Sorting In Linear Time Research Articles

DCJ-Indel sorting revisited

BackgroundThe introduction of the double cut and join operation (DCJ) caused a flurry of research into the study of multichromosomal rearrangements. However, little of this work has incorporated indels (i.e., insertions and deletions of chromosomes and chromosomal intervals) into the calculation of genomic distance functions, with the exception of Braga et al., who provided a linear time algorithm for the problem of DCJ-indel sorting. Although their algorithm only takes linear time, its derivation is lengthy and depends on a large number of possible cases.ResultsWe note the simple idea that a deletion of a chromosomal interval can be viewed as a DCJ that creates a new circular chromosome. This framework will allow us to amortize indels as DCJs, which in turn permits the application of the classical breakpoint graph to obtain a simplified indel model that still solves the problem of DCJ-indel sorting in linear time via a more concise formulation that relies on the simpler problem of DCJ sorting. Furthermore, we can extend this result to fully characterize the solution space of DCJ-indel sorting.ConclusionsEncoding indels as DCJ operations offers a new insight into why the problem of DCJ-indel sorting is not ultimately any more difficult than that of sorting by DCJs alone. There is still room for research in this area, most notably the problem of sorting when the cost of indels is allowed to vary with respect to the cost of a DCJ and we demand a minimum cost transformation of one genome into another.

Sorting in linear time - variations on the bucket sort

The Bucket Sort is an algorithm for sorting in linear time. That is, we can sort a list of n numbers in expected time O(n). Implementing the Bucket Sort shows off several important aspects of data ...

Sorting in Linear Time?

We show that a unit-cost RAM with a word length ofwbits can sortnintegers in the range 0…2w−1 inO(nloglogn) time for arbitraryw⩾logn, a significant improvement over the bound ofO(nlogn) achieved by the fusion trees of Fredman and Willard. Provided thatw⩾(logn)2+εfor some fixedε>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length ofwbits. The first one yields an algorithm that usesO(logn) time andO(nloglogn) operations on a deterministic CRCW PRAM. The second one yields an algorithm that usesO(logn) expected time andO(n) expected operations on a randomized EREW PRAM, provided thatw⩾(logn)2+εfor some fixedε>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words.

Theoretical bounds on the complexity of inexact computations

This paper considers the reduction in algorithmic complexity that can be achieved by permitting approximate answers to computational problems. It is shown that Shannon's rate-distortion function could, under quite general conditions, provide lower bounds on the mean complexity of inexact computations. As practical examples of this approach, we show that partial sorting of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> items, insisting on matching any nonzero fraction of the terms with their correct successors, requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O (N \log N)</tex> comparisons. On the other hand, partial sorting in linear time is feasible (and necessary) if one permits any finite fraction of pairs to remain out of order. It is also shown that any error tolerance below 50 percent can neither reduce the state complexity of binary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -sequences from the zero-error value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N)</tex> nor reduce the combinational complexity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -variable Boolean functions from the zero-error level of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(2^{N}/N)</tex> .