Shortest-Path Property 4.1

1
Shortest-Path Property 4.1

• If the path s -gt n1 -gt n2 -gt -gt nk is a
  shortest path from s to k, then the subpath s -gt
  n1 -gt n2 -gt -gt ni is a shortest path from s to
  i for i 1, 2, 3, , k-1.

2
Proof of Prop. 4.1 Idea
1) P1 P2 is a shortest path from s to k
P1
P2
s
k
i
If P3 is shorter than P1, then the directed walk
P3 P2 contains a shorter path from s to k than
P1 P2. Contradicts statement 1.
3
Proof of Prop. 4.1 Details
Suppose P1 P2 is a shortest path from s to k,
but P1 is not a shortest path from s to i.
P1
P2
s
k
i
Case 1 P1 and P3 are node-disjoint Case 2 P1
and P3 have at least one node in common
4
Proof of Prop. 4.1 Case 1
If P1 is not a shortest path from s to node i and
P1 and P3 are node-disjoint, then the path
composed of P3 followed by P2 is a shorter path
from s to k then the path composed of P1 followed
by P2. This is a contradiction.
P1
P2
s
k
i
Case 1 P1 and P3 are node-disjoint
5
Proof of Prop. 4.1 Case 2
s
k
i
j
P3 and P2 pass have node j in common.
6
Directed Path Contained in the Directed Walk P3
P2
s
k
i
P4
j
7
Proof by Contradiction

• We assume there are no negative cycles.
• Length (P4) ? Length(P3 P2).
• If P3 is a shortest path from s to i, but P1 is
  not then Length(P3 P2) lt Length(P1 P2).
• Implies Length (P4) lt Length(P1 P2).
• Implies P1 P2 is not a shortest path from s to
  k which is a contradiction.

8
A shortest path from 1 to 5
From 4.1 d(2) c12 c12 0 c12 d(1)
d(3) d(2) c23
d(4) d(3) c34
d(5) d(4) c45
9
Shortest Path Tree
7

6


7
5
?
2
4
5
7
1
1
6
2
?
9
8
0


1
6
5
2
3
4
-2
?
5
3
?
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Corollary to 4.1

• Let d be the vector of shortest path distances
  for a given network and source node s.
• Let P be a shortest path from s to some node k.
• d(j) d(i) cij for every arc (i,j) in P

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Converse Property

• Suppose d(j) d(i) cij for every arc (i,j) on
  a path P from s to t.
• Claim
• P is a shortest path from s to t.

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A path where d(j) d(i) cij
Length of path c12 c23 c34 c45
d(2) d(1) c12 c12
d(3) d(2) c23 c12 c23
d(4) d(3) c34 c12 c23 c34
d(5) d(4) c45 c12 c23 c34 c45
Length of path d(5) (shortest path length)
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Shortest-Path Property 4.2

• Let d be the vector of shortest path distances
  for a given network.
• The directed path P from the source to node k is
  a shortest path if and only if
• d(j) d(i) cij for every arc (i,j) in P.

14
Optimality Condition for Shortest Path Trees

• Let d be the vector of shortest path distances
  for a given network.
• The directed-out tree T rooted at s is a shortest
  path tree if and only if
• d(j) d(i) cij for every arc (i,j) in T and
• d(j) ? d(i) cij for every arc (i,j) not in T.

15
Correctness of the Reaching Algorithm

• Proof by induction.
• Suppose the algorithm has reached node k-1 and
  that the labels on nodes 1,2,,k-1 are optimal.
• Show that when the algorithm reaches node k that
  the label on node k is optimal.

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A shortest path to node k
By 4.1, the subpath to ni is a shortest path.
Since we have a T.O., ni lt k.
Inductive hypothesis the label on is ni is a
correct shortest path label.
When the algorithm reached ni it scanned
arc (ni,k) and set the label on k to d(ni)
c(ni,k) length of shortest path from s to node
k.