Search

Given a real number α ∈ (0 , 1) , we define the Webster sequence of density α to be W α = (⌈ (n − 1 / 2) / α ⌉) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N. On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ. We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O (1) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O (1) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogeneous m -machine scheduling problem where unit-time jobs have to be scheduled within their time windows so that, for any subset of machines, the set of the time units at which at least one machine is busy, is an interval. For this problem, a time assignment of the jobs satisfying the time windows constraints is a schedule if it has a pyramidal profile and is a k -schedule if this profile property is satisfied up to level k only. We develop a generic algorithm to solve the problem and show it is correct using a proof that mainly relies on a necessary and sufficient condition for a schedule to exist proved in a previous paper. We finally show that the generic algorithm is polynomial. We conclude by giving the directions of ongoing works and by bringing open questions related to different variants of the basic non-idling m -machine scheduling problem. [ABSTRACT FROM AUTHOR]

Linear feedback shift registers (LFSRs) play a significant role in communications security and we investigate design of a selected class of word-based LFSRs known as σ -LFSRs. Both the search algorithm and the construction algorithm generate efficient primitive σ -LFSRs. The search algorithm first constructs the σ -polynomial and then checks the primitiveness of the σ -polynomial, whereas the construction algorithm for the σ -LFSR, first finds a primitive polynomial f ( x ) and then constructs the primitive σ -LFSR from f ( x ) . In this paper, we present some novel results pertaining to the search algorithm for primitive σ -LFSR along with the exhaustive search space complexity of the search algorithm for σ -LFSRs. Then we investigate and compare the performance of the construction algorithm with the search algorithm for the primitive σ -LFSR. Finally, the number of σ -LFSRs similar to the σ -LFSRs generated by the construction algorithm is provided. [ABSTRACT FROM AUTHOR]