductive step: once we prove the shortest path to G is correct, we add it to the cloud

Negative weights blow this proof away!

Revenge of the Dijkstra Pseudocode

Initialize the cost of each node to s.cost = 0;

heap.insert(s);

While (! heap.empty)

n = heap.deleteMin()

For (each node a which is adjacent to n)

if (n.cost + edge[n,a].cost < a.cost) then

a.cost = n.cost + edge[n,a].cost

a.path = n;

if (a is in the heap) then heap.decreaseKey(a)

else heap.insert(a)

Data Structures for Dijkstra’s Algorithm

Select the unknown node with the lowest cost

findMin/deleteMin

a’s cost = min(a’s old cost, …)

decreaseKey

find by name

|V| times:

|E| times:

runtime:

O(log |V|)

O(log |V|)

O(1)