1. 15.082J & 6.855J & ESD.78J Visualizations
Label Correcting Algorithm
Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

2. An Example ∞ ∞ 4 2 ∞ ∞ 1 5 7 Initialize d(1) := 0; d(j) := ∞ for j ≠ 13 3 3 1 ∞ ∞ -2 6 1 5 7
Initialize 2 3 4 3
d(1) := 0; d(j) := ∞ for j ≠ 1 -4 3 6 ∞ ∞
In next slides: the number inside the node will be d(j).
Violating arcs will be in thick lines.

3. An arc (i,j) is violating if d(j) > d(i) + cij. ∞ ∞2 ∞ 3 ∞ 3 3 3 1 -2 6 ∞ ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 ∞ ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

4. An arc (i,j) is violating if d(j) > d(i) + cij. ∞ ∞2 ∞ 3 3 3 3 1 -2 6 ∞ 6 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 ∞ ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

5. An arc (i,j) is violating if d(j) > d(i) + cij. ∞ 3 ∞2 ∞ 3 3 3 3 1 -2 6 6 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 ∞ 3 ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

6. An arc (i,j) is violating if d(j) > d(i) + cij. 3 ∞2 5 ∞ 3 3 3 3 1 -2 6 6 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 3 ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

7. An arc (i,j) is violating if d(j) > d(i) + cij. 3 ∞2 5 3 3 3 3 1 -2 6 6 4 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 3 ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

8. An arc (i,j) is violating if d(j) > d(i) + cij. 3 6 ∞2 5 3 3 3 3 1 -2 6 6 4 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 3 6 ∞
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

9. An arc (i,j) is violating if d(j) > d(i) + cij. 2 3 ∞ 65 3 3 3 3 1 -2 6 6 4 ∞
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 2 3 ∞ 6
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

10. An arc (i,j) is violating if d(j) > d(i) + cij. 2 65 3 3 3 3 1 -2 6 4 ∞ 9
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 2 6
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.

11. An arc (i,j) is violating if d(j) > d(i) + cij. 2 65 3 3 3 3 1 -2 6 4 9
Generic Step 2 3 4 3
An arc (i,j) is violating if d(j) > d(i) + cij. -4 2 6
Pick a violating arc (i,j) and replace d(j) by d(i) + cij.
No arc is violating
We now show the predecessor arcs.
The distance labels are optimal

12. 15.082J / 6.855J / ESD.78J Network Optimization
MITOpenCourseWare
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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