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Advanced Tree Partitioning Shifting Algorithm

Explore the shifting algorithm technique for tree partitioning, including max-min, min-max, and size-constrained solutions. Optimal partitions ensure algorithm termination within a finite number of steps, guaranteeing an optimal result. Lemmas provide insights into down-shifting and component weights. Experience the power of the shifting algorithm in tree partitioning problems.

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Advanced Tree Partitioning Shifting Algorithm

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  1. The shifting algorithm technique for the partitioning of trees

  2. Tree partition 1 1 6 4 3 1 1 7 6 4 3 7

  3. Problem • Max-min • Min-max • Size constrained min-max • Height-constrained min-max • Most uniform problem

  4. Shifting algorithm

  5. 4-partition 1 1 6 4 3 1 1 7 6 4 3 7

  6. 1 1 6 4 3 1 1 7 6 4 3 7

  7. 1 1 6 4 3 1 1 7 6 4 3 7

  8. 1 1 6 4 3 1 1 7 6 4 3 7

  9. 1 1 6 4 3 1 1 7 6 4 3 7

  10. 1 1 6 4 3 1 1 7 6 4 3 7

  11. Definition • A above Q

  12. Optimality • If A above Q, the algorithm continues. • If A is not optimal, the algorithm does not terminate at A. • Since the algorithm must terminate in a finite number of steps, the result is an optimal partition

  13. Lemma1 Let A > Q. Then the algorithm does not terminate. It makes a down-shift with resulting down-component of weight ≥ Wmin(Q).

  14. Lemma2 • Let A > Q. Let the application of one down-shift of the algorithm change A to A’. Then there exists an optimal partition Q’ such that A’≥ Q’.

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