1. Shortest Paths C B A E D F 3 2 8 5 4 7 1 9 Shortest Paths 1
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Shortest Paths C B A E D F 3 2 8 5 4 7 1 9
Shortest Paths 1 1

2. Weighted Graphs PVD ORD SFO LGA HNL LAX DFW MIA
Shortest Path
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Weighted Graphs
In a weighted graph, each edge has an associated numerical value, called the weight of the edge
Edge weights may represent, distances, costs, etc.
Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
849
PVD
1843
ORD
142
SFO
802
LGA
1743
1205
337
1387
HNL
2555
1099
LAX
1233
DFW
1120
MIA
Shortest Paths 2

3. Shortest Paths PVD ORD SFO LGA HNL LAX DFW MIA
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Shortest Paths
Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v.
Length of a path is the sum of the weights of its edges.
Example:
Shortest path between Providence and Honolulu
Applications
Internet packet routing
Flight reservations
Driving directions
849
PVD
1843
ORD
142
SFO
802
LGA
1743
1205
337
1387
HNL
2555
1099
LAX
1233
DFW
1120
MIA
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4. Shortest Path Properties
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Shortest Path Properties
Property 1:
A subpath of a shortest path is itself a shortest path
Property 2:
There is a tree of shortest paths from a start vertex to all the other vertices
Example:
Tree of shortest paths from Providence
849
PVD
1843
ORD
142
SFO
802
LGA
1743
1205
337
1387
HNL
2555
1099
LAX
1233
DFW
1120
MIA
Shortest Paths 4

