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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
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
4-partition 1 1 6 4 3 1 1 7 6 4 3 7
1 1 6 4 3 1 1 7 6 4 3 7
1 1 6 4 3 1 1 7 6 4 3 7
1 1 6 4 3 1 1 7 6 4 3 7
1 1 6 4 3 1 1 7 6 4 3 7
1 1 6 4 3 1 1 7 6 4 3 7
Definition • A above Q
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
Lemma1 Let A > Q. Then the algorithm does not terminate. It makes a down-shift with resulting down-component of weight ≥ Wmin(Q).
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’.