1. Discrete Math 2 Shortest Paths Using Matrix
2001
Discrete Math 2 Shortest Paths Using Matrix
CIS112
February 14, 2007
Daniel L. Silver

2. Overview Previously: In weighted graph . .
Shortest path from #7 to all others
Search matrix method
Now:
Problem 8.6.2
Implement Floyd’s Algorithm
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3. Strategy 3 nested loops i ≔ 1 to 6 j ≔ 1 to 6 k ≔ 1 to 6 Basic rule
If d(j,i) + d(i,k) < d(j,k) . .
Then d(j,k)  d(j,i) + d(i,k)
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4. Interpretation of Matrix
Row designates from vertices
Column designates to vertices
Suppose entry [2,3] is 7
From vertex #2 to vertex #3 . .
cost of travel = 7
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5. Seek – best cost route from j to k
The Basic Operation
Seek – best cost route from j to k
Initial entries = cost of direct route, i.e., cost of edge jk
No edge  cost = ∞
Guarantees that any route found is better
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6. Old cost from j to k compared to . . Cost of 2 step hop
The Basic Operation
Old cost from j to k compared to . .
Cost of 2 step hop
j to i and
then i to k
If 2 step hop has better cost . .
Then it becomes the new j to k cost
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7. Initially old cost is edge cost Later . .
The Basic Operation
Initially old cost is edge cost
Later . .
Old cost is cost of best route found so far
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8. Denotations Outer loop – the i loop Middle loop – the j loop
Inner loop – the k loop
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9. Operation Proceed Loops work in tandem 2 outer loops {i & j loops}
2 inner loops {j & k loops}
Proceed
row by row &
column by column
Update travel cost . .
from vertex j to vertex k
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10. Matrix for Weighted Graph
Vtx A B C D E Z ∞ 2 3 5 1 4
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11. First Outer Loop i = 1 j = ≔ 1 to 6 k ≔ 1 to 6 2007
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12. First Inner Loop
i = 1
j = 1
k ≔ 1 to 6
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13. First Middle Loop d(1,1) = ∞ d(1,1) + d(1,k) ≮ d(1,k), ∀k
Therefore, no change
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14. Second Middle Loop
i = 1
j = 2
k ≔ 1 to 6
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15. Second Middle Loop k = 1 d(1,1) = ∞ d(2,1) + d(1,1) = ∞
Therefore, no change
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16. i=1; j=2; k = 2 (2,1) + (1,2) < (2,2) ≡ 4 < ∞
Vtx A B C D E Z ∞ 2 3 4 5 1
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17. i=1; j=2; k = 3 (2,1) + (1, 3) < (2,3) ≡ 5 < ∞
Vtx A B C D E Z ∞ 2 3 4 5 1
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18. i=1; j=2; k ≔ 4,5,6 (2,1) + (1,k) < (2,k) ≡ ∞ ≮ x
Vtx A B C D E Z ∞ 2 3 4 5 1
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19. Third Middle Loop
i = 1
j = 3
k ≔ 1 to 6
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20. Third Middle Loop k = 1 d(1,1) = ∞ d(3,1) + d(1,1) = ∞
Therefore, no change
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21. i=1; j=3; k = 2 (3,1) + (1, 2) < (3,2) ≡ 5 < ∞
Vtx A B C D E Z ∞ 2 3 4 5 1
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22. i=1; j=3; k = 3 (3,1) + (1, 3) < (3,3) ≡ 6 < ∞
Vtx A B C D E Z ∞ 2 3 4 5 6 1
