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GraphTerminology

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Apath is a sequence of nodesv1, v2, , vN such that (vi,vi+1)E for 0

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PIT

BOS

JFK

DTW

LAX

SFO

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In a directed graph:

Theindegree of a node v is thenumber ofdistinct edges (w,v)E.

Theoutdegree of a node v is thenumber of

distinct edges (v,w)

E.

A node withindegree 0 is a root.

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A dag is a directed acyclicgraph.

A tree is a connected acyclicundirected graph.

Aforest is an acyclicundirected graph (notnecessarilyconnected), i.e., eachconnectedcomponent is a tree.

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Watch

Socks

Shoes

Undershorts

Pants

Belt Tie

Shirt

Jacket

a DAG implies an

ordering on events

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Watch

Socks

Shoes

Undershorts

Pants

Belt Tie

Shirt

Jacket

In a complex DAG, it

can be hard to find a

schedule that obeys

all the constraints.

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Graph Traversals

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Both take time: O(V+E)

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It is verycommon to use a stack to keep trackof:

nodes to be visited next, or

nodes that wehave already visited.

Typically, use of a stack leads to a depth-firstvisit order.

Depth-first visit order is aggressivein thesense that it examinescomplete paths.

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It is verycommon to use a queue to keep trackof: nodes to be visited next, or

nodes that wehave already visited. Typically, use of a queue leads to a breadth-first

visit order.

Breadth-first visit order is cautiousin the

sense that it examinesevery path of lengthibeforegoing on to paths of lengthi+1.

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Central office

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Central office

Expensive!

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Central office

Minimize the total length of wire connecting the customers

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(see Weiss, Section 24.2.2)

it is a tree (i.e., it is acyclic) it covers all the vertices V

contains |V| - 1 edges

the total cost associated with tree edges is the

minimum among all possible spanning trees not necessarily unique

A minimum spanning tree is a subgraph of an

undirected weighted graph G, such that

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Any timeyou want to visit all verticesin a graph atminimumcost (e.g., wire routing on printed circuitboards, sewer pipe layout, road planning)

Internet content distribution Idea: publisher produces web pages, content distribution

network replicates web pages to many locationsso consumerscan access at higher speed

Provides a heuristic for travelingsalesman problems.

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a

ce

d

b2

45

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a

ce

d

b2

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Initialization

a. Pick a vertex r to be the root

b. Set D(r) = 0, parent(r) =null

c. For all vertices v V, v{ r, set D(v) =g

d. Insert all vertices into priority queue P,using distances as the keys

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e a b c d

0 g g g g

Vertex Parent

e -

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While P is not empty:

1. Select the next vertex u to add to the tree

u=P.deleteMin()

2. Update the weight of each vertex w adjacent to

u which is not in the tree (i.e., w P)

Ifweight(u,w) < D(w),

a.parent(w) = u

b. D(w) = weight(u,w)c. Update the priority queue to reflect

new distance forw

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a

ce

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d b c a

4 5 5 g

Vertex Parent

e -

b e

c e

d e

The MST initially consists of the vertex e, and we update

the distances and parent for its adjacent vertices

Vertex Parent

e -

b -

c -

d -

d b c a

g g g g

e

0

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ce

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a c b

2 4 5

Vertex Parent

e -

b e

c d

d e

a d

d b c a

4 5 5 g

Vertex Parent

e -

b e

c e

d e

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a

ce

d

b2

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c b

4 5

erte are t

e -

b ec d

d e

a d

a c b

2 4 5

erte are t

e -b e

c d

d e

a d

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a

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b

5

Vertex Parent

e -

b e

c dd e

a d

c b

4 5

Vertex Parente -

b e

c d

d e

a d

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Vertex Parent

e -

e

c

e

a

a

ce

d

b

2

45

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The final minimum spanning tree

b

5

Vert

ex Par

ente -

b e

c d

d e

a d

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At eachstep, we add theedge(u,v) s.t. theweight of (u,v) isminimum among all edgeswhereu isin the tree and v isnot in the tree

Eachstep maintains a minimumspanning treeof the vertices that havebeenincluded thus far

When all verticeshavebeenincluded, wehavea MST for thegraph!

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a

ce

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Create a forest of trees from the vertices

Repeatedly merge trees by adding safe edges

until only one tree remains

A safe edge is an edge of minimum weight which

does not create a cycle

forest: {a}, {b}, {c}, {d}, {e}

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Initialization

a. Create a set for each vertex v V

b. Initialize the set of safe edges A

comprising the MST to the empty set

c. Sort edges by increasing weight

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F={a}, {b}, {c}, {d}, {e}

A=

E= {(a,d), (c,d), (d,e), (a,c),

(b,e), (c,e), (b,d), (a,b)}

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For each edge (u,v) E in increasing orderwhile more than one set remains:

Ifu and v, belong to different sets Uand V

a. add edge (u,v) to the safe edge set

A=A {(u,v)}

b. merge the sets Uand V

F=F - U - V + (U V)

Return A

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E= {(a,d), (c,d), (d,e), (a,c),

(b,e), (c,e), (b,d), (a,b)}

Forest

{a}, {b}, {c}, {d}, {e}

{a,d}, {b}, {c}, {e}

{a,d,c}, {b}, {e}{a,d,c,e}, {b}

{a,d,c,e,b}

A

{(a,d)}

{(a,d), (c,d)}{(a,d), (c,d), (d,e)}

{(a,d), (c,d), (d,e), (b,e)}

a

c e

d

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