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Modified Label Correcting Algorithm for Optimizing Distance Labels

This document outlines the Modified Label Correcting Algorithm, a method for optimizing distance labels in graph structures. By sequentially updating distance values for nodes based on their arcs and predecessors, the algorithm ensures that the shortest paths are accurately calculated. The algorithm initializes distances and iteratively refines them to reach an optimal solution. Key examples illustrate the process of node selection and distance updating, emphasizing how the algorithm adapts to the graph's structure. This technique is especially useful for solving routing and network optimization problems.

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Modified Label Correcting Algorithm for Optimizing Distance Labels

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  1. 15.082 and 6.855J Modified Label Correcting Algorithm

  2. The Modified Label Correcting Algorithm   2 5 2 3 3 3 1   -2 6 1 4 7 0 Initialize 2 3 4 3 d(1) := 0; d(j) :=  for j  1 -4 3 6   LIST := {1} In next slides: the number inside the node will be d(j).

  3. An Example 2  LIST := { 1 } LIST := { 2, 3 } LIST := { 2 } LIST := { } LIST := {1} LIST := { 2, 3, 4 }  3 3 3 3 1 -2 6 0 0  6  2 3 4 3 Generic Step -4  3  Take a node i from LIST Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  4. An Example 2  5 LIST := { 3, 4, 5 } LIST := {1} LIST := { 2 } LIST := { } LIST := { 2, 3, 4 } LIST := { 3, 4 } LIST := { 2, 3 } 3 3  3 3 3 1 -2 6 0 4 6   2 3 4 3 -4 Take a node i from LIST 3   Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  5. An Example 2 5  LIST := { 4, 5 } 3  LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0 6 4   2 3 4 3 -4 Take a node i from LIST 3  3  Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  6. An Example 2 5  LIST := { 5 } LIST := { 4, 5 } LIST := { 5, 6 } 3  LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0  4 4 6  2 3 4 3 -4 Take a node i from LIST 3  6  Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  7. An Example 2 5  5 LIST := { 5, 6 } LIST := { 5 } LIST := { 4, 5 } LIST := { 6 }  3 LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0 4  6  2 3 4 3 -4 Take a node i from LIST  3  6 Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  8. An Example 2  5 LIST := { 3, 7 } LIST := { 5 } LIST := { 5, 6 } LIST := { 4, 5 } LIST := { } LIST := { 3 } LIST := { 6 } 3  LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0  6 4  9 2 3 4 3 -4 Take a node i from LIST 2  3  6 6 Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  9. An Example 2  5 LIST := { 7 } LIST := { 4, 5 } LIST := { 5, 6 } LIST := { 5 } LIST := { } LIST := { 3 } LIST := { 3, 7 } LIST := { 6 }  3 LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0  4 6  9 2 3 4 3 -4 Take a node i from LIST 3  2 2  6 Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  10. An Example 2  5 LIST := { } LIST := { 4, 5 } LIST := { 5 } LIST := { 6 } LIST := { 5, 6 } LIST := { 3 } LIST := { 3, 7 } LIST := { 7 } LIST := { }  3 LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0 6 4  9 9  2 3 4 3 -4 Take a node i from LIST  2 3  6 Update(i): for each arc (i,j) with d(j) > d(i) + cijreplace d(j) by d(i) + cij.

  11. An Example 2 5  LIST := { 7 } LIST := { 3, 7 } LIST := { 5, 6 } LIST := { 3 } LIST := { 5 } LIST := { } LIST := { 6 } LIST := { 4, 5 } LIST := { }  3 LIST := { 3, 4, 5 } 3 3 3 1 -2 6 0 6 4  9  2 3 4 3 -4 LIST is empty. The distance labels are optimal 3 2   6 Here are the predecessors

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