5. Shortest Path
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Dijkstra’s Algorithm
The distance of a vertex v from a vertex s is the length of a shortest path between s and v
Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s
Assumptions:
the graph is connected
the edges are undirected
the edge weights are nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices
We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices
At each step
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
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6. Edge Relaxation Consider an edge e  (u,z) such that d(z) 75
Shortest Path
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Edge Relaxation
Consider an edge e  (u,z) such that
u is the vertex most recently added to the cloud
z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:
d(z)  min{d(z),d(u) weight(e)}
d(u) 50 10
d(z) 75 e u s z
d(u) 50 10
d(z) 60 u e s z
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7. Shortest Path
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Example C B A E D F 3 2 8 5 4 7 1 9 A 4 8 2 8 2 4 7 1 B C D 3 9   2 5 E F A 4 A 4 8 8 2 2 8 2 3 7 2 3 7 1 7 1 B C D B C D 3 9 3 9 5 11 5 8 2 5 2 5 E F E F
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8. Example (cont.) A 4 8 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F A 4 8 2 7 2 3
Shortest Path
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Example (cont.) A 4 8 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F A 4 8 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F
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9. Shortest Path
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Dijkstra’s Algorithm
Algorithm DijkstraDistances(G, s)
Q  new heap-based priority queue
for all v  G.vertices()
if v s
v.setDistance(0)
else
v.setDistance()
l  Q.insert(v.getDistance(), v)
v.setEntry(l)
while Q.empty()
l  Q.removeMin()
u  l.getValue()
for all e  u.incidentEdges() { relax e }
z  e.opposite(u)
r  u.getDistance() e.weight()
if r z.getDistance()
z.setDistance(r) Q.replaceKey(z.getEntry(), r)
A heap-based adaptable priority queue with location-aware entries stores the vertices outside the cloud
Key: distance
Value: vertex
Recall that method replaceKey(l,k) changes the key of entry l
We store two labels with each vertex:
Distance
Entry in priority queue
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10. Analysis of Dijkstra’s Algorithm
Shortest Path
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Analysis of Dijkstra’s Algorithm
Graph operations
Method incidentEdges is called once for each vertex
Label operations
We set/get the distance and locator labels of vertex z O(deg(z)) times
Setting/getting a label takes O(1) time
Priority queue operations
Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time
The key of a vertex in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time
Dijkstra’s algorithm runs in O((n  m) log n) time provided the graph is represented by the adjacency list structure
Recall that v deg(v) 2m
The running time can also be expressed as O(m log n) since the graph is connected
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11. Shortest Path
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Shortest Paths Tree
Using the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices
We store with each vertex a third label:
parent edge in the shortest path tree
In the edge relaxation step, we update the parent label
Algorithm DijkstraShortestPathsTree(G, s) …
for all v  G.vertices()
v.setParent()
for all e  u.incidentEdges()
{ relax edge e }
z  e.opposite(u)
r  u.getDistance() e.weight()
if r z.getDistance()
z.setDistance(r)
z.setParent(e)
Q.replaceKey(z.getEntry(),r)
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12. Why Dijkstra’s Algorithm Works
Shortest Path
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Why Dijkstra’s Algorithm Works
Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance.
Suppose it didn’t find all shortest distances. Let F be the first wrong vertex the algorithm processed.
When the previous node, D, on the true shortest path was considered, its distance was correct
But the edge (D,F) was relaxed at that time!
Thus, so long as d(F)>d(D), F’s distance cannot be wrong. That is, there is no wrong vertex A 8 4 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F
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13. Why It Doesn’t Work for Negative-Weight Edges
Shortest Path
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Why It Doesn’t Work for Negative-Weight Edges
Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. A 8 4
If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud. 6 7 5 4 7 1 B C D -8 5 9 2 5 E F
C’s true distance is 1, but it is already in the cloud with d(C)=5!
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14. Bellman-Ford Algorithm (not in book)
Shortest Path
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Bellman-Ford Algorithm (not in book)
Works even with negative- weight edges
Must assume directed edges (for otherwise we would have negative- weight cycles)
Iteration i finds all shortest paths that use i edges.
Running time: O(nm).
Can be extended to detect a negative-weight cycle if it exists
How?
Algorithm BellmanFord(G, s)
for all v  G.vertices()
if v s
v.setDistance(0)
else
v.setDistance()
for i  1 to n1 do
for each e  G.edges()
{ relax edge e }
u  e.origin()
z  e.opposite(u)
r  u.getDistance() e.weight()
if r z.getDistance()
z.setDistance(r)
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15. Nodes are labeled with their d(v) values
Shortest Path
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Bellman-Ford Example
Nodes are labeled with their d(v) values 4 4 8 8 -2 -2 8 -2 4 7 1 7 1       3 9 3 9 -2 5 -2 5     4 4 8 8 -2 -2 5 7 1 -1 7 1 8 -2 4 5 -2 -1 1 3 9 3 9 6 9 4 -2 5 -2 5   1 9
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16. DAG-based Algorithm (not in book)
Shortest Path
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DAG-based Algorithm (not in book)
Algorithm DagDistances(G, s)
for all v  G.vertices()
if v s
v.setDistance(0)
else
v.setDistance()
{ Perform a topological sort of the vertices }
for u  1 to n do {in topological order}
for each e  u.outEdges()
{ relax edge e }
z  e.opposite(u)
r  u.getDistance() e.weight()
if r z.getDistance()
z.setDistance(r)
Works even with negative-weight edges
Uses topological order
Doesn’t use any fancy data structures
Is much faster than Dijkstra’s algorithm
Running time: O(n+m).
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17. Nodes are labeled with their d(v) values
Shortest Path
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DAG Example
Nodes are labeled with their d(v) values 1 1 4 4 8 8 -2 -2 4 8 -2 4 4 3 7 2 1 3 7 2 1       3 9 3 9 -5 5 -5 5     6 5 6 5 1 1 4 4 8 8 -2 -2 5 3 7 2 1 4 -1 3 7 2 1 4 8 -2 4 5 -2 -1 9 3 3 9 1 7 4 -5 5 -5 5   1 7 6 5 6 5
Shortest Paths 17
(two steps)

