Graph Algorithms shortest paths, minimum spanning trees, etc.
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Transcript of Graph Algorithms shortest paths, minimum spanning trees, etc.
![Page 1: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/1.jpg)
Graph Algorithms shortest paths, minimum spanning trees, etc.
Nancy AmatoParasol Lab, Dept. CSE, Texas A&M University
Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004)
ORD
DFW
SFO
LAX
802
1743
1843
1233
337
http://parasol.tamu.edu
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Graphs 2
Minimum Spanning Trees
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
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Outline and Reading
• Minimum Spanning Trees (§13.6)• Definitions• A crucial fact
• The Prim-Jarnik Algorithm (§13.6.2)
• Kruskal's Algorithm (§13.6.1)
3Graphs
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Reminder: 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
4Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 5: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/5.jpg)
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
5Graphs
ORD
PIT
ATL
STL
DEN
DFW
DCA
101
9
8
6
3
25
7
4
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Exercise: MST
Show an MSF of the following graph.
6Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
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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 let 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
7Graphs
84
2 36
7
7
9
8e
C
f
84
2 36
7
7
9
8
C
e
f
Replacing f with e yieldsa better spanning tree
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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 eProof:• 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 8Graphs
U V
74
2 85
7
3
9
8 e
f
74
2 85
7
3
9
8 e
f
Replacing f with e yieldsanother MST
U V
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Prim-Jarnik’s Algorithm
• (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 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
9Graphs
• 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 (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
10Graphs
Algorithm PrimJarnikMST(G)Q new heap-based priority queues a vertex of Gfor all v G.vertices()
if v = ssetDistance(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)
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Example
11Graphs
BD
C
A
F
E
74
28
5
7
3
9
8
07
2
8
BD
C
A
F
E
74
28
5
7
3
9
8
07
2
5
7
BD
C
A
F
E
74
28
5
7
3
9
8
07
2
5
7
BD
C
A
F
E
74
28
5
7
3
9
8
07
2
5 4
7
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Example (contd.)
12Graphs
BD
C
A
F
E
74
28
5
7
3
9
8
03
2
5 4
7
BD
C
A
F
E
74
28
5
7
3
9
8
03
2
5 4
7
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Exercise: Prim’s MST alg
• Show how Prim’s MST algorithm works on the following graph, assuming you start with SFO, I.e., s=SFO.
• Show how the MST evolves in each iteration (a separate figure for each iteration).
13Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
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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 v deg(v) 2m
• The running time is O(m log n) since the graph is connected14
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15Graphs
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 keyT
while T has fewer than n-1 edges edge e = T.removeMin()
Let u, v be the endpoints of eif Cloud(v) Cloud(u) then
Add edge e to TMerge Cloud(v) and Cloud(u)
return T
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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
Graphs 16
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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
Graphs 17
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Partition-Based Implementation
• A partition-based version of Kruskal’s Algorithm performs cloud merges as unions and tests as finds.
Graphs 18
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
Add (u,v) to T
P.union(u,v)
return TRunning time: O((n+m) log
n)
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Graphs 19
Kruskal Example
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 20: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/20.jpg)
Example
20Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 21: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/21.jpg)
Example
21Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 22: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/22.jpg)
Example
22Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 23: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/23.jpg)
Example
23Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 24: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/24.jpg)
Example
24Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 25: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/25.jpg)
Example
25Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 26: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/26.jpg)
Example
26Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 27: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/27.jpg)
Example
27Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 28: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/28.jpg)
Example
28Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 29: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/29.jpg)
Example
29Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 30: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/30.jpg)
Example
30Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 31: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/31.jpg)
Example
31Graphs
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 32: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/32.jpg)
Graphs 32
Example
JFK
BOS
MIA
ORD
LAXDFW
SFO BWI
PVD
8672704
187
1258
849
144740
1391
184
946
1090
1121
2342
1846 621
802
1464
1235
337
![Page 33: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/33.jpg)
Exercise: Kruskal’s MST alg
• Show how Kruskal’s MST algorithm works on the following graph.• Show how the MST evolves in each iteration (a separate figure for each
iteration).
33Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 34: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/34.jpg)
Shortest Paths
38Graphs
CB
A
E
D
F
0
328
5 8
48
7 1
2 5
2
3 9
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Outline and Reading
• Weighted graphs (§13.5.1)• Shortest path problem• Shortest path properties
• Dijkstra’s algorithm (§13.5.2)• Algorithm• Edge relaxation
• The Bellman-Ford algorithm • Shortest paths in DAGs • All-pairs shortest paths
39Graphs
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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
40Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 37: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/37.jpg)
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
41Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 38: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/38.jpg)
Shortest Path Problem
• If there is no path from v to u, we denote the distance between them by d(v, u)=+
• What if there is a negative-weight cycle in the graph?
42Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
-802
1387-17431099
1120-1233
2555
142
12
05
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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
43Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 40: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/40.jpg)
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(single-source shortest paths)
• 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
• The label D[v] is initialized to positive infinity
• At each step• We add to the cloud the vertex u
outside the cloud with the smallest distance label, D[v]
• We update the labels of the vertices adjacent to u (i.e. edge relaxation)
44Graphs
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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)}
45Graphs
D[z] 75
D[u] 5010
zsu
D[z] 60
D[u] 5010
zsu
e
e
y
y
D[y] 2055
D[y] = 20 55
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Example
46Graphs
CB
A
E
D
F
0
428
48
7 1
2 5
2
3 9
CB
A
E
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
55
1. Pull in one of the vertices with red labels
2. The relaxation of edges updates the labels of LARGER font size
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Example (cont.)
