FIFO preflow-push: Difference between revisions
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The complexity considerations for the [[Preflow-push|generic preflow-push algorithm]] yield <math>\mathcal{O}(n^2)</math> relabel operations and <math>\mathcal{O}(n\cdot m)</math> saturating push operations, where <math>m=|A|</math>. Both function classes are subsets of <math>\mathcal{O}(n^3)</math>. There it was also shown that the total number of changes of outgoing arcs is in <math>\mathcal{O}(n^3)</math>. Hence, it rmains to show that the total number of non-saturating push operations is in <math>\mathcal{O}(n^3)</math>. | The complexity considerations for the [[Preflow-push|generic preflow-push algorithm]] yield <math>\mathcal{O}(n^2)</math> relabel operations and <math>\mathcal{O}(n\cdot m)</math> saturating push operations, where <math>m=|A|</math>. Both function classes are subsets of <math>\mathcal{O}(n^3)</math>. There it was also shown that the total number of changes of outgoing arcs is in <math>\mathcal{O}(n^3)</math>. Hence, it rmains to show that the total number of non-saturating push operations is in <math>\mathcal{O}(n^3)</math>. | ||
Conceptually, we may partition the | Conceptually, we may partition the iterations main loop into ''phases''. The first phase commences with the very first iteration. A new phase commences as soon as all nodes that were in <math>Q</math> at the beginning fof the last phase, have been extracted once from <matH>Q</math>. | ||
Revision as of 18:14, 27 October 2014
Abstract view
This is a specialization of Preflow-push:
- The set of all active nodes is stored in a FIFO queue [math]\displaystyle{ Q }[/math].
- While the first node in [math]\displaystyle{ Q }[/math] is active and has an admissible arc, choose this node.
Complexity
Statement: The asymptotic complexity is in [math]\displaystyle{ \mathcal{O}(n^3) }[/math], where [math]\displaystyle{ n=|V| }[/math].
Proof: The complexity considerations for the generic preflow-push algorithm yield [math]\displaystyle{ \mathcal{O}(n^2) }[/math] relabel operations and [math]\displaystyle{ \mathcal{O}(n\cdot m) }[/math] saturating push operations, where [math]\displaystyle{ m=|A| }[/math]. Both function classes are subsets of [math]\displaystyle{ \mathcal{O}(n^3) }[/math]. There it was also shown that the total number of changes of outgoing arcs is in [math]\displaystyle{ \mathcal{O}(n^3) }[/math]. Hence, it rmains to show that the total number of non-saturating push operations is in [math]\displaystyle{ \mathcal{O}(n^3) }[/math].
Conceptually, we may partition the iterations main loop into phases. The first phase commences with the very first iteration. A new phase commences as soon as all nodes that were in [math]\displaystyle{ Q }[/math] at the beginning fof the last phase, have been extracted once from [math]\displaystyle{ Q }[/math].