Graph Algorithms (III)
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COMP9024: Data Structures and
Algorithms
Graphs (III)
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Contents
Shortest Paths
Minimum Spanning Trees
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Shortest Paths
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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
ORD PVD
MIA
DFW
SFO
LAX
LGA
HNL
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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
ORD PVD
MIA
DFW
SFO
LAX
LGA
HNL
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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
ORD PVD
MIA
DFW
SFO
LAX
LGA
HNL
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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 that are adjacent to u
and not in the cloud
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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(z) = 75
D(u) = 50
zs
u
D(z) = 60
D(u) = 50
zs
u
e
e
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Example (1/2)
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D
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428
∞ ∞
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7 1
2 5
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3 9
CB
A
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D
F
0
328
5 11
48
7 1
2 5
2
3 9
CB
A
E
D
F
0
328
5 8
48
7 1
2 5
2
3 9
CB
A
E
D
F
0
327
5 8
48
7 1
2 5
2
3 9
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Example (2/2)
CB
A
E
D
F
0
327
5 8
48
7 1
2 5
2
3 9
CB
A
E
D
F
0
327
5 8
48
7 1
2 5
2
3 9
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Dijkstra’s Algorithm
Algorithm DijkstraDistances(G, s)
Input: A simple undirected weighted graph G with nonnegative edge weights,
and a distinguished vertex s.
Output: A label D[u], for each vertex u, such that D[u] is the length of a shortest
path from s to u in G.
{ for each v ∈ G do
if ( v = s )
D[v] = 0;
else
D[v] = +∞;
Create a priority queue Q containing all the vertices of G using the D labels as keys;
while Q is not empty do
{ u = Q.removeMin();
for each z ∈ Q such that z is adjacent to u do
if ( D[u] + w((u,z)) < D[z] ) // relax edge e
{ D[z] = D[u] + w((u,z));
Change to D[z] the key of vertex z in Q;
}
}
return the label D[u] of each vertex u;
}
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Analysis of Dijkstra’s Algorithm
Creating the priority queue Q takes O(n log n) time if using an
adaptable priority queue, or O(n) time by using bottom-up heap
construction.
At each iteration of the while loop, we spend O(log n) time to
remove vertex u from Q and O(degree(u) log n) time to perform
the relaxation procedure on the edges incident on u.
The overall running time of the while loop is
O(Σu (1+degree(u)) log n) = O((n + m) log n)
(Recall that Σu degree(u) = 2m)
The running time can also be expressed as O(m log n) since the
graph is connected
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Why Dijkstra’s Algorithm Works
Dijkstra’s algorithm is based on the greedy
method. It adds vertices by increasing distance.
CB
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u
v
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327
5 8
48
7 1
2 5
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3 9
Suppose it didn’t find all shortest
distances. Let v be the first wrong
vertex the algorithm processed.
When the previous node, u, on the
true shortest path was considered,
its distance was correct.
But the edge (u,v) was relaxed at
that time!
Thus, so long as D(v)>D(u), v’s
distance cannot be wrong. That is,
there is no wrong vertex.
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Why It Doesn’t Work for
Negative-Weight Edges
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.
CB
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5 -3
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0 -8
Dijkstra’s algorithm is based on the greedy
method. It adds vertices by increasing distance.
C’s true distance is 1,
but it is already in the
cloud with D(C)=5!
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Bellman-Ford Algorithm
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 each v ∈ G do
if ( v = s )
D[v] = 0;
else
D[v] = +∞;
for ( i = 1; i ≤ n-1; i ++ )
for each e ∈ G.edges()
// relax edge e
{ u = G.origin(e);
z = G.opposite(u,e);
r = D[u] + weight(e);
if ( r < D[z] )
D[z] = r;
}
}
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∞
-2
Bellman-Ford Example
∞∞
0
∞
∞
∞
48
7 1
-2 5
-2
3 9
∞
0
∞
∞
∞
48
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-2 5
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Nodes are labeled with their D(v) values
-2
-28
0
4
∞
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-2 5
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∞
8 -2 4
-15
6
1
9
-25
0
1
-1
9
48
7 1
-2 5
-2
3 9
4
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DAG-based Algorithm
(not in book)
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).
