Data Structures and algorithms (IS ZC361)
Weighted Graphs – Shortest path algorithms – MST
S.P.Vimal
BITS-Pilani
http://discovery.bits-pilani.ac.in/discipline/csis/vimal/
Source :This presentation is composed from the presentation materials provided by the
authors (GOODRICH and TAMASSIA) of text book -1 specified in the handout
Data Structures and Algorithms
1
Topics today
1.
2.
DAGs, topological sorting
Weighted Graphs




3.
Dijkstra’s algorithm
The Bellman-Ford algorithm
Shortest paths in dags
All-pairs shortest paths
Minimum Spanning Trees
Data Structures and Algorithms
2
DAGS, Topological Sorting
(Text book Reference 6.4.4)
Data Structures and Algorithms
3
DAGs and Topological Ordering
D
•
A directed acyclic graph (DAG) is a
digraph that has no directed cycles
• A topological ordering of a digraph is a
numbering
v1 , …, vn
of the vertices such that for every edge (vi ,
vj), we have i < j
• Example: in a task scheduling digraph, a
topological ordering a task sequence that
satisfies the precedence constraints
Theorem
A digraph admits a topological ordering if
and only if it is a DAG
E
B
C
DAG G
A
v2
v1
D
B
C
A
Data Structures and Algorithms
v4
E
v5
v3
Topological
ordering of G
4
Topological Sorting
• Number vertices, so that (u,v) in E implies u < v
wake up
1
A typical student day
3
2
study computer sci.
eat
4
7
play
nap
8
write c.s. program
9
make cookies
for professors
5
more c.s.
6
work out
10
11
sleep
dream about graphs
Data Structures and Algorithms
5
Algorithm for Topological Sorting
• Note: This algorithm is different than the one in
Goodrich-Tamassia
Method TopologicalSort(G)
HG
// Temporary copy of G
n  G.numVertices()
while H is not empty do
Let v be a vertex with no outgoing edges
Label v  n
nn-1
Remove v from H
• Running time: O(n + m). How…?
Data Structures and Algorithms
6
Topological Sorting Algorithm using DFS
•
Simulate the algorithm by using depthfirst search
Algorithm topologicalDFS(G)
Input dag G
Output topological ordering of G
n  G.numVertices()
for all u  G.vertices()
setLabel(u, UNEXPLORED)
for all e  G.edges()
setLabel(e, UNEXPLORED)
for all v  G.vertices()
if getLabel(v) = UNEXPLORED
topologicalDFS(G, v)
• O(n+m) time.
Algorithm topologicalDFS(G, v)
Input graph G and a start vertex v of G
Output labeling of the vertices of G
in the connected component of v
setLabel(v, VISITED)
for all e  G.incidentEdges(v)
if getLabel(e) = UNEXPLORED
w  opposite(v,e)
if getLabel(w) = UNEXPLORED
setLabel(e, DISCOVERY)
topologicalDFS(G, w)
else
{e is a forward or cross edge}
Label v with topological number n
nn-1
Data Structures and Algorithms
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Topological Sorting Example
Data Structures and Algorithms
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Topological Sorting Example
9
Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Topological Sorting Example
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Data Structures and Algorithms
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Weighted graphs (Minimum Spanning Tree)
(Text book Reference 7.3)
Data Structures and Algorithms
18
Minimum Spanning Tree
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
DEN
PIT
9
6
STL
4
8
DFW
Data Structures and Algorithms
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3
DCA
5
2
ATL
19
Partition Property
U
f
Partition Property:
V
7
4
– Consider a partition of the vertices of G
9
into subsets U and V
5
2
8
– Let e be an edge of minimum weight across
the partition
3
8
e
– There is a minimum spanning tree of G
7
containing edge e
Proof:
Replacing f with e yields
– Let T be an MST of G
another MST
– If T does not contain e, consider the cycle
U
V
C formed by e with T and let f be an edge
7
f
of C across the partition
4
– weight(f)  weight(e)
9
– Thus, weight(f) = weight(e)
5
2
– We obtain another MST by replacing f
8
with e
8
Data Structures and Algorithms
e
7
3
20
Prim-Jarnik’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 a label d(v) = 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

