We study the problem of computing a longest increasing subsequence in a sequence @math of @math distinct elements in the presence of persistent comparison errors. In this model, every comparison between two elements can return the wrong result with some fixed (small) probability @math , and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically tight upper and lower bounds on both the approximation factor and the running time. In particular, we present an algorithm that computes an @math -approximation in time @math , with high probability. This approximation relies on the fact that that we can approximately sort @math elements in @math time such that the maximum dislocation of an element is at most @math . For the lower bounds, we prove that (i) there is a set of sequences, such that on a sequence picked randomly from this set every algorithm must return an @math -approximation with high probability, and (ii) any @math -approximation algorithm for longest increasing subsequence requires @math comparisons, even in the absence of errors.
There are several algorithms to compute a longest increasing subsequence of a sequence @math , if no comparison errors happen. Typically, they are based on a common underlying algorithmic idea: They process the elements one by one and maintain for each length found so far the increasing subsequence of this length that ends with the smallest possible element seen so far. We shall call this algorithmic idea the to compute a longest increasing subsequence. The running time of the Core-Algorithm is @math in the decision-tree model (see for instance @cite_15 @cite_0 @cite_5 ). This time complexity is tight, as shown in @cite_5 . In the RAM model, where one can also inspect the values, the algorithm can be implemented to run in @math time @cite_10 @cite_11 . All the results can be parameterized to @math or @math , respectively, where @math is the length of the longest increasing subsequence.
@cite_0 @cite_5 @cite_15 @cite_10 @cite_11
2009853785 2069893028 2010260400 1989633256
We consider the complexity of computing a longest increasing subsequence (LIS) parameterised by the length of the output. Namely, we show that the maximal length k of an increasing subsequence of a permutation of the set of integers 1,2,...,n can be computed in time O(nloglogk) in the RAM model, improving the previous 30-year bound of O(nlogk). The algorithm also improves on the previous O(nloglogn) bound. The optimality of the new bound is an open question. Reducing the computation of a longest common subsequence (LCS) between two strings to an LIS computation leads to a simple O(rloglogk)-time algorithm for two sequences having r pairs of matching symbols and an LCS of length k. Let S = x"1, x"2, ... x"n be a sequence of n distinct elements from a linearly ordered set. We consider the problem of determining the length of the longest increasing subsequences of S. An algorithm which performs this task is described and is shown to perform n log n-n log log n + O(n) comparisons in its worst case. This worst case behavior is shown to be best possible. The problems of finding shortest common supersequences (SCS) and longest common subsequences (LCS) are two well-known NP -hard problems that have applications in many areas including computational molecular biology, data compression, robot motion planning and scheduling, text editing, etc. A lot of fruitless effort has been spent in searching for good approximation algorithms for these problems. In this paper, we show that these problems are inherently hard to approximate in the worst case. In particular, we prove that (i) SCS does not have a polynomial time linear approximation algorithm, unless P = NP ; (ii) There exists a constant @math such that, if SCS has a polynomial time approximation algorithm with ratio @math , where @math is the number of input sequences, then NP is contained in DTIME @math ; (iii) There exists a constant @math such that, if LCS has a polynomial time approximation algorithm with performance ratio @math , then P = NP . The proofs utilize the recent results of [Proc. 23rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 14-23] on the complexity of approximation problems. In the second part of the paper, we introduce a new method for analyzing the average-case performance of algorithms for sequences, based on Kolmogorov complexity. Despite the above nonapproximability results, we show that near optimal solutions for both SCS and LCS can be found on the average. More precisely, consider a fixed alphabet @math and suppose that the input sequences are generated randomly according to the uniform probability distribution and are of the same length @math . Moreover, assume that the number of input sequences is polynomial in @math . Then, there are simple greedy algorithms which approximate SCS and LCS with expected additive errors @math and @math for any @math , respectively. Incidentally, our analyses also provide tight upper and lower bounds on the expected LCS and SCS lengths for a set of random sequences, solving a generalization of another well-known open question on the expected LCS length for two random sequences [K. Alexander, The rate of convergence of the mean length of the longest common subsequence, 1992, manuscript],[V. Chvatal and D. Sankoff, J. Appl. Probab., 12 (1975), pp. 306-315], [D. Sankoff and J. Kruskall, eds., Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, Addison-Wesley, Reading, MA, 1983]. We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation ( S_n ) is proportional to (q^ Inv ( ) ) where (q ) is a real parameter and ( Inv ( ) ) is the number of inversions in ( ). The case (q=1 ) corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that (q<1 ), our results apply when (q ) is a function of (n ) satisfying (n(1-q) ). The case that (n(1-q)=O(1) ) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order (n 1-q ), whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of (n ) and (q ). We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order (1 (1-q) ). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.