graphs topic 21
DESCRIPTION
Graphs Topic 21. - PowerPoint PPT PresentationTRANSCRIPT
GraphsTopic 21
" Hopefully, you've played around a bit with The Oracle of Bacon at Virginia and discovered how few steps are necessary to link just about anybody who has ever been in a movie to Kevin Bacon, but could there be some actor or actress who is even closer to the center of the Hollywood universe?.By processing all of the almost half of a million people in the Internet Movie Database I discovered that there are currently 1160 people who are better centers than Kevin Bacon! … By computing the average of these numbers we see that the average (Sean) Connery Number is about 2.682 making Connery a better center than Bacon"
-Who is the Center of the Hollywood Universe?, University of VirginiaThat was in 2001. In 2013 Harvey Keitel has become the center of the Hollywood Universe. Connery is 136th. Bacon has moved up to 370th.
CS314 2
An Early Problem in Graph Theory
8 Leonhard Euler (1707 - 1783)– One of the first mathematicians to study graphs
8 The Seven Bridges of Konigsberg Problem8 A puzzle for the residents of the city8 The river Pregel flows through the city8 7 bridges cross the river8Can you cross all bridges while crossing
each bridge only once?
Graphs
CS314 3
Konigsberg and the River Pregel
Graphs
Clicker Question 18How many solutions does the Seven Bridges
of Konigsberg Problem have?A. 0B. 1C. 2D. 3E. 4 or more
CS314 Graphs 4
CS314 5
How to Solve8 Brute Force?8 Euler's Solution
– Redraw the map as a graph (really a multigraph)
a
b
c
d
Graphs
CS314 6
Euler's Proposal 8 A connected graph has an Euler tour
(possible to cross each edge only once while traversing each edge only once and returning to starting point) if and only if every vertex has an even number of edges– Eulerian Circuit
8What if we reduce the problem to only crossing each exactly once edge once?– Doesn't matter if we end up where we started– Eulerian Trail
Graphs
CS314 7
Graph Definitions8 A graph is comprised of a set of vertices
(nodes) and a set of edges (links, arcs) connecting the vertices
8 in a directed graph edges are one-way– movement allowed from first node to second, but
not second to first– directed graphs also called digraphs
8 in an undirected graph edges are two-way– movement allowed in either direction
Graphs
Definitions8 In a weighted graph the edge has cost or weight
that measures the cost of traveling along the edge8 A path is a sequence of vertices connected by
edges– The path length is the number of edges– The weighted path length is the sum of the cost of the
edges in a path8 A cycle is a path of length 1 or more that starts and
ends at the same vertex– a directed acyclic graph is a directed graph with no
cycles
CS314 Graphs 8
CS314 9
Graphs We've Seen
link link link link
link link link
19
12 35
3 16 5621
Graphs
Example Graph8Computer Scientists use graphs to model all
kinds of things
CS314 Graphs 10
Arpanet 1969, 1971
Example Graph
CS314 Graphs 11
Enron emails 2001
Example Graph
CS314 Graphs 12
Example Graph
CS314 Graphs 13"Jefferson" High School, Ohio http://researchnews.osu.edu/archive/chains.htm, 2005,
CS314 14
8How to store a graph as a data structure?8
Representing Graphs
Graphs
CS314 15
Adjacency Matrix Representation
8 A Br Bl Ch Co E FG G Pa Pe S U VA 0 1 1 1 0 0 0 0 1 0 0 1 0Br 1 0 1 0 1 0 1 1 1 1 1 1 1 Bl 1 1 0 1 0 0 0 0 1 1 0 0 0 Ch 1 0 1 0 0 0 0 0 0 1 0 0 0Co 0 1 0 0 0 1 0 0 0 1 0 0 1E 0 0 0 0 1 0 0 0 0 1 0 0 0 FG 0 1 0 0 0 0 0 0 0 0 1 0 0G 0 1 0 0 0 0 0 0 0 0 1 0 1Pa 1 1 1 0 0 0 0 0 0 0 0 0 0Pe 0 1 1 1 1 1 0 0 0 0 0 0 0S 0 1 0 0 0 0 1 1 0 0 0 0 0U 1 1 0 0 0 0 0 0 0 0 0 0 0V 0 1 0 0 1 0 0 1 0 0 0 0 0
Country Code
Argentina A
Brazil Br
Bolivia Bl
Chile Ch
Columbia Co
Ecuador E
French Guiana FG
Guyana G
Paraguay Pa
Peru Pe
Suriname S
Uruguay U
Venezuela V
Graphs
CS314 16
The Map Coloring Problem8How many colors do you need to color a
map, so that no 2 countries that have a common border (not a point) are colored the same?
8How to solve using Brute Force?