Data Structures and Algorithms
21
Prim-Jarnik’s Algorithm (cont.)
•
A priority queue stores the
vertices outside the cloud
– Key: distance
– Element: vertex
•
Locator-based methods
– insert(k,e) returns a locator
– replaceKey(l,k) changes the
key of an item
•
We store three labels with each
vertex:
– Distance
– Parent edge in MST
– Locator 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
setDistance(v, 0)
else
setDistance(v, )
setParent(v, )
l  Q.insert(getDistance(v), v)
setLocator(v,l)
while Q.isEmpty()
u  Q.removeMin()
for all e  G.incidentEdges(u)
z  G.opposite(u,e)
r  weight(e)
if r < getDistance(z)
setDistance(z,r)
setParent(z,e)
Q.replaceKey(getLocator(z),r)
Data Structures and Algorithms
22
Example
2
7
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B
5
C
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0
A
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9
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C
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
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
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0
Data Structures and Algorithms
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
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4
Example (contd.)
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Data Structures and Algorithms
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9
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0
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3
24
Analysis
•
Graph operations
– Method incidentEdges is called once for each vertex
•
Label operations
– We 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 Sv deg(v) = 2m
•
The running time is O(m log n) since the graph is connected
Data Structures and Algorithms
25
Kruskal’s Algorithm
A priority queue stores
the edges outside the
cloud


Key: weight
Element: edge
At the end of the
algorithm


We are left with one
cloud that encompasses
the MST
A tree T which is our
MST
Algorithm KruskalMST(G)
for each vertex V in G do
define a Cloud(v) of  {v}
let Q be a priority queue.
Insert all edges into Q using their
weights as the key
T
while T has fewer than n-1 edges do
edge e = T.removeMin()
Let u, v be the endpoints of e
if Cloud(v)  Cloud(u) then
Add edge e to T
Merge Cloud(v) and Cloud(u)
return T
Data Structures and Algorithms
26
Data Structure for Kruskal Algortihm
• The algorithm maintains a forest of trees
• An edge is accepted it if connects distinct trees
• We need a data structure that maintains a partition, i.e., a collection of
disjoint sets, with the operations:
-find(u): return the set storing u
-union(u,v): replace the sets storing u and v with their union
Data Structures and Algorithms
27
Representation of a
Partition
• Each set is stored in a sequence
• Each element has a reference back to the set
– operation find(u) takes O(1) time, and returns the set of which
u is a member.
– in operation union(u,v), we move the elements of the smaller
set to the sequence of the larger set and update their references
– the time for operation union(u,v) is min(nu,nv), where nu and nv
are the sizes of the sets storing u and v
• Whenever an element is processed, it goes into a set of size at least
double, hence each element is processed at most log n times
Data Structures and Algorithms
28
Partition-Based Implementation
• A partition-based version of Kruskal’s Algorithm performs cloud
merges as unions and tests as finds.
Algorithm Kruskal(G):
Input: A weighted graph G.
Output: An MST T for G.
Let P be a partition of the vertices of G, where each vertex forms a separate set.
Let Q be a priority queue storing the edges of G, sorted by their weights
Let T be an initially-empty tree
while Q is not empty do
(u,v)  Q.removeMinElement()
if P.find(u) != P.find(v) then
Running time:
Add (u,v) to T
O((n+m)log n)
P.union(u,v)
return T
Data Structures and Algorithms
29
Kruskal Example
2704
BOS
867
849
ORD
LAX
1391
1464
1235
144
JFK
1258
184
802
SFO
337
187
740
621
1846
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
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Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
31
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
32
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
33
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
34
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
35
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
36
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
37
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
38
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
39
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
40
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
41
Example
2704
BOS
867
849
ORD
740
621
1846
337
LAX
1391
1464
1235
187
144
JFK
1258
184
802
SFO
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
42
Example
2704
BOS
867
849
ORD
LAX
1391
1464
1235
144
JFK
1258
184
802
SFO
337
187
740
621
1846
PVD
BWI
1090
DFW
946
1121
MIA
2342
Data Structures and Algorithms
43
Weighted graphs (Shortest Path Algorithms)
(Text book Reference 6.44)
Data Structures and Algorithms
44
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
PVD
ORD
SFO
LGA
HNL
LAX
DFW
Data Structures and Algorithms
MIA
45
Shortest Path Problem
•
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
PVD
ORD
SFO
LGA
HNL
LAX
DFW
Data Structures and Algorithms
MIA
46
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
PVD
ORD
SFO
LGA
HNL
LAX
DFW
Data Structures and Algorithms
MIA
47
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
Data Structures and Algorithms
48
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
•
d(u) = 50
s
u
e
d(z) = 75
z
The relaxation of edge e updates
distance d(z) as follows:
d(z)  min{d(z),d(u) + weight(e)}
d(u) = 50
s
Data Structures and Algorithms
u
d(z) = 60
e
z
49
Example
A
8
0
4
A
8
2
B
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7
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D
9
E