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23. i=1; j=3; k ≔ 4,5,6 (3,1) + (1, k) < (3,k) ≡ ∞ ≮ x
Vtx A B C D E Z ∞ 2 3 4 5 6 1
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24. 4th, 5th & 6th Middle Loops i = 1 j ≔ 4 to 6 k ≔ 1 to 6 2007
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25. 4th, 5th & 6th Middle Loops j ≔ 4 to 6 d(j,1) = ∞ d(j,1) + d(i,k) = ∞
d(j,1) + d(i,1) = ∞ ≮ d(j,k)
Therefore, no change
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26. Second Outer Loop i = 2 j = ≔ 1 to 6 k ≔ 1 to 6 2007
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27. i=2; j=1; k = 1 (1,2) + (2, 1) < (1,1) ≡ 4 < ∞
Vtx A B C D E Z 4 2 3 ∞ 5 6 1
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28. i=2; j=1; k = 2 (1,2) + (2, 2) < (1,2) ≡ 6 ≮ 2
Vtx A B C D E Z 4 2 3 ∞ 5 6 1
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29. i=2; j=1; k = 3 (1,2) + (2, 3) < (1,3) ≡ 7 ≮ 3
Vtx A B C D E Z 4 2 3 ∞ 5 6 1
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30. i=2; j=1; k = 4 (1,2) + (2, 4) < (1,4) ≡ 7 < ∞
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 1
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31. i=2; j=1; k = 5 (1,2) + (2, 5) < (1,5) ≡ 4 < ∞
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 1
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32. i=2; j=1; k = 6 (1,2) + (2, 6) < (1,6) ≡ ∞ ≮ ∞
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 1
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33. Second Middle Loop
i = 2
j = 2
k ≔ 1 to 6
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34. Second Middle Loop k ≔ 1 to3
i j k
2 2 1
(2,2) + (2,1) < (2,1) ?
4 + 2 ≮ 2
2 2 2
(2,2) + (2,2) < (2,2) ?
4 + 4 ≮ 4
2 2 3
(2,2) + (2,3) < (2,3) ?
4 + 5 ≮ 5
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35. Second Middle Loop k ≔ 4 to6
i j k
2 2 4
(2,2) + (2,4) < (2,4) ?
4 + 5 ≮ 5
2 2 5
(2,2) + (2,5) < (2,5) ?
4 + 2 ≮ 2
2 2 6
(2,2) + (2,6) < (2,6) ?
4 + ∞ ≮ ∞
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36. Matrix After 2nd Middle Loop {no change}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 1
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37. Third Middle Loop k ≔ 1 to3 i j k 2 3 1 2 3 2 2 3 3
2 3 1
(3,2) + (2,1) < (3,1) ?
5 + 2 ≮ 3
2 3 2
(3,2) + (2,2) < (3,2) ?
5 + 4 ≮ 5
2 3 3
(3,2) + (2,3) < (3,3) ?
5 + 5 ≮ 6
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38. Third Middle Loop k ≔ 4 to6 i j k 2 3 4 2 3 5 2 3 6
2 3 4
(3,2) + (2,4) < (3,4) ?
5 + 5 < ∞
2 3 5
(3,2) + (2,5) < (3,5) ?
5 + 2 ≮ 5
2 3 6
(3,2) + (2,6) < (3,6) ?
5 + ∞ ≮ ∞
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39. Matrix After 3rd Middle Loop {1 cell changed}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1
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40. Fourth Middle Loop k ≔ 1 to3
i j k
2 4 1
(4,2) + (2,1) < (4,1) ?
5 + 2 < ∞
2 4 2
(4,2) + (2,2) < (4,2) ?
5 + 4 ≮ 5
2 4 3
(4,2) + (2,3) < (4,3) ?
5 + 5 < ∞
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41. Fourth Middle Loop k ≔ 4 to6
i j k
2 4 4
(4,2) + (2,4) < (4,4) ?
5 + 5 < ∞
2 4 5
(4,2) + (2,5) < (4,5) ?
5 + 2 ≮ 1
2 4 6
(4,2) + (2,6) < (4,6) ?
5 + ∞ ≮ 2
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42. Matrix After 4th Middle Loop {3 cells changed}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1
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43. Fifth Middle Loop k ≔ 1 to3 i j k 2 5 1 2 5 2 2 5 3
2 5 1
(5,2) + (2,1) < (5,1) ?