Algorithm DagDistances(G, s)
{ for all v ∈ G.vertices()
if ( v = s )
D[v] = 0;
else
D[v] = +∞;
Perform a topological sort of the vertices;
for ( u = 1; u ≤ n; u ++ )
// in topological order
for each e ∈ G.outEdges(u) do
// relax edge e
{ z = G.opposite(u,e);
r = D[u] + weight(e);
if ( r < D[z] )
D[z] = r;
}
}
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∞
-2
DAG Example
∞∞
0
∞
∞
∞
48
7 1
-5 5
-2
3 9
∞
0
∞
∞
∞
48
7 1
-5 5
3 9
Nodes are labeled with their D(v) values
-2
-28
0
4
∞
48
7 1
-5 5
3 9
∞
-2 4
-1
1 7
-25
0
1
-1
7
48
7 1
-5 5
-2
3 9
4
1
2 43
6 5
1
2 43
6 5
8
1
2 43
6 5
1
2 43
6 5
5
0
(two steps)
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Minimum Spanning Trees
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
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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
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ATL
STL
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DFW
DCA
10
1
9
8
6
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4
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Cycle Property
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 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
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2 3
6
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7
9
8
e
C
f
8
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2 3
6
7
7
9
8
C
e
f
Replacing f with e yields
a better spanning tree
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U V
Partition Property
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
7
4
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9
8 e
f
7
4
2 8
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8 e
f
Replacing f with e yields
another MST
U V
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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 = Q.removeMin();
Let u, v be the endpoints of e;
if ( Cloud(v) ≠ Cloud(u) )
{ Add edge e to T;
Merge Cloud(v) and Cloud(u);
}
}
return T;
}
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Data Structure for Kruskal
Algorithm
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
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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
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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) )
{ Add (u,v) to T;
P.union(u,v);
}
}
return T;
}
Running time: O((m+n) log n)
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Kruskal Example (1/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
28
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
Example (2/14)
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Example (3/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
30
Example (4/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
31
Example (5/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
32
Example (6/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
33
Example (7/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
34
Example (8/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
35
Example (9/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
36
Example (10/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
37
Example (11/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
38
Example (12/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
39
Example (13/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
40
Example (14/14)
JFK
BOS
MIA
ORD
LAX
DFW
SFO BWI
PVD
867
2704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
41
Prim-Jarnik’s Algorithm (1/2)
Similar to Dijkstra’s algorithm (for a connected graph)
We pick an arbitrary vertex s and we grow the MST as a
cloud of vertices, starting from v
We store with each vertex u a label D(u) = the smallest
weight of an edge connecting u 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
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Prim-Jarnik’s Algorithm (2/2)
Algorithm PrimJarnikMST(G)
{
Pick any vertex v of G;
D[v] = 0;
for each u ∈ G with u ≠ v do
D[u] = + ∞;
T = ∅ ;
Create a priority queue Q with an entry
((u, null), D[u]) for each vertex u, where
(u, null) is the element and D[u] is the key;
while Q is not Empty do
{ (u, e) = Q.removeMin();
add vertex u and edge e to T;
for each vertex z in Q such that z is adjacent to u do
if ( w((u, z)) < D[z] ) { D[z] = w((u, z)); Change to (z, (u,z)) the element of vertex z in Q; Change to D[z] the key of vertex z in Q; } } return T; } 43 Example (1/2) B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 8 ∞ ∞ B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5 ∞ 7 B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5 ∞ 7 B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5 4 7 44 Example (2/2) B D C A F E 7 4 2 8 5 7 3 9 8 0 3 2 5 4 7 B D C A F E 7 4 2 8 5 7 3 9 8 0 3 2 5 4 7 45 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 Σv deg(v) = 2m The running time is O(m log n) since the graph is connected 46 Boruvka’s Algorithm Like Kruskal’s Algorithm, Boruvka’s algorithm grows many “clouds” at once. Each iteration of the while-loop halves the number of connected components in T. The running time is O(m log n). Algorithm BoruvkaMST(G) { T = V; // just the vertices of G while T has fewer than n-1 edges do for each connected component C in T do { Let edge e be the smallest-weight edge from C to another component in T; if e is not already in T then Add edge e to T; } return T; } 47 JFK BOS MIA ORD LAX DFW SFO BWI PVD 867 2704 187 1258 849 144740 1391 184 946 1090 1121 2342 1846 621 802 1464 1235 337 Boruvka Example (1/3) 48 Example (2/3) JFK BOS MIA ORD LAX DFW SFO BWI PVD 867 2704 187 1258 849 144740 1391 184 946 1090 1121 2342 1846 621 802 1464 1235 337 49 Example (3/3) JFK BOS MIA ORD LAX DFW SFO BWI PVD 867 2704 187 1258 849 144740 1391 184 946 1090 1121 2342 1846 621 802 1464 1235 337 50 Summary • Shortest path problem Dijkstra’s algorithm Bellman-Ford algorithm • Minimum spanning tree problem Kruskal’s algorithm Prim-Jarnik’s algorithm Boruvk’s algorithm Suggested reading (Sedgewick): Ch.20 Ch.21 COMP9024: Data Structures and Algorithms Contents Shortest Paths Weighted Graphs Shortest Paths Shortest Path Properties Dijkstra’s Algorithm Edge Relaxation Example (1/2) Example (2/2) Dijkstra’s Algorithm Analysis of Dijkstra’s Algorithm Why Dijkstra’s Algorithm Works Why It Doesn’t Work for Negative-Weight Edges Bellman-Ford Algorithm � Bellman-Ford Example DAG-based Algorithm �(not in book) DAG Example Minimum Spanning Trees Minimum Spanning Trees Cycle Property Partition Property Slide Number 23 Data Structure for Kruskal Algorithm Representation of a Partition Partition-Based Implementation Kruskal Example (1/14) Example (2/14) Example (3/14) Example (4/14) Example (5/14) Example (6/14) Example (7/14) Example (8/14) Example (9/14) Example (10/14) Example (11/14) Example (12/14) Example (13/14) Example (14/14) Prim-Jarnik’s Algorithm (1/2) Prim-Jarnik’s Algorithm (2/2) Example (1/2) Example (2/2) Analysis Boruvka’s Algorithm � Boruvka Example (1/3) Example (2/3) Example (3/3) Summary