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
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7
2
Data Structures and Algorithms
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5
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50
Example (cont.)
A
8
0
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2
B
2
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7
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3
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2
1
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Data Structures and Algorithms
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51
Dijkstra’s Algorithm
•
A priority queue stores the
vertices outside the cloud
– Key: distance
– Element: vertex
•
Locator-based methods
– insert(k,e) returns a
locator
– replaceKey(l,k) changes
the key of an item
•
We store two labels with each
vertex:
– Distance (d(v) label)
– locator in priority queue
Algorithm DijkstraDistances(G, s)
Q  new heap-based priority queue
for all v  G.vertices()
if v = s
setDistance(v, 0)
else
setDistance(v, )
l  Q.insert(getDistance(v), v)
setLocator(v,l)
while Q.isEmpty()
u  Q.removeMin()
for all e  G.incidentEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
Q.replaceKey(getLocator(z),r)
Data Structures and Algorithms
52
Analysis
•
•
•
•
•
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 Sv deg(v) = 2m
The running time can also be expressed as O(m log n) since the graph is
connected
Data Structures and Algorithms
53
Extension
•
•
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()
…
setParent(v, )
…
for all e  G.incidentEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
setParent(z,e)
Q.replaceKey(getLocator(z),r)
Data Structures and Algorithms
54
Bellman-Ford Algorithm
•
•
•
•
•
Works even with negative-weight Algorithm BellmanFord(G, s)
edges
for all v  G.vertices()
Must assume directed edges (for
if v = s
otherwise we would have
setDistance(v, 0)
negative-weight cycles)
else
Iteration i finds all shortest paths
setDistance(v, )
that use i edges.
for i  1 to n-1 do
Running time: O(nm).
for each e  G.edges()
{ relax edge e }
Can be extended to detect a
u  G.origin(e)
negative-weight cycle if it exists
z  G.opposite(u,e)
– How?
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
Data Structures and Algorithms
55
Bellman-Ford Example
Nodes are labeled with their d(v) values
0
8
4
0
8
-2
7

-2
1

3
-2
8

7

9


3
5

0
8
-2
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7
1
-2
6

1


5
4
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0
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-2
1
-2
3
-2

-2
5 8
4
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4
9

-1
5
7
3
5
-2
Data Structures and Algorithms
1
1
-2
9
4
9
-1
5
56
DAG-based Algorithm
•
•
•
•
•
Works even with negativeweight edges
Uses topological order
Doesn’t use any fancy data
structures
Is much faster than Dijkstra’s
algorithm
Running time: O(n+m).
Algorithm DagDistances(G, s)
for all v  G.vertices()
if v = s
setDistance(v, 0)
else
setDistance(v, )
Perform a topological sort of the vertices
for u  1 to n do {in topological order}
for each e  G.outEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
Data Structures and Algorithms
57
1
DAG Example
Nodes are labeled with their d(v) values
1
1
0
8
4
0
8
-2
3
2
7

-2

4
1

3
-5
3


8
2
7
9
5

6
-5

5
0
8
-5
2
1

3
6
4
1

4
9

6
4
5
5
0
8
-2
7
-2
1
8
5

3
1
3
4
4
-2
4
1
-2
4
-1
3
5
2
7
9
7

5
-5
5
Data
Structures and Algorithms
0
1
3
6
4
1
-2
9
4
7
(two steps)
-1
5
5
58
This presentation & the practice problems will be available in the course page.
http://discovery.bits-pilani.ac.in/discipline/csis/vimal/course%2007-08%20First/DSA_Offcampus/DSA.htm
Questions ?
Data Structures and Algorithms
59