We present an interprocedural flow-insensitive points-to analysis based on type inference methods with an almost linear time cost complexity. To our knowledge, this is the asymptotically fastest non-trivial interprocedural points-to analysis algorithm yet described. The algorithm is based on a non

this paper, we study and improve three recently proposed algorithms for identifying loops in an irreducible graph. The first algorithm we study is due to Havlak [1997]. We show that the running time of this algorithm is quadratic in the worst-case, and not almost-linear as claimed. We then show how

in the worst case. For a given sequence of length n, we present an almost linear-time algorithm for this problem. Our algorithm uses a disjoint-set data structure and requires O(nα(n, n)) time in the worst case, where α(n, n) is the inverse Ackermann function.

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We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V (G) with |X | ≤ k such that G − X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number

reserves the right to add new problems once a problem set has already been posted. Questions from last time. • What is a level set? The level set of a function corresponding to a (fixed) constant c is the set of points in the function’s domain whose image equals c. • What is a good reference

Let F be a perfect field, and let X = X1,..., Xn be indeterminates over F. A (monic) triangular set T = (T1,..., Tn) is a family of polynomials in F[X] such that for all i, Ti is in F[X1,..., Xi], monic in Xi, and reduced modulo 〈T1,..., Ti−1〉. The degree of T is the product deg(T1, X1) · · · deg(Tn, Xn). These objects allow one to solve a variety of problems for systems of polynomial equations, see [7, 1, 10, 6, 12]. We are interested here in the complexity of operations modulo a given triangular set T. The first question is modular multiplication: given polynomials A, B reduced modulo T, compute AB mod T. Further operations involve families of triangular sets. The lexicographic Gröbner basis of an ideal I for a given variable order may not be triangular. The workaround is to decompose I as I = I1∩ · · ·∩Is, with pairwise coprime Ij, where each Ij admits a triangular basis. The decomposition is in general not unique, but there exists a canonical choice, the equiprojectable decomposition [4]. That said, the most useful notion of “inversion ” is quasi-inverses: given A reduced modulo T, we decompose the ideal 〈T 〉 as I0 ∩ I1, where A is zero modulo I0 and invertible modulo I1; the output is the equiprojectable decompositions of I0, I1, and the inverse of A modulo the triangular sets that define I1. The next question is change of order: starting from T, we output the equiprojectable

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires Ω(n2) time [1], [2], [3]. A key ingredient in our algorithm is a novel combination

We present improvements to a greedy decoding algorithm for statistical machine translation that reduce its time complexity from at least cubic (O(n^6) when applied navely) to practically linear time without sacrificing translation quality. We achieve this by integrating hypothesis

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