Note on Max Lin-2 above Average

In the Max Lin-2 problem we are given a system S of m linear equations in n variables over F₂ in which equation j is assigned a positive integral weight wj for each j. We wish to find an assignment of values to the variables which maximizes the total weight of satisfied equations. This problem generalizes Max Cut. The expected weight of satisfied equations is W/2, where W=w₁+?+wm; W/2 is a tight lower bound on the optimal solution of Max Lin-2.Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153) stated the following parameterized version of Max Lin-2: decide whether there is an assignment of values to the variables that satisfies equations of total weight at least W/2+k, where k is the parameter. They asked whether this parameterized problem is fixed-parameter tractable, i.e., can be solved in time f(k)(nm)^O(1), where f(k) is an arbitrary computable function in k only. Their question remains open, but using some probabilistic inequalities and, in one case, a Fourier analysis inequality, Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245) proved that the problem is fixed-parameter tractable in three special cases.In this paper we significantly extend two of the three special cases using only tools from combinatorics. We show that one of our results can be used to obtain a combinatorial proof that another problem from Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153), Max r-SAT above Average, is fixed-parameter tractable for each r?2. Note that Max r-SAT above Average has been already shown to be fixed-parameter tractable by Alon et al. (Solving MAX-r-SAT above a tight lower bound, in: Proc. SODA 2010, pp. 511-517), but the paper used the approach of Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245).     

机群环境下傅立叶变换的并行算法研究

并行FFT是解决大数据量FFT运算耗时过久的重要途径,在PC机群上实现并行FFT是一种低成本、高效率的解决方案。本文讨论了PC机群环境下MPI并行FFT实现,并利用建立的平台,对并行算法进行了测试,得出了一些有意义的结论和方法。     

Bottleneck flows in unit capacity networks

The bottleneck network flow problem (BNFP) is a generalization of several well-studied bottleneck problems such as the bottleneck transportation problem (BTP), bottleneck assignment problem (BAP), bottleneck path problem (BPP), and so on. The BNFP can easily be solved as a sequence of O(logn) maximum flow problems on almost unit capacity networks. We observe that this algorithm runs in O(min{m3/2,n2/3m}logn) time by showing that the maximum flow problem on an almost unit capacity graph can be solved in O(min{m3/2,n2/3m}) time. We then propose a faster algorithm to solve the unit capacity BNFP in time, an improvement by a factor of at least . For dense graphs, the improvement is by a factor of . On unit capacity simple graphs, we show that BNFP can be solved in time, an improvement by a factor of . As a consequence we have an algorithm for the BTP with unit arc capacities.     

Optimal algorithms for the average-constrained maximum-sum segment problem

Given a real number sequence A=(a₁,a₂,…,an), an average lower bound L, and an average upper bound U, the Average-Constrained Maximum-Sum Segment problem is to locate a segment A(i,j)=(ai,ai+1,…,aj) that maximizes _∑i?k?jak subject to . In this paper, we give an O(n)-time algorithm for the case where the average upper bound is ineffective, i.e., U=∞. On the other hand, we prove that the time complexity of the problem with an effective average upper bound is Ω(nlogn) even if the average lower bound is ineffective, i.e., L=−∞.     

Range mode and range median queries in constant time and sub-quadratic space

Given a list of n items and a function defined over sub-lists, we study the space required for computing the function for arbitrary sub-lists in constant time.For the function mode we improve the previously known space bound O(n²/logn) to O(n²loglogn/log²n) words.For median the space bound is improved to O(n²loglog²n/log²n) words from O(n²⋅log(k)n/logn), where k is an arbitrary constant and log(k) is the iterated logarithm.     

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g=3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g∈{4,5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g?6, both problems are APX-hard already for graphs with maximum degree 3.     

Improving multikey Quicksort for sorting strings with many equal elements

Bentley and Sedgewick proposed multikey Quicksort with ‘split-end’ partitioning for sorting strings. But it can be slow in case of many equal elements because it adopted ‘split-end’ partitioning that moves equal elements to the ends and swaps back to the middle. We present ‘collect-center’ partitioning to improve multikey Quicksort in that case. It moves equal elements to the middle directly like the ‘Dutch National Flag Problem’ partitioning approach and it uses two inner loops like Bentley and McIlroy's. In case of many equal elements such as DNA sequences, HTML files, and English texts, multikey Quicksort with ‘collect-center’ partitioning is faster than multikey Quicksort with ‘split-end’ partitioning.     

Unary finite automata vs. arithmetic progressions

We point out a subtle error in the proof of Chrobak's theorem that every unary NFA can be represented as a union of arithmetic progressions that is at most quadratically large. We propose a correction for this and show how Martinez's polynomial time algorithm, which realizes Chrobak's theorem, can be made correct accordingly. We also show that Martinez's algorithm cannot be improved to have logarithmic space, unless L = NL.