Online unit clustering in higher dimensions

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of $n$ points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in $\mathbb{R}^d$ using the $L_\infty$ norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension $d$. We also give a randomized online algorithm with competitive ratio $O(d^2)$ for Unit Clustering}of integer points (i.e., points in $\mathbb{Z}^d$, $d\in \mathbb{N}$, under $L_{\infty}$ norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least $2^d$. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

Endothelial Progenitor Cell Number and Colony-forming Capacity in Overweight and Obese Adults

OBJECTIVE: To investigate whether adiposity influences endothelial progenitor cell (EPC) number and colony-forming capacity.DESIGN: Cross-sectional study of normal weight, overweight and obese adult humans.PARTICIPANTS: Sixty-seven sedentary adults (aged 45-65 years): 25 normal weight (body mass index (BMI) or=30 kg/m(2); 18 males/6 females). All participants were non-smokers and free of overt cardiometabolic disease.MEASUREMENTS: Peripheral blood samples were collected and circulating EPC number was assessed by flow cytometry. Putative EPCs were defined as CD45(-)/CD34(+)/VEGFR-2(+)/CD133(+) or CD45(-)/CD34(+) cells. EPC colony-forming capacity was measured in vitro using a colony-forming unit (CFU) assay.RESULTS: Number of circulating putative EPCs (either CD45(-)/CD34(+)/VEGFR-2(+)/CD133(+) or CD45(-)/CD34(+) cells) was lower (P\u3c0.05) in obese (0.0007±0.0001%; 0.050±0.006%) compared with overweight (0.0016±0.0004%; 0.089±0.019%) and normal weight (0.0015±0.0003%; 0.082±0.008%) adults. There were no differences in EPC number between the overweight and normal weight groups. EPC colony-formation was significantly less in the obese (6±1) and overweight (4±1) compared with normal weight (9±2) adults.CONCLUSION: These results indicate that: (1) the number of circulating EPCs is lower in obese compared with overweight and normal weight adults; and (2) EPC colony-forming capacity is blunted in overweight and obese adults compared with normal weight adults. Impairments in EPC number and function may contribute to adiposity-related cardiovascular risk

Endothelial progenitor cells and integrins: adhesive needs

In the last decade there have been multiple studies concerning the contribution of endothelial progenitor cells (EPCs) to new vessel formation in different physiological and pathological settings. The process by which EPCs contribute to new vessel formation in adults is termed postnatal vasculogenesis and occurs via four inter-related steps. They must respond to chemoattractant signals and mobilize from the bone marrow to the peripheral blood; home in on sites of new vessel formation; invade and migrate at the same sites; and differentiate into mature endothelial cells (ECs) and/or regulate pre-existing ECs via paracrine or juxtacrine signals. During these four steps, EPCs interact with different physiological compartments, namely bone marrow, peripheral blood, blood vessels and homing tissues. The success of each step depends on the ability of EPCs to interact, adapt and respond to multiple molecular cues. The present review summarizes the interactions between integrins expressed by EPCs and their ligands: extracellular matrix components and cell surface proteins present at sites of postnatal vasculogenesis. The data summarized here indicate that integrins represent a major molecular determinant of EPC function, with different integrin subunits regulating different steps of EPC biology. Specifically, integrin α4β1 is a key regulator of EPC retention and/or mobilization from the bone marrow, while integrins α5β1, α6β1, αvβ3 and αvβ5 are major determinants of EPC homing, invasion, differentiation and paracrine factor production. β2 integrins are the major regulators of EPC transendothelial migration. The relevance of integrins in EPC biology is also demonstrated by many studies that use extracellular matrix-based scaffolds as a clinical tool to improve the vasculogenic functions of EPCs. We propose that targeted and tissue-specific manipulation of EPC integrin-mediated interactions may be crucial to further improve the usage of this cell population as a relevant clinical agent

Online unit covering in Euclidean space

We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in Rd, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in R d using Euclidean distance. The current best competitive ratio of an online algorithm, O(2 d d log d), is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio O(1.321 d ), thereby improving on the earlier record by an exponential factor. In particular, the competitive ratios are 5 for the plane and 12 for 3-space (the previous ratios were 7 and 21, respectively). For d = 3, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual). (II) We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in Rd under the L 2 norm is at least d + 1 for every d ≥ 1. This greatly improves upon the previous best lower bound, Ω(log d/ log log log d), due to Charikar et al. (2004). (III) We obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online Unit Covering in R 2 and respectively R 3 ; the previous best lower bounds were both 3. (IV) When the input points are taken from the square or hexagonal lattices in R 2 , we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of 3

Mobilization of stem and progenitor cells in cardiovascular diseases

Circulating bone marrow (BM)-derived stem and progenitor cells (SPCs) participate in turnover of vascular endothelium and myocardial repair after acute coronary syndromes. Acute myocardial infarction (MI) produces a generalized inflammatory reaction, including mobilization of SPCs, increased local production of chemoattractants in the ischemic myocardium, as well as neural and humoral signals activating the SPC egress from the BM. Several types of circulating BM cells were identified in the peripheral blood, including hematopoietic stem cells, endothelial progenitor cells, mesenchymal stromal cells, circulating angiogenic cells and pluripotent very small embryonic-like cells; however, the contribution of circulating cells to the myocardial and endothelial repair is still unknown. The number and function of these cells is impaired in patients with diabetes and other cardiovascular risk factors, but can be improved by physical exercise and use of statins. The mobilization of SPCs in acute coronary syndromes and stable coronary artery disease seems to predict the clinical outcomes in selected groups of patients. Interpretation of the findings has to incorporate other factors that modulate the process of mobilization, such as coexisting diseases, age and medications. This review discusses the mobilization of SPCs in acute ischemia (MI, stroke), as well as in stable cardiovascular disease, and highlights the possibility of using the SPC as a marker of cardiovascular risk