47Graphs
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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Exercise: Dijkstra’s alg
• Show how Dijkstra’s algorithm works on the following graph, assuming you start with SFO, I.e., s=SFO.
• Show how the labels are updated in each iteration (a separate figure for each iteration).
48Graphs
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
![Page 45: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/45.jpg)
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
49Graphs
Algorithm DijkstraDistances(G, s)Q new heap-based priority queuefor 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()
{ pull a new vertex u into the cloud }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)
O(n) iter’s
O(logn)
O(logn)∑v deg(u)
iter’s
O(n) iter’s
O(logn)
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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 v deg(v) 2m
• The running time can also be expressed as O(m log n) since the graph is connected
• The running time can be expressed as a function of n, O(n2 log n)50Graphs
![Page 47: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/47.jpg)
Exercise: Analysis
51Graphs
Algorithm DijkstraDistances(G, s)Q new unsorted-sequence-based priority queuefor 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()
{ pull a new vertex u into the cloud }u Q.removeMin() for all e G.incidentEdges(u) //use adjacency
list{ 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)
O(1)
![Page 48: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/48.jpg)
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
52Graphs
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)
![Page 49: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/49.jpg)
Why Dijkstra’s Algorithm Works
• Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance.
53Graphs
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.
CB
s
E
D
F
0
327
5 8
48
7 1
2 5
2
3 9
![Page 50: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/50.jpg)
Why It Doesn’t Work for Negative-Weight Edges
54Graphs
Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. 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.
C’s true distance is 1,
but it is already in the cloud
with D[C]=2!
CB
A
E
D
F
0
428
48
7 -3
2 5
2
3 9
CB
A
E
D
F
0
028
5 11
48
7 -3
2 5
2
3 9
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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?
55Graphs
Algorithm BellmanFord(G, s)for all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
for i 1 to n-1 dofor each e G.edges()
{ relax edge e }u G.origin(e)z G.opposite(u,e)r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r)
![Page 52: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/52.jpg)
Bellman-Ford Example
56Graphs
-2
0
48
7 1
-2 5
-2
3 9
0
48
7 1
-2 53 9
Nodes are labeled with their d(v) values
-2
-28
0
4
48
7 1
-2 53 9
8 -2 4
-15
61
9
-25
0
1
-1
9
48
7 1
-2 5
-2
3 94
First round
Second round
Third round
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Exercise: Bellman-Ford’s alg
• Show how Bellman-Ford’s algorithm works on the following graph, assuming you start with the top node
• Show how the labels are updated in each iteration (a separate figure for each iteration).
57Graphs
0
48
7 1
-5 5
-2
3 9
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DAG-based Algorithm
• Works even with negative-weight edges
• Uses topological order• Is much faster than
Dijkstra’s algorithm• Running time: O(n+m).
58Graphs
Algorithm DagDistances(G, s)for all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
Perform a topological sort of the verticesfor 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)
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DAG Example
59Graphs
-2
0
48
7 1
-5 5
-2
3 9
0
48
7 1
-5 53 9
Nodes are labeled with their d(v) values
-2
-28
0
4
48
7 1
-5 53 9
-2 4
-1
1 7
-25
0
1
-1
7
48
7 1
-5 5
-2
3 94
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)
![Page 56: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/56.jpg)
Exercize: DAG-based Alg
• Show how DAG-based algorithm works on the following graph, assuming you start with the second rightmost node
• Show how the labels are updated in each iteration (a separate figure for each iteration).
60Graphs
∞ 0 ∞ ∞∞ ∞5 2 7 -1 -2
6 1
3 4
2
1
2
3
4
5
![Page 57: Graph Algorithms shortest paths, minimum spanning trees, etc.](https://reader033.fdocuments.net/reader033/viewer/2022061504/56812a55550346895d8da9e0/html5/thumbnails/57.jpg)
Summary of Shortest-Path Algs
• Breadth-First-Search• Dijkstra’s algorithm (§13.5.2)
• Algorithm• Edge relaxation
• The Bellman-Ford algorithm • Shortest paths in DAGs
61Graphs
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All-Pairs Shortest Paths
• Find the distance between every pair of vertices in a weighted directed graph G.
• We can make n calls to Dijkstra’s algorithm (if no negative edges), which takes O(nmlog n) time.
• Likewise, n calls to Bellman-Ford would take O(n2m) time.
• We can achieve O(n3) time using dynamic programming (similar to the Floyd-Warshall algorithm).
62Graphs
Algorithm AllPair(G) {assumes vertices 1,…,n} for all vertex pairs (i,j)
if i jD0[i,i] 0
else if (i,j) is an edge in GD0[i,j] weight of edge (i,j)
elseD0[i,j] +
for k 1 to n do for i 1 to n do for j 1 to n do
Dk[i,j] min{Dk-1[i,j], Dk-1[i,k]+Dk-1[k,j]} return Dn
k
j
i
Uses only verticesnumbered 1,…,k-1 Uses only vertices
numbered 1,…,k-1
Uses only vertices numbered 1,…,k(compute weight of this edge)