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The shifting algorithm technique for the partitioning of trees

The shifting algorithm technique for the partitioning of trees. Tree partition. 1. 1. 6. 4. 3. 1. 1. 7. 6. 4. 3. 7. Problem. Max-min Min-max Size constrained min-max Height-constrained min-max Most uniform problem. Shifting algorithm. 4-partition. 1. 1. 6. 4. 3. 1. 1.

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The shifting algorithm technique for the partitioning of trees

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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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