Graphs
CS314 17
8
A Solution
Green
Green
Green
Blue
Yellow
Blue
Yellow
Blue
Yellow
Yellow
Blue
Red
Graphs
CS314 18
What About the Ocean? A Br Bl Ch Co E FG G Pa Pe S U V OcA 0 1 1 1 0 0 0 0 1 0 0 1 0 1Br 1 0 1 0 1 0 1 1 1 1 1 1 1 1 Bl 1 1 0 1 0 0 0 0 1 1 0 0 0 0 Ch 1 0 1 0 0 0 0 0 0 1 0 0 0 1Co 0 1 0 0 0 1 0 0 0 1 0 0 1 1E 0 0 0 0 1 0 0 0 0 1 0 0 0 1 FG 0 1 0 0 0 0 0 0 0 0 1 0 0 1G 0 1 0 0 0 0 0 0 0 0 1 0 1 1Pa 1 1 1 0 0 0 0 0 0 0 0 0 0 0Pe 0 1 1 1 1 1 0 0 0 0 0 0 0 1S 0 1 0 0 0 0 1 1 0 0 0 0 0 1U 1 1 0 0 0 0 0 0 0 0 0 0 0 1V 0 1 0 0 1 0 0 1 0 0 0 0 0 1Oc 1 1 0 1 1 1 1 1 0 1 1 1 1 0
Graphs
CS314 19
8
Make the Ocean Blue
Green
Green
Green
Blue
Yellow
Blue
Yellow
BlueYellow
Yellow
Red
Graphs
Red
Red
Red
More Definitions8 A dense graph is one with a large number of
edges– maximum number of edges?
8 A sparse graph is one in which the number of edges is much less than the maximum possible number of edges– No standard cutoff for dense and sparse graphs
CS314 Graphs 20
Graph Representation8 For dense graphs the adjacency matrix is a
reasonable choice– For weighted graphs change booleans to cost– Can the adjacency matrix handle directed
graphs?8Most graphs are sparse, not dense8 For sparse graphs an adjacency list is an
alternative that uses less space8 Each vertex keeps a list of vertices it is
connected to.CS314 Graphs 21
Graph Implementationpublic class Graph
private static final double INFINITY = Double.MAX_VALUE;
private Map<String, Vertex> vertices;
public Graph() // create empty Graph
public void addEdge(String source, String dest, double cost)
// find all paths from given vertexpublic void findUnweightedShortestPath
(String startName)
// called after findUnweightedShortestPathpublic void printPath(String destName)
Graph Class8 This Graph class stores vertices8 Each vertex has an adjacency list
– what vertices does it connect to?8 shortest path method finds all paths from
start vertex to every other vertex in graph8 after shortest path method called queries can
be made for path length from start node to destination node
CS314 Graphs 23
Vertex Class (nested in Graph)
CS314 Graphs 24
private static class Vertexprivate String name;private List<Edge> adjacent;
public Vertex(String n)
// for shortest path algorithmsprivate double distance;private Vertex prev; private int scratch;
// call before finding new pathspublic void reset()
Edge Class (nested in Graph)
CS314 Graphs 25
private static class Edgeprivate Vertex dest;private double cost;
private Edge(Vertex d, double c)
Unweighted Shortest Path8Given a vertex, S (for start) find the shortest
path from S to all other vertices in the graph8Graph is unweighted (set all edge costs to 1)
CS314 Graphs 26
S
V5V3
V1 V6
V4
V2
V7
V8
Word Ladders8 Agree upon dictionary8 Start word and end word of
same length8Change one letter at a time to
form step8 Step must also be a word8 Example: Start = silly, end =
funny
CS314 Graphs 27
sillysullysulkyhulkyhunkyfunkyfunny
Graph Representation8What are the vertices and when does
an edge exist between two vertices?
Vertices EdgesA. Letters WordsB. Words Words that share one or more
lettersC. Letters Words that share one or more
lettersD. Words Words that differ by one letterE. Words Letters
CS314 Graphs 28
CS314 Graphs 29
smart
swart
start
smarm
smalt
scart
Portion of Graph
Size of Graph8 Number of vertices and edges depends on dictionary8 Modified Scrabble dictionary, 5 letter words8 Words are vertices
– 8660 words8 Edge exists between word if they are one letter
different– 24,942 edges
Is this graph sparse or dense?A. SparseB. Dense
CS314 Graphs 30
Max number of edges = N * (N - 1) / 237,493,470
Unweighted Shortest Path Algorithm8 Problem: Find the shortest word ladder
between two words if one exists8What kind of search should we use?
