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

Overview Previously: In weighted graph . . Shortest path from #7 to all others Search matrix method Now: Problem 8.6.2 Implement Floyd’s Algorithm 2007 Kutztown University

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) 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

Denotations Outer loop – the i loop Middle loop – the j loop Inner loop – the k loop 2007 Kutztown University

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 2007 Kutztown University

Matrix for Weighted Graph Vtx A B C D E Z ∞ 2 3 5 1 4 2007 Kutztown University

First Outer Loop i = 1 j = ≔ 1 to 6 k ≔ 1 to 6 2007 Kutztown University

First Inner Loop i = 1 j = 1 k ≔ 1 to 6 2007 Kutztown University

First Middle Loop d(1,1) = ∞ d(1,1) + d(1,k) ≮ d(1,k), ∀k Therefore, no change 2007 Kutztown University

Second Middle Loop i = 1 j = 2 k ≔ 1 to 6 2007 Kutztown University

Second Middle Loop k = 1 d(1,1) = ∞ d(2,1) + d(1,1) = ∞ Therefore, no change 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

Third Middle Loop i = 1 j = 3 k ≔ 1 to 6 2007 Kutztown University

Third Middle Loop k = 1 d(1,1) = ∞ d(3,1) + d(1,1) = ∞ Therefore, no change 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

4th, 5th & 6th Middle Loops i = 1 j ≔ 4 to 6 k ≔ 1 to 6 2007 Kutztown University

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 2007 Kutztown University

Second Outer Loop i = 2 j = ≔ 1 to 6 k ≔ 1 to 6 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

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 2007 Kutztown University

Second Middle Loop i = 2 j = 2 k ≔ 1 to 6 2007 Kutztown University

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 2007 Kutztown University

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 + ∞ ≮ ∞ 2007 Kutztown University

Matrix After 2nd Middle Loop {no change} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 1 2007 Kutztown University

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 2007 Kutztown University

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 + ∞ ≮ ∞ 2007 Kutztown University

Matrix After 3rd Middle Loop {1 cell changed} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1 2007 Kutztown University

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 < ∞ 2007 Kutztown University

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 2007 Kutztown University

Matrix After 4th Middle Loop {3 cells changed} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1 2007 Kutztown University

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 2007 Kutztown University

Fifth Middle Loop k ≔ 4 to6 i j k 2 5 4 2 5 5 2 5 4 (5,2) + (2,4) < (5,4) ? 2 + 5 ≮ 1 2 5 5 (5,2) + (2,5) < (5,5) ? 2 + 2 < ∞ 2 5 6 (5,2) + (2,6) < (5,6) ? 2 + ∞ ≮ 4 2007 Kutztown University

Matrix After 5th Middle Loop {2 cells changed} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1 2007 Kutztown University

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 ≮ ∞ 2007 Kutztown University

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) ? ∞ + ∞ ≮ ∞ 2007 Kutztown University

Matrix After 6th Middle Loop {0 cells changed} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1 2007 Kutztown University

Third Outer Loop i = 3 j = ≔ 1 to 6 k ≔ 1 to 6 Step by step details given here 2007 Kutztown University

Matrix After 3rd Outer Loop {0 cells changed} Vtx A B C D E Z 4 2 3 7 ∞ 5 6 10 1 2007 Kutztown University

Fourth Outer Loop i = 4 j = ≔ 1 to 6 k ≔ 1 to 6 Step by step details given here 2007 Kutztown University

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 2007 Kutztown University

Fifth Outer Loop i = 5 j = ≔ 1 to 6 k ≔ 1 to 6 Step by step details given here 2007 Kutztown University

Matrix After 5th Outer Loop {12 cells changed} Vtx A B C D E Z 4 2 3 5 7 6 8 1 2007 Kutztown University

Sixth Outer Loop i = 6 j = ≔ 1 to 6 k ≔ 1 to 6 Step by step details given here 2007 Kutztown University

Matrix After 6th Outer Loop {0 cells changed} Vtx A B C D E Z 4 2 3 5 7 6 8 1 2007 Kutztown University

Final Matrix Diagonal values are removed Vtx A B C D E Z - 2 3 5 4 7 6 8 1 2007 Kutztown University

Path Costs from A Given by row entries A  B :: 2 A  C :: 3 A  D :: 5 A  E :: 4 A  Z :: 7 2007 Kutztown University

Path Costs from B Given by row entries B  A :: 2 B  C :: 5 B  D :: 3 B  E :: 2 B  Z :: 5 2007 Kutztown University

Path Costs from C Given by row entries C  A :: 3 C  B :: 5 C  D :: 6 C  E :: 5 C  Z :: 8 2007 Kutztown University

Path Costs from D Given by row entries D  A :: 5 D  B :: 3 D  C :: 6 D  E :: 1 D  Z :: 2 2007 Kutztown University

Path Costs from E Given by row entries E  A :: 4 E  B :: 2 E  C :: 5 E  D :: 1 E  Z :: 3 2007 Kutztown University

Path Costs from Z Given by row entries Z  A :: 7 Z  B :: 5 Z  C :: 8 Z  D :: 2 Z  E :: 3 2007 Kutztown University

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 Kutztown University