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  • 1.
    Arvestad, Lars
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
    Lagergren, Jens
    KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
    Sennblad, Bengt
    The Gene Evolution Model and Computing Its Associated Probabilities2009Ingår i: Journal of the ACM, ISSN 0004-5411, E-ISSN 1557-735X, Vol. 56, nr 2Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Phylogeny is both a fundamental tool in biology and a rich source of fascinating modeling and algorithmic problems. Today's wealth of sequenced genomes makes it increasingly important to understand evolutionary events such as duplications, losses, transpositions, inversions, lateral transfers, and domain shuffling. We focus on the gene duplication event, that constitutes a major force in the creation of genes with new function [Ohno 1970; Lynch and Force 2000] and, thereby also, of biodiversity. We introduce the probabilistic gene evolution model, which describes how a gene tree evolves within a given species tree with respect to speciation, gene duplication, and gene loss. The actual relation between gene tree and species tree is captured by a reconciliation, a concept which we generalize for more expressiveness. The model is a canonical generalization of the classical linear birth-death process, obtained by replacing the interval where the process takes place by a tree. For the gene evolution model, we derive efficient algorithms for some associated probability distributions: the probability of a reconciled tree, the probability of a gene tree, the maximum probability reconciliation, the posterior probability of a reconciliation, and sampling reconciliations with respect to the posterior probability. These algorithms provides the basis for several applications, including species tree construction, reconciliation analysis, orthology analysis, biogeography, and host-parasite co-evolution.

  • 2.
    Henzinger, Monika
    et al.
    Univ Vienna, Fac Comp Sci, Wahringer Str 29, A-1090 Vienna, Austria..
    Krinninger, Sebastian
    Univ Salzburg, Dept Comp Sci, Jakob Haringer Str 2, A-5020 Salzburg, Austria..
    Nanongkai, Danupon
    KTH, Skolan för datavetenskap och kommunikation (CSC).
    Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time2018Ingår i: Journal of the ACM, ISSN 0004-5411, E-ISSN 1557-735X, Vol. 65, nr 6, artikel-id 36Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the decremental single-source shortest paths (SSSP) problem, we want to maintain the distances between a given source node s and every other node in an n-node m-edge graph G undergoing edge deletions. While its static counterpart can be solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic O(mn) total update time of Even and Shiloach [16] has been the fastest known algorithm for three decades. At the cost of a (1 + is an element of)-approximation factor, the running time was recently improved to n(2)(+o(1)) by Bernstein and Roditty [9]. In this article, we bring the running time down to near-linear: We give a (1 + is an element of)-approximation algorithm with m(1+)(o(1)) expected total update time, thus obtaining near-linear time. Moreover, we obtain m(1)(+)(o(1)) log W time for the weighted case, where the edge weights are integers from 1 to W. The only prior work on weighted graphs in o(mn) time is the mn(0.9+o(1))-time algorithm by Henzinger et al. [18, 19], which works for directed graphs with quasi-polynomial edge weights. The expected running time bound of our algorithm holds against an oblivious adversary. In contrast to the previous results, which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse (h, is an element of)-hop set introduced by Cohen [12] in the PRAM literature. An (h, is an element of)-hop set of a graph G = (V, E) is a set F of weighted edges such that the distance between any pair of nodes in G can be (1 + is an element of)-approximated by their h-hop distance (given by a path containing at most h edges) on G' = (V, E boolean OR F). Our algorithm can maintain an (n(o(1)), is an element of)-hop set of near-linear size in near-linear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain approximate distances using this hop set, we extend the monotone Even-Shiloach tree of Henzinger et al. [20] and combine it with the bounded-hop SSSP technique of Bernstein [4, 5] and Madry [27]. These two new tools might be of independent interest.

  • 3.
    Håstad, Johan
    KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
    Some optimal inapproximability results2001Ingår i: Journal of the ACM, ISSN 0004-5411, E-ISSN 1557-735X, Vol. 48, nr 4, s. 798-859Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove optimal, up to an arbitrary epsilon > 0, inapproximability results for Max-Ek-Sat for k greater than or equal to 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.

  • 4.
    Håstad, Johan
    et al.
    KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
    Naslund, M.
    The security of all RSA and discrete log bits2004Ingår i: Journal of the ACM, ISSN 0004-5411, E-ISSN 1557-735X, Vol. 51, nr 2, s. 187-230Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the security of individual bits in an RSA encrypted message E-N(x). We show that given E-N(x), predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy, is (through a polynomial-time reduction) as hard as breaking RSA. Moreover, we prove that blocks of O (log log N) bits of x are computationally indistinguishable from random bits. The results carry over to the Rabin encryption scheme. Considering the discrete exponentiation function g(x) modulo p, with probability 1 - o(1) over random choices of the prime p, the analog results are demonstrated. The results do not rely on group representation, and therefore applies to general cyclic groups as well. Finally, we prove that the bits of ax + b modulo p give hard core predicates for any one-way function f. All our results follow from a general result on the chosen multiplier hidden numberproblem: given an integer N, and access to an algorithm P-x, that on input a random a epsilon Z(N), returns a guess of the ith bit of ax mod N, recover x. We show that for any i, if P-x has at least a nonnegligible advantage in predicting the ith bit, we either recover x, or, obtain a nontrivial factor of N in polynomial time. The result also extends to prove the results about simultaneous security of blocks of O (log log N) bits.

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