lecture12 graphs

7/25/2019 Lecture12 Graphs

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Lecture 12: Graphs


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Graph Terminology

vertex, node, point

edge, line, arc

G = (V, E)

V is set of vertices

E is set of edges

Each edge joins two different vertices

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Undirected Graph edges do not have a direction

The edge from 1 to 2 is also an edge

from 2 to 1 Edge (1, 2) implies that there is also

an edge (2, 1) [ The same edge ]

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Directed Graph edges have a direction

Edge (2, 1) means only that there is an

edge from 2 to 1 In this example,there is no edge from 1

to 2

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Weighted Graph

weights (values) are associated with the

edges in the graph

may be directed for undirected

Weighted graphs are also referred to as

networks

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Complete Graph

For each pair of vertices, there is one

edge

If G = (V, E) is a complete graph, and

|V| = n, then can you calculate |E|?

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Subgraph

A subgraph G of graph G = (V, E) is a

graph (V, E) that V V and E E.

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Path

the sequence of edges (i1, i2), (i2,

i3), ,(ik-1, ik).

Denoted as path i1, i2, ..., ik

Simple pathall vertices (expect

possibly first and last) are different

Length of path is sum of the lengths of

the edges


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Examples of graph


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Representation of Graphs

Adjacency matrix

Incidence matrix

Adjacency lists: Table, Linked List

Space/time trade-offs depending on

which operation are most frequent as

well as properties of the graph


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Can we use tree traversal

algorithms to traverse graphs?

Why?

Loops in graphs, parent/childrelation in trees


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Depth first search

DFS(v)

num(v)=i++;

for all vertices u adjacent to v

if num(u) is 0

attach edge(uv) to edges;DFS(u);

depthFirstSearch()

for all vertices v

num(v)=0;edges=null; i=1;

while there is a vertex v such that num(v) is 0

DFS(v);

output edges


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Breadth first searchbreadthFirstSearch()

for all vertices u

num(u)=0;

Edges=null; i=1;

While there is a vertex v such that num(v)==0

num(V)=i++;

enqueue(v);

while queue is not empty

v=dequeue();

for all vertices u adjacent to vif num(u) is 0

num(u)=i++;

enqueue(u);

attach edge(uv) to edges;

Output edges;


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An Example

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What would the visit orders for

DFS(1), DFS(5), BFS(1), BFS(5)

look like?


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Unweighted Shortest Path

Find the shortest path (measured by

number of edges) from a designated

vertex S to every vertex Simplified case of weighted shortest

path


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Algorithm

Starting from node S

Distance from S to S is 0, so label as 0

Find all nodes which are distance 1 from S Label as distance 1

Find all nodes which are distance 2 from S

These are 1 step from those labeled 1 This is precisely a breadth first search


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An Example

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What are the values of distance[2], distance[3], ,

distance[7] after FindShortestPath(G, 1) is executed?


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Positive Weighted Shortest Path

Length is sum of the edges costs on thepath

All edges have nonnegative cost Find shortest paths from some start

vertex to all vertices

similar process to unweighted case Dijkstra's Algorithm(Single source

shortest path)


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Distance at each node v is shortest path

distance from s to v using only knownvertices as intermediates

An example of a Greedy Algorithm

Solve problem in stages by doing whatappears to be the best thing at eachstage

Decisionin one stage is not changed

laterA simple greedy example is counting

change (quarters, then dimes,)


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Greedy algorithms do not always work

Change breaks down with addition

of .12 coin ( try .15 value )

They do work for certain problems and

are fairly simple


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DijkstraAlgorithm(weighted Simpledigraph, vertexfirst)

for all verticesv

currDist(v)=infinity;currDist(first)=0;

toBeChecked=all vertices;

WhiletoBeChecked is not empty

v = a vertex intoBeChecked with minimalcurrDist(v);

removev fromtoBeChecked;

for all verticesu adjacentto v and intoBeChecked

if currDist(u)>currDist(v)+weight(edge(uv))

currDist(u)=currDist(v)+weight(edge(uv));

predecessor(u)=v;


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start at v1, all distances are infinity

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mark v1 as known, with distance 0

0


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Now adjust distances for v2 and v4

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Select v4 as known, since it has the lowest cost among unknown

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Examine adjacent, adjust v3, v5, v6 and v7

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Select v2 as known

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Check v4, v5. v4 is final, v5 has lower cost of 3 vs. 12

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Select v5 as known

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v3 is selected as known

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Check v1 and v6. v1 known, v6 adjusted from 9 to 8


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Select v7 as known

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Check v6, change from 8 to 6


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Select v6 as known

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Nothing to check

Done


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Spanning tree

subgraph of G

contains all vertices of G

connected graph with no cycles


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Examples of spanning trees


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Minimum spanning tree

spanning tree with minimum cost

only exists if G is connected

number of edges is |V|-1 three greedy methods

Kruskal's algorithm

Dijkstras method Prim's algorithm

differ in how next edge is selected


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Kruskal's algorithm

KruskalAlgorithm( weighted connectedundirectedgraph)

tree = null;edges = sequence of all edges ofgraphsorted by weight;

for(i=1;i


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Construct MST for this graph using Kruskal's algorithm starting

v1

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v1

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v1

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v1

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Dijkstras method

DijkstraMethod( weighted connected undirectedgraph)

tree = null;

edges = an unsorted sequence of all edges ofgraph;

for j=1 to |E|

addei to tree;

if there is a cycle in treeremove an edge with maximum

weight from this only cycle;


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Prim's algorithm

Similar to the Dijkstras algorithm inshortest paths

grow the tree in successive stages in each stage, one node is picked as the

root, we add an edge, and thus a vertexis added to the tree

have a set on vertices in the tree and aset that is not in the tree


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Prim's algorithm (cont.)

at each stage, a new vertex to add tothe tree is selected by choosing edge (u,

v) such that the cost of (u,v) is thesmallest among all edges where u is inthe tree and v is not

Build spanning tree starting from v1

Result in the same spanning tree asthat given by the Kruskal algorithm


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Construct MST from v1 for this graph using Prim's algorithm

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Same MST as Kruskal

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