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Lecture 25 Partition

Lecture 25 Partition. Partition. Subsum. S=33333333333333333333 1111111111111111111111. 1. 1. Puzzle. Answer. Knapsack. Decision Version. Theorem. Proof. Theorem. Algorithm. Classify: for i < m, c i < a= c G , for i > m+1, c i > a. Sort For.

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Lecture 25 Partition

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  1. Lecture 25 Partition

  2. Partition

  3. Subsum

  4. S=33333333333333333333 1111111111111111111111 1 1

  5. Puzzle Answer

  6. Knapsack

  7. Decision Version

  8. Theorem Proof.

  9. Theorem

  10. Algorithm • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For

  11. Proof.

  12. Time

  13. Fully PTAS • A problem has a fully PTAS if for any ε>0, it has (1+ε)-approximation running in time poly(n,1/ε).

  14. Fully FTAS for Knapsack

  15. Time • outside loop: O(n) • Inside loop: O(nM) where M=max ci • Core: O(n log (MS)) • Total O(n M log (MS)) • Since input size is O(n log (MS)), this is a pseudo-polynomial-time due to M=2 3 log M

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