Dynamic Time Warping (DTW) and Geometric Edit Distance (GED) are basic similarity measures between curves or general temporal sequences (e.g., time series) that are represented as sequences of points in some metric space ( X , dist). The DTW and GED measures are massively used in various fields of computer science and computational biology. Consequently, the tasks of computing these measures are among the core problems in P. Despite extensive efforts to find more efficient algorithms, the best-known algorithms for computing the DTW or GED between two sequences of points in X = R d are long-standing dynamic programming algorithms that require quadratic runtime, even for the one-dimensional case d = 1, which is perhaps one of the most used in practice. In this article, we break the nearly 50-year-old quadratic time bound for computing DTW or GED between two sequences of n points in R by presenting deterministic algorithms that run in O ( n 2 log log log n / log log n ) time. Our algorithms can be extended to work also for higher-dimensional spaces R d , for any constant d , when the underlying distance-metric dist is polyhedral (e.g., L 1 , L infin ).
Given a set of n real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Grønlund and Pettie [FOCS'14], a simple Θpn 2 q-time deterministic algorithm for this problem was conjectured to be optimal. Over the years many algorithmic problems have been shown to be reducible from the 3SUM problem or its variants, including the more generalized forms of the problem, such as k-SUM and k-variate linear degeneracy testing (k-LDT). The conjectured hardness of these problems have become extremely popular for basing conditional lower bounds for numerous algorithmic problems in P.In this paper, we show that the randomized 4-linear decision tree complexity 1 of 3SUM is Opn 3{2 q, and that the randomized p2k ´2q-linear decision tree complexity of k-SUM and k-LDT is Opn k{2 q, for any odd k ě 3. These bounds improve (albeit randomized) the corresponding Opn 3{2 ? log nq and Opn k{2 ? log nq decision tree bounds obtained by Grønlund and Pettie. Our technique includes a specialized randomized variant of fractional cascading data structure. Additionally, we give another deterministic algorithm for 3SUM that runs in Opn 2 log log n{ log nq time. The latter bound matches a recent independent bound by Freund [Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use of the word-RAM model.
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation t. This property is captured by the notion of extremal-distance spanners. Given a graph G = (V, E), a subgraph H = (V, EH ) is defined to be a t-diameter spanner if the diameter of H is at most t times the diameter of G.We show that for any n-vertex and m-edges directed graph G, we can compute a sparse subgraph H that is a (1.5)-diameter spanner of G, such that H contains at most O(n 1.5 ) edges. We also show that the stretch factor cannot be improved to (1.5 − ). For a graph whose diameter is bounded by some constant, we show the existence of 5 3 -diameter spanner that contains at most O(n 4 3 ) edges. We also show that this bound is tight. Additionally, we present other types of extremal-distance spanners, such as 2-eccentricity spanners and 2-radius spanners, both contain only O(n) edges and are computable in O(m) time.Finally, we study extremal-distance spanners in the dynamic and fault-tolerant settings. An interesting implication of our work is the first O(m)-time algorithm for computing 2-approximation of vertex eccentricities in general directed weighted graphs. Backurs et al. [STOC 2018] gave an O(m √ n) time algorithm for this problem, and also showed that no O(n 2−o(1) ) time algorithm can achieve an approximation factor better than 2 for graph eccentricities, unless SETH fails; this shows that our approximation factor is essentially tight.
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