15.082J / 6.855J
February 27, 2003
The Label Correcting Algorithm

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2
Overview of the Lecture
A generic algorithm for solving shortest path
problems
z
negative costs permitted
z
but no negative cost cycle (at least for now)
The use of reduced costs
All pair shortest path problem
INPUT
G = (N, A) with costs c
Node 1 is the source node
There is no negative cost cycle
z
We will relax that assumption later

3
Optimality Conditions
Lemma.
Let d*(j) be the shortest path length from
node 1 to node j, for each j.
Let d( ) be node labels
with the following properties:
d(j)
≤
d(i) + c
ij
for i
∈
N
for j
≠
1.
(1)
d(1) = 0.
(2)
Then d(j)
≤
d*(j) for each j.
Proof
.
Let P be any path from node 1 to node j,
with length c(P), and suppose P has k arcs.
Claim
:
d(j)
≤
c(P).
Note: if P is the shortest path from 1 to j, then
d(j)
≤
c(P) = d*(j), which is what we want to prove.

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