2 + 2 < ∞
2 5 2
(5,2) + (2,2) < (5,2) ?
2 + 4 ≮ 2
2 5 3
(5,2) + (2,3) < (5,3) ?
2 + 5 ≮ 5
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44. Fifth Middle Loop k ≔ 4 to6 i j k 2 5 4 2 5 5
2 5 4
(5,2) + (2,4) < (5,4) ?
≮ 1
2 5 5
(5,2) + (2,5) < (5,5) ?
2 + 2 < ∞
2 5 6
(5,2) + (2,6) < (5,6) ?
2 + ∞ ≮ 4
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45. Matrix After 5th Middle Loop {2 cells changed}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1
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46. Sixth Middle Loop k ≔ 1 to3 i j k 2 6 1 2 6 3
2 6 1
(6,2) + (2,1) < (6,1) ?
∞ + 2 ≮ ∞
2 6 2
(6,2) + (2,2) < (6,2) ?
∞ + 4 ≮ ∞
2 6 3
(6,2) + (2,3) < (6,3) ?
∞ + 5 ≮ ∞
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47. Sixth Middle Loop k ≔ 4 to6 i j k 2 6 4 2 6 5
2 6 4
(6,2) + (2,4) < (6,4) ?
∞ + 5 ≮ ∞
2 6 5
(6,2) + (2,5) < (6,5) ?
∞ + 2 ≮ ∞
2 6 6
(6,2) + (2,6) < (6,6) ?
∞ + ∞ ≮ ∞
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48. Matrix After 6th Middle Loop {0 cells changed}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1
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49. Third Outer Loop i = 3 j = ≔ 1 to 6 k ≔ 1 to 6
Step by step details given here
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50. Matrix After 3rd Outer Loop {0 cells changed}
Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1
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51. Fourth Outer Loop i = 4 j = ≔ 1 to 6 k ≔ 1 to 6
Step by step details given here
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52. Matrix After 4th Outer Loop {10 cells changed}
Vtx A B C D E Z 4 2 3 7 9 5 6 10 12 1
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53. Fifth Outer Loop i = 5 j = ≔ 1 to 6 k ≔ 1 to 6
Step by step details given here
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54. Matrix After 5th Outer Loop {12 cells changed}
Vtx A B C D E Z 4 2 3 5 7 6 8 1
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55. Sixth Outer Loop i = 6 j = ≔ 1 to 6 k ≔ 1 to 6
Step by step details given here
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56. Matrix After 6th Outer Loop {0 cells changed}
Vtx A B C D E Z 4 2 3 5 7 6 8 1
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57. Final Matrix Diagonal values are removed
Vtx A B C D E Z - 2 3 5 4 7 6 8 1
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58. Path Costs from A Given by row entries A  B :: 2 A  C :: 3
A  D :: 5
A  E :: 4
A  Z :: 7
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59. Path Costs from B Given by row entries B  A :: 2 B  C :: 5
B  D :: 3
B  E :: 2
B  Z :: 5
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60. Path Costs from C Given by row entries C  A :: 3 C  B :: 5
C  D :: 6
C  E :: 5
C  Z :: 8
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61. Path Costs from D Given by row entries D  A :: 5 D  B :: 3
D  C :: 6
D  E :: 1
D  Z :: 2
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62. Path Costs from E Given by row entries E  A :: 4 E  B :: 2
E  C :: 5
E  D :: 1
E  Z :: 3
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63. Path Costs from Z Given by row entries Z  A :: 7 Z  B :: 5
Z  C :: 8
Z  D :: 2
Z  E :: 3
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64. Shortest Paths Floyd’s Algorithm gives the path costs
Finding the actual paths reuqires additional work
We will need two matrices
One to hold travel costs
Other to hold actual paths
2007
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