A. Breadth First SearchB. Depth First SearchC. Either one
CS314 Graphs 31
Unweighted Shortest Path Algorithm8 Set distance of start to itself to 08Create a queue and add the start vertex8while the queue is not empty
– remove front – loop through all edges of current vertex
• get node edge connects to• if this node has not been visited
– sets its distance to current distance + 1– sets its previous node to current node– add new node to queue
CS314 Graphs 32
CS314 Graphs 33
smart
swart
start
smarm
smalt
scart
Portion of Graph
CS314 Graphs 34
smart
swart
start
smarm
smalt
scart
Start at "smart" and enqueue it[smart]
CS314 Graphs 35
smart
swart
start
smarm
smalt
scart
Dequeue (smart), loop through edges[swart]
CS314 Graphs 36
smart
swart
start
smarm
smalt
scart
Dequeue (smart), loop through edges[swart, start]
CS314 Graphs 37
smart
swart
start
smarm
smalt
scart
Dequeue (smart), loop through edges[swart, start, scart]
CS314 Graphs 38
smart
swart
start
smarm
smalt
scart
Dequeue (smart), loop through edges[swart, start, scart, smalt]
CS314 Graphs 39
smart
swart
start
smarm
smalt
scart
Dequeue (smart), loop through edges[swart, start, scart, smalt, smarm]
CS314 Graphs 40
smart
swart
start
smarm
smalt
scart
Done with smart, dequeue (swart)[start, scart, smalt, smarm]
CS314 Graphs 41
smart
swart
start
smarm
smalt
scart
loop through edges of swart (start already present)[start, scart, smalt, smarm]
CS314 Graphs 42
smart
swart
start
smarm
smalt
scart
loop through edges of swart (scart already present)[start, scart, smalt, smarm]
CS314 Graphs 43
smart
swart
start
smarm
smalt
scart
loop through edges of swart[start, scart, smalt, smarm, swarm]
swarm
CS314 Graphs 44
smart
swart
start
smarm
smalt
scart
loop through edges of swart[start, scart, smalt, smarm, swarm, sware]
swarm
sware
Unweighted Shortest Path8 Implement method8 demo8 how is path printed?8 The diameter of a graph is the longest
shortest past in the graph8How to find?8How to find center of graph?
– vertex connected to the largest number of other vertices with the shortest average path length
CS314 Graphs 45
Positive Weighted Shortest Path8 Edges in graph are weighted and all weights
are positive8 Similar solution to unweighted shortest path8Dijkstra's algorithhm8 Edsger W. Dijkstra (1930–2002)8UT Professor 1984 - 20008 Algorithm developed in 1956
and published in 1959.
CS314 Graphs 46
Dijkstra's Algorithm8 Pick the start vertex8 Set the cost of the start vertex to 0 and all other
vertices to INFINITY8 While there are unvisited vertices:
– Let the current vertex be the lowest cost vertex that has not yet been visited
– mark current vertex as visited– for each edge from the current vertex
• if the sum of the cost of the current vertex and the cost of the edge is less than the cost of the destination vertex
– update the cost of the destination vertex– set the previous of the destination vertex to the current vertex
47
Dijkstra's Algorithm8 Example of a Greedy Algorithm
– A Greedy Algorithm does what appears to be the best thing at each stage of solving a problem
8Gives best solution in Dijkstra's Algorithm8Does NOT always lead to best answer8 Fair teams:
– (10, 10, 8, 8, 8), 2 teams8Making change with fewest coins
(1, 5, 10) 15 cents(1, 5, 12) 15 cents
CS314 Graphs 48
49
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
What is the lowest cost path from A to E?A. 3B. 17C. 20D. 28E. 37
50
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
A is start vertexSet cost of A to 0, all others to INFINITYPlace A in a priority queue
51
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(A,0)] pqdequeue (A,0)Mark A as visited
52
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[ ] pqcurrent vertex A:loop through A's edgesif sum of cost from A to edge is less than current cost
update cost and prev
53
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[ ] pqA -> B, 0 + 1 < INFINITY[(B,1)] pq
54
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(B,1)] pqA -> C, 0 + 7 < INFINITY[(B,1), (C, 7)] pq
55
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(B,1), (C, 7)] pqA -> G, 0 + 17 < INFINITY[(B,1), (C, 7), (G, 17)] pq
56
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(B,1), (C, 7), (G, 17)] pqcurrent vertex B:loop through B's edgesif sum of cost from B to edge is less than current cost
update cost and prev
57
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 7), (G, 17)] pqB -> C, 1 + 3 < 7update C's cost and previous[(C, 4), (C, 7), (G, 17)] pq
58
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 4), (C, 7), (G, 17)] pqB -> D, 1 + 21 < INFINITY[(C, 4), (C, 7), (G, 17), (D, 22)] pq
59
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 4), (C, 7), (G, 17), (D, 22)] pqcurrent vertex is C, cost 4loop through C's edges
60
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 7), (G, 17), (D, 22)] pqC -> A, A already visited so skip
61
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 7), (G, 17), (D, 22)] pqC -> B, B already visited so skip
