We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class.We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge e in an m-edge graph the number of triangles through e. This natural triangle problem can be solved using the best known triangle counting algorithm,running time is known to be tight (within m o(1) factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound.To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the (min, max)-product. * Partially supported by the National Science Center, Poland under grants 2017/27/N/ST6/01334 and 2018/28/T/ST6/00305. O m 2ω/(ω+1) time by the Alon-Yuster-Zwick [4] algorithm.Definition 1 (EdgeTriangleCounting). Given an undirected graph G = (V, E), with n nodes and m edges, compute for every edge e ∈ E the number of triangles in G which contain e.
We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X1, . . . , X k of length at most n, the task is to determine the length of the longest common subsequence of X1, . . . , X k that is also strictly increasing. Especially for the case of k = 2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case.Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound (1) to rule out O (nL) 1−ε time algorithms for LCIS, where L denotes the solution size, (2) to rule out O n k−ε time algorithms for k-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.
We prove that the difference between the paint number and the choice number of a complete bipartite graph KN,N is Θ(log log N ). That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with Θ(2 k ) edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.
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