- May 2023
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Local file Local file
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Proposition 5.4
Consequence of Fejer Monotonicity
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- Apr 2023
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www.dl.behinehyab.com www.dl.behinehyab.com
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Theorem 5.7.
Complexity of the Floyd warshall Algorithm.
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- Mar 2023
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www.dl.behinehyab.com www.dl.behinehyab.com
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If we apply themodified label-correcting algorithm to a problem with nonnegative arc lengths andwe always examine a node from LIST with the minimum distance label, the resultingalgorithm is the same as Dijkstra's algorithm discussed in Section 4.5
Dijkstra's interpretation from the generic labeling algorithm.
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Modified Label-Correcting Algorithm
Not Bellmand ford, but it's an generic version of the FIFO it looks like.
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Consequently, we can store all nodeswhose distance labels change during a pass, and consider (or examine) only thosenodes in the next pass.
this is the key that helps to improve the labeling algorithm and develope the idea of FIFO.
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In the Floyd-Warshall algorithm, we detect the presence of a negative cyclesimply by checking the condition d[i, i] < 0
Detecting the negative cycles using Floyd Warshall Algorithm.
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- Feb 2023
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www.dl.behinehyab.com www.dl.behinehyab.com
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5.41.
Graduate base, proof. Description:
Gives a proof if the answer is yes or a no.
We would need an arc that goes over $O(n^2)$ many pairs of paths for all the possible path, and we need to firstly show that the previous algorithm 5.40 is not going to work anymore and, if would have no choice to look over almost all the possible path to figure out the optimal path for the pertrubed costs!
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5.40
Graduate based, proof. Description:
Use the answers from 4.20 to help.
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5.4
Undergraduate homework question. Description from the assignment sheet:
Order the nodes 1,...,7, and order the arcs with tail node i, A(i), increasing by their head nodes. See next page for the algorithm.
The graph is at the same page. The FIFO Label-Correcting algorithm is exactly the same as Moore Bellman Ford Algorithm. Labels at the end should be:
[0, -5, 15, -5, -10] -
5.56
Undergraduate. Proof based. Take note that the converse is definitely not true. But if we assume cycle exists in the predecessor graph, then there has to be a negative cycle cost directed cycle on the original graph.
Key: Page 137, last 10 lines and keep reading. The invariant of the label correcting algorithm is the key.
The reduce costs of all arcs \((i, j)\) on the predecessor graph have to have a reduced cost that is less than zero.\(\forall (j, pred(j)): d(j) \ge d(i) + c_{i, j}\).
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For those familiar with linear programming, we point out that the shortest pathoptimality conditions can also be viewed as the linear programming optimality con-ditions. In the linear programming formulation of the shortest path problem, thenegative of the shortest path distances [i.e., - d(j)] define the optimal dual variables,and the conditions (5.2) are equivalent to the fact that in the optimal solution, reducedcosts of all primal variables are nonnegative. The presence of a negative cycle impliesthe unboundedness of the primal problem and hence the infeasibility of the dualproblem.
Reduced costs perspective of the shortest path optimal labels, such an interpretation exists due to the reduction from the shortest path to network flow.
- The distance is the dual variable.
- The shortest path label is the negative of the potential, the negative of the dual variables.
- The non-negative condition of the reduced costs comes from the simplex interpretation of the matter.
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Theorem 5.1 (Shortest Path Optimality Conditions).
There is a linear programming interpretation via the network flow standard form and the reduction of shortest path to the network flow problem at the end of chapter 5.2.
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Property 5.2
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algorithm Floyd-Warshall
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