Abstract
A vector merging problem is introduced where two vectors of length n are merged such that the k-th entry of the new vector is the minimum over ℓ of the ℓ-th entry of the first vector plus the sum of the first k − ℓ + 1 entries of the second vector. For this problem a new algorithm with O(n log n) running time is presented thus improving upon the straightforward O(n 2) time bound.
The vector merging problem can appear in different settings of dynamic programming. In particular, it is applied for a recent fully polynomial time approximation scheme (FPTAS) for the classical 0–1 knapsack problem by the same authors.
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References
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Kellerer, H., Pferschy, U. Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem. Journal of Combinatorial Optimization 8, 5–11 (2004). https://doi.org/10.1023/B:JOCO.0000021934.29833.6b
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DOI: https://doi.org/10.1023/B:JOCO.0000021934.29833.6b