18. Minimum Spanning Trees
Shortest Path
Minimum Spanning Tree
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Minimum Spanning Trees
2704
867
BOS
849
PVD
ORD
187
740
144
1846
621
JFK
184
1258
802
SFO
1391
BWI
1464
337
1090
DFW
946
LAX
1235
1121
MIA
2342
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19. Minimum Spanning Trees
Shortest Path
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Minimum Spanning Trees
Spanning subgraph
Subgraph of a graph G containing all the vertices of G
Spanning tree
Spanning subgraph that is itself a (free) tree
Minimum spanning tree (MST)
Spanning tree of a weighted graph with minimum total edge weight
Applications
Communications networks
Transportation networks
ORD 10 1
PIT
DEN 6 7 9 3
DCA
STL 4 8 5 2
DFW
ATL
Minimum Spanning Trees 19

20. Cycle Property Cycle Property: C C 8 f
Shortest Path
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Cycle Property 8 4 2 3 6 7 9 e C f
Cycle Property:
Let T be a minimum spanning tree of a weighted graph G
Let e be an edge of G that is not in T and let C be the cycle formed by e with T.
For every edge f of C, weight(f) weight(e).
Proof:
By contradiction
If weight(f) weight(e) we can get a spanning tree of smaller weight by replacing e with f.
Replacing f with e yields a better spanning tree 8 4 2 3 6 7 9 C e f
Minimum Spanning Trees 20

21. Partition Property Partition Property: U V 7 f
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Partition Property U V 7 f
Partition Property:
Consider a partition of the vertices of G into subsets U and V.
Let e be an edge of minimum weight across the partition.
There is a minimum spanning tree of G containing edge e.
Proof:
Let T be an MST of G.
If T does not contain e, consider the cycle C formed by e with T and let f be an edge of C across the partition.
By the cycle property, weight(f) weight(e).
Thus, weight(f) weight(e).
We obtain another MST by replacing f with e. 4 9 5 2 8 8 3 e 7
Replacing f with e yields another MST. U V 7 f 4 9 5 2 8 8 3 e 7
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22. Minimum Spanning Trees
Shortest Path
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Kruskal’s Algorithm
Maintain a partition of the vertices into clusters.
Initially, single-vertex clusters.
Keep an MST for each cluster.
Merge “closest” clusters and their MSTs.
A priority queue stores the edges outside clusters.
Key: weight
Element: edge
At the end of the algorithm,
there are one cluster and MST.
Algorithm KruskalMST(G)
for each vertex v in G do
Create a cluster consisting of v
let Q be a priority queue.
Insert all edges into Q
T  
{T is the union of the MSTs of the clusters}
while T has fewer than n 1 edges do e  Q.removeMin().getValue()
[u, v]  G.endVertices(e)
A  getCluster(u)
B  getCluster(v)
if A  B then
Add edge e to T
mergeClusters(A, B)
return T
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23. Shortest Path
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Example G G 8 8 B 4 B 4 E 9 E 6 9 6 5 1 F 5 1 F C 3 C 3 11 11 2 2 7 H 7 H D D A 10 A 10 G G 8 8 B 4 B 4 9 E E 6 9 6 5 1 F 5 1 F C 3 11 C 3 11 2 2 7 H 7 H D D A A 10 10
Campus Tour 23

24. Example (contd.) four steps two steps G G 8 8 B 4 B 4 E 9 E 6 9 6 5 F
Shortest Path
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Example (contd.) G G 8 8 B 4 B 4 E 9 E 6 9 6 5 F 5 1 1 F C 3 11 C 3 11 2 2 7 H 7 H D D A 10 A 10
four steps
two steps G G 8 8 B 4 B 4 E E 9 6 9 6 5 5 1 F 1 F C 3 3 11 C 11 2 2 7 H 7 H D D A A 10 10
Campus Tour 24

