Improved Bounds for 3SUM, kk-SUM, and Linear Degeneracy

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 Gr{\o}nlund and Pettie [FOCS'14], a simple $\Theta(n^2)$-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 of 3SUM is $O(n^{3/2})$, and that the randomized $(2k-2)$-linear decision tree complexity of $k$-SUM and $k$-LDT is $O(n^{k/2})$, for any odd $k\ge 3$. These bounds improve (albeit randomized) the corresponding $O(n^{3/2}\sqrt{\log n})$ and $O(n^{k/2}\sqrt{\log n})$ decision tree bounds obtained by Gr{\o}nlund 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 $O(n^2 \log\log n / \log n )$ 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 word-RAM model

Dominance Product and High-Dimensional Closest Pair under L∞L_\infty

Given a set $S$ of $n$ points in $\mathbb{R}^d$, the Closest Pair problem is to find a pair of distinct points in $S$ at minimum distance. When $d$ is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general $d$. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional $L_\infty$ Closest Pair problem, where $d=n^r$ for some $r > 0$, and the underlying metric is $L_\infty$. We improve and simplify previous results for $L_\infty$ Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in $O(DP(n,d)\log n)$ time, and by a randomized algorithm that runs in $O(DP(n,d))$ expected time, where $DP(n,d)$ is the time bound for computing the {\em dominance product} for $n$ points in $\mathbb{R}^d$. That is a matrix $D$, such that $D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|$; this is the number of coordinates at which $p_j$ dominates $p_i$. For integer coordinates from some interval $[-M, M]$, we obtain an algorithm that runs in $\tilde{O}\left(\min\{Mn^{\omega(1,r,1)},\, DP(n,d)\}\right)$ time, where $\omega(1,r,1)$ is the exponent of multiplying an $n \times n^r$ matrix by an $n^r \times n$ matrix. We also give slightly better bounds for $DP(n,d)$, by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths)

Dynamic Time Warping and Geometric Edit Distance: Breaking the Quadratic Barrier

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 paper, we break the nearly 50 years 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_infty)

Near-optimal Linear Decision Trees for k-SUM and Related Problems

We construct near-optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant k , we construct linear decision trees that solve the k -SUM problem on n elements using O ( n log 2 n ) linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two k -subsets; when viewed as linear queries, comparison queries are 2 k -sparse and have only { −1,0,1} coefficients. We give similar constructions for sorting sumsets A+B and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms. Our constructions are based on the notion of “inference dimension,” recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension

"Acute pseudo-pericardial tamponade": the compression of the thoracal inferior vena cava – a case report

We describe a case of 68-year-old woman which was admitted to our hospital for mitral valve replacement (MVR), in whom acute compresion of the vena cava inferior developed after repair of lacerated atrio-caval junction with hemostatic tissue sealant, biologic glue (BioGlue, Cryolife, ınc, Kennesaw, Ga). Removal of the BioGlue relieved the unexpected problem

Global Respiratory Syncytial Virus-Related Infant Community Deaths.

BACKGROUND: Respiratory syncytial virus (RSV) is a leading cause of pediatric death, with >99% of mortality occurring in low- and lower middle-income countries. At least half of RSV-related deaths are estimated to occur in the community, but clinical characteristics of this group of children remain poorly characterized. METHODS: The RSV Global Online Mortality Database (RSV GOLD), a global registry of under-5 children who have died with RSV-related illness, describes clinical characteristics of children dying of RSV through global data sharing. RSV GOLD acts as a collaborative platform for global deaths, including community mortality studies described in this supplement. We aimed to compare the age distribution of infant deaths <6 months occurring in the community with in-hospital. RESULTS: We studied 829 RSV-related deaths <1 year of age from 38 developing countries, including 166 community deaths from 12 countries. There were 629 deaths that occurred <6 months, of which 156 (25%) occurred in the community. Among infants who died before 6 months of age, median age at death in the community (1.5 months; IQR: 0.8-3.3) was lower than in-hospital (2.4 months; IQR: 1.5-4.0; P < .0001). The proportion of neonatal deaths was higher in the community (29%, 46/156) than in-hospital (12%, 57/473, P < 0.0001). CONCLUSIONS: We observed that children in the community die at a younger age. We expect that maternal vaccination or immunoprophylaxis against RSV will have a larger impact on RSV-related mortality in the community than in-hospital. This case series of RSV-related community deaths, made possible through global data sharing, allowed us to assess the potential impact of future RSV vaccines