62
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 7), (G, 17), (D, 22)] pqC -> F, 4 + 3 < INFINITY[(C, 7), (F, 7), (G, 17), (D, 22)] pq
63
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(C, 7), (F, 7), (G, 17), (D, 22)] pqcurrent vertex is CAlready visited so skip
64
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(F, 7), (G, 17), (D, 22)] pqcurrent vertex is Floop through F's edges
65
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (D, 22)] pqF -> C, already visited so skip
66
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (D, 22)] pqF -> D, 7 + 4 < 22update D's cost and previous[(D, 11), (G, 17), (D, 22)] pq
67
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(D, 11), (G, 17), (D, 22)] pqcurrent vertex is Dloop through D's edges
68
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (D, 22)] pqD -> B, already visited so skip
69
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (D, 22)] pqD -> E, 11 + 6 < INFINITYupdate E's cost and previous[(G, 17), (E, 17), (D, 22)] pq
70
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[ (G, 17), (E, 17), (D, 22)] pqD -> F, already visited so skip
71
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (E, 17), (D, 22)] pqD -> G, 11 + 5 < 17update G's cost and previous[(G, 16), (G, 17), (E, 17), (D, 22)] pq
72
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(G, 17), (E, 17), (D, 22)] pqcurrent vertex is Gloop though edges, already visited all neighbors
73
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
[(E, 17), (D, 22)] pqcurrent vertex is Eloop though edges, already visited all neighbors
74
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
No unvisited vertices.Done.
Implementing Dijkstra's8Create a Path class to allow for multiple
distances to a given vertexprivate static class Path
implements Comparable<Path> {
private Vertex dest;private double cost;
8Use a priority queue of Paths to store the vertices and distances
CS314 Graphs 75
Alternatives to Dijkstra's Algorithm 8 A*, pronounced "A Star"8 A heuristic, goal of finding shortest weighted path
from single start vertex to goal vertex8 Uses actual distance like Dijkstra's but also
estimates remaining cost or distance– distance is set to current distance from start PLUS the
estimated distance to the goal8 For example when finding a path between towns,
estimate the remaining distance as the straight-line (as the crow files) distance between current location and goal.
CS314 Graphs 76
Spanning Tree8 Spanning Tree: A tree of edges that connects
all the vertices in a graph
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Minimum Spanning Tree8Minimum Spanning Tree: A spanning tree in
a weighted graph with the lowest total cost8 used in network design, taxonomy, Image registration,
and more!
78
A
B
C
F
DE
G
1
3
3
214
6
17 5
7Cost of spanningtree shown?A. 6B. 7C. 29D. 61E. None of These
Prim's Algorithm8 Pick a vertex arbitrarily from graph
– In other words, it doesn't matter which one8 Add lowest cost edge between the tree and a
vertex that is not part of the graph UNTIL every vertex is part of the tree
8Greedy Algorithm, very similar to Dijkstra's
CS314 Graphs 79
Prim's Algorithm
CS314 Graphs 80
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Pick D as root
5 8 4
Prim's Algorithm
CS314 Graphs 81
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?2 from D to A (or C)
5 8 4
Prim's Algorithm
CS314 Graphs 82
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?2 from D to C (OR from A o B)
5 8 4
Prim's Algorithm
CS314 Graphs 83
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?2 from A to B
5 8 4
Prim's Algorithm
CS314 Graphs 84
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?5 from D to G
5 8 4
Prim's Algorithm
CS314 Graphs 85
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?1 from G to F
5 8 4
Prim's Algorithm
CS314 Graphs 86
A B
C
F
D E
G
2 311
2
7
6
2
1
4
Lowest cost edge from tree to vertex not in Tree?6 from G to E
5 8 4
Prim's Algorithm
CS314 Graphs 87
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Pick D as root
Prim's Algorithm
CS314 Graphs 88
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?4 from D to F
Prim's Algorithm
CS314 Graphs 89
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?3 from F to C
Prim's Algorithm
CS314 Graphs 90
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?3 from C to B
Prim's Algorithm
CS314 Graphs 91
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?1 from B to A
Prim's Algorithm
CS314 Graphs 92
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?5 from D to G
Prim's Algorithm
CS314 Graphs 93
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Lowest cost edge from tree to vertex not in Tree?6 from D to E
Prim's Algorithm
CS314 Graphs 94
A
B
C
F
DE
G
1
3
3
214
6
17 5
7
Cost of Spanning Tree?
Other Graph Algorithms8 Lots!8 http://en.wikipedia.org/wiki/Category:Graph_algorithms
CS314 Graphs 95