25. Minimum Spanning Trees
Shortest Path
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Data Structure for Kruskal’s Algorithm
The algorithm maintains a forest of trees.
A priority queue extracts the edges by increasing weight.
An edge is accepted it if connects distinct trees.
We need a union-find data structure, which maintains a collection of disjoint sets, with operations:
makeSet(u): create a set consisting of u
find(u): return the set storing u
union(A, B): replace sets A and B with their union
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26. Minimum Spanning Trees
Shortest Path
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Running Time
Cluster merges as unions.
Cluster locations as finds.
Running time
O((n + m) log n)
PQ operations
O(m log n)
UF operations
O(n log n) with lists
O(n log* n) with trees
Algorithm KruskalMST(G)
Initialize a partition P
for each vertex v in G do
P.makeSet(v)
let Q be a priority queue.
Insert all edges into Q
T  
{T is the union of the MSTs of the clusters}
while T has fewer than n 1 edges do e  Q.removeMin().getValue()
[u, v]  G.endVertices(e)
A  P.find(u)
B  P.find(v)
if A  B then
Add edge e to T
P.union(A, B)
return T
Minimum Spanning Trees 26

27. Minimum Spanning Trees
Shortest Path
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Prim-Jarnik’s Algorithm
Similar to Dijkstra’s algorithm.
We pick an arbitrary vertex s and we grow the MST as a cloud of vertices, starting from s.
We store with each vertex v label d(v) representing the smallest weight of an edge connecting v to a vertex in the cloud.
At each step:
We add to the cloud the vertex u outside the cloud with the smallest distance label.
We update the labels of the vertices adjacent to u.
Minimum Spanning Trees 27

28. Minimum Spanning Trees
Shortest Path
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Prim-Jarnik’s Algorithm (cont.)
A heap-based adaptable priority queue with location- aware entries stores the vertices outside the cloud
Key: distance
Value: vertex
Recall that method replaceKey(l,k) changes the key of entry l
We store three labels with each vertex:
Distance
Parent edge in MST
Entry in priority queue
Algorithm PrimJarnikMST(G)
Q  new heap-based priority queue
s  a vertex of G
for all v  G.vertices()
if v s
v.setDistance(0)
else
v.setDistance()
v.setParent()
l  Q.insert(v.getDistance(), v)
v.setLocator(l)
while Q.empty()
l  Q.removeMin()
u  l.getValue()
for all e  u.incidentEdges()
z  e.opposite(u)
r  e.weight()
if r z.getDistance()
z.setDistance(r)
z.setParent(e) Q.replaceKey(z.getEntry(), r)
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29. Minimum Spanning Trees
Shortest Path
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Example  7 7 D 7 D 2 2 B 4 B 4 8 9  5 9  5 5 2 F 2 F C C 8 8 3 3 8 8 E E A 7 A 7 7 7 7 7 7 D 2 7 D 2 B 4 B 4 5 9  5 5 9 4 2 F 5 C 2 F 8 C 8 3 8 3 8 E A E 7 7 A 7 7
Minimum Spanning Trees 29

30. Minimum Spanning Trees
Shortest Path
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Example (contd.) 7 7 D 2 B 4 9 4 5 5 2 F C 8 3 8 E A 3 7 7 7 D 2 B 4 5 9 4 5 2 F C 8 3 8 E A 3 7
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31. Minimum Spanning Trees
Shortest Path
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Analysis
Graph operations
Method incidentEdges is called once for each vertex.
Label operations
Set/get the distance, parent and locator labels of vertex z O(deg(z)) times.
Setting/getting a label takes O(1) time.
Priority queue operations
Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time.
The key of a vertex w in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time.
Prim-Jarnik’s algorithm runs in O((n  m) log n) time provided the graph is represented by the adjacency list structure.
Recall that v deg(v) 2m
The running time is O(m log n) since the graph is connected.
Minimum Spanning Trees 31