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## Lecture 8 - Shortest-Path Trees (SPT)

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### Dijkstra's Algorithm. 1. Mark every node as unscanned and give each node a ... If we run Dijkstra's algorithm on a sparse graph, we will get a tree with a fair ... – PowerPoint PPT presentation

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Title: Lecture 8 - Shortest-Path Trees (SPT)

1
Lecture 8 - Shortest-Path Trees (SPT)
• Definition 3.19
• Given a weighted graph (G,W) and nodes n1 and
n2, the shortest path from n1 to n2 is a path P
such that
is a minimum.
• Definition 3.20
• Given a weighted graph (G,W) and a node n1, a
shortestpath tree rooted at n1 is a tree T such
that, for any other node n2 G, the path from
n1 to n2 in the tree T is a shortest path between
the nodes.

2
Dijkstras Algorithm
• 1. Mark every node as unscanned and give each
node a label of
• 2. Set the predecessor of the root to itself. The
root will be the only node that is its own
predecessor.

3
• 3. Loop until you have scanned all the nodes.
• -Find the node n with the smallest label. Since
the label represents the distance to the root we
call it d_min.
• -Mark the node as scanned.
• -Scan all the adjacent nodes m and see if the
distance to the root through n is better than the
distance stored in the label of m. If it is,
update the label and update predmn.
• 4. When the loop finishes, we have a tree stored
in pred format rooted at root.

4
1
2
Example
5
6
4
3
7
4
9
8
MST
SPT Rooted at 4
1
2
1
2
6
4
3
6
8
9
Total Cost 16
Total Cost 26
5
SPT vs. MST
• Lower utilization of the links
• More cost
• Important
• Smaller average number of hops

Actually we will compare star (a kind of SPT)
with MST, because
6
Star vs. MST
• If we run Dijkstras algorithm on a sparse graph,
we will get a tree with a fair number of nodes
not connected directly to the root.
• If we run Dijkstras algorithm on a complete
graph (exactly what were studying now), then we
usually get a star.

7
Star vs. MST (contd)
• Design Name Ave Hops MAX_UTIL Cost
• MST 13.9479 0.493
325.516
• Star 1.9800 0.09
453.861

Prims algorithm produces much shorter paths but
can produce very expensive networks. SPT is not
good, either. Is there some middle ground between
MST and SPT?
8
PrimDijkstra Trees
• Algorithm Label
• Prims
• 2) Dijkstras
• 3) Prim-Dijkstras
• build a tree by starting at a node and bringing
in nodes by selecting the one with smallest label
• Use the label
• (a)dist(root,neighbor) (1-a)dist(neighbor,node)
• for 0 lt a lt 1

9
PrimDijkstra Trees
• a 1 gt Prim (cheep)
• a 1 gt Dijkstra (short hops)
• 0 lt a lt 1 gt Mixture

2
1
Example with a 0.7 higher priority to small
hop count
5
7
3
4
6
8
9
10
Step 1 begin at node 4
(.7)(6)(.3)(6)6
(.7)(7)(.3)(7)7
(.7)(8)(.3)(8)8
(.7)(9)(.3)(9)9
The winner is 6
11
Step 2
(.7)(8)(.3)(2)6.2
(.7)(7)(.3)(1)5.2
(.7)(11)(.3)(5)8.2
7
8
9
The winner is 5.2
12
Step 3
6.2
8.2
(.7)(11)(.3)(4)8.9
7
8
9
The winner is 6.2
13
Step 4
8.2
(.7)(11)(.3)(3)8.6
8.9
8
9
The winner is 8
14
Step 5
8.6
9
The winner is 8.6
15
Summary
SPT - Cost 26
MST - Cost 16
Prim-Dijkstra Cost 20
16
Prim Dijkstra Trees (contd)
a Ave Hops Link delay Cost 0 13.9479 0.3066 325,
516 0.1 10.5717 0.1451 280,162 0.2 7.8640 0.1067
247,217 0.3 6.7762 0.0913 243,551 0.7 3.0186 0.
0380 295,012 0.8 2.2879 0.0277 378,792 0.9 1.980
0 0.0233 453,861
17

Proceed to HW 5