Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Koml\'os problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in $T$ compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of $O(\sqrt{\log d+ \log\!\log T})$ with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. We also introduce a generalization of the prefix discrepancy problem where the discrepancy constraints correspond to paths on a DAG on $T$ vertices. We show that an analog of Banaszczyk's $O(\sqrt{\log d+ \log T})$ bound continues to hold in this setting for adversarially given unit vectors and that the $\sqrt{\log T}$ factor is unavoidable for DAGs. We also show that the dependence on $T$ cannot be improved significantly in the smoothed case for DAGs. We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.Comment: 22 pages. Appear in ITCS 202

Online discrepancy minimization for stochastic arrivals

In the stochastic online vector balancing problem, vectors v1, v2,..., vT chosen independently from an arbitrary distribution in Rn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where kvtk2 ≤ 1 for each t, our algorithm achieves Oe(1) discrepancy with high probability, improving upon the previous Oe(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(logd+4 T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Oe(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multicolor discrepancy

Online discrepancy minimization for stochastic arrivals

In the stochastic online vector balancing problem, vectors v1,v2,…,vT chosen independently from an arbitrary distribution in Rn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlos problem where ∥v_t∥_2≤1 for each t, our algorithm achieves ˜O(1) discrepancy with high probability, improving upon the previous ˜O(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(log^{d+4}T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log^{2d+1}T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves \tilde{O}(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multi-color discrepancy.</p

Casale's tube with VQZ stoma: An alternative to “double Monti”

In situations requiring an ileal segment for performing a Mitrofanoff cathetrisable urinary diversion, occasionally a “Double Monti” is needed to achieve a length for the cathetrisable channel to conveniently reach the abdomen of an adult. Casale's tube is an alternative where it can provide a jointless tube with adequate length. The video demonstrates the procedure in an adult with a neurogenic acontractile bladder who had developed a panurethral stricture because of years of self catheterization. “VQZ” technique of stoma formation is helpful in prevention of stenosis

Role of tranexamic acid on blood loss in laparoscopic cholecystectomy

Context: Nonsurgical uses of tranexamic acid include the management of bleeding associated with leukemia, ocular bleeding, recurrent hemoptysis, menorrhagia, hereditary angioneurotic edema, and numerous other medical problems. However, there is hardly any documentation of the use of tranexamic acid in laparoscopic cholecystectomy. Aims: This study was conducted to evaluate the role of tranexamic acid in limiting blood loss in laparoscopic cholecystectomy and to evaluate the effect of blood loss on morbidity in terms of hospital stay and mortality of the patient. Subjects and Methods: The study was conducted on sixty patients admitted with gallstones, candidates for laparoscopic cholecystectomy. Thirty patients received an intravenous 20 mg/kg bolus dose of tranexamic acid at induction of anesthesia (Group A), and another thirty did not receive the aforementioned drug at induction (Group B). Statistical Analysis: The two groups were compared, and the data collected were entered and tabulated using Microsoft Office Excel and analyzed using appropriate statistical tests. Results: The mean postoperative hospital stay (2.4 vs. 2.63, P = 0.4147), drain fluid hemoglobin (Hb) (0.83 vs. 0.90, P = 0.2087), drain fluid hematocrit (0.2434 vs. 0.2627, P = 0.3787), mean drain output (85 vs. 87.23, P = 0.9271), mean pulse rate at the start of surgery (74.2 vs. 75, P > 0.999), mean pulse rate 24 h after surgery (75.9 vs. 76.4, P = 0.5775), and mean change in Hb (0.240 vs. 0.266, P = 0.2502) in both the groups were not significant. Conclusions: There is no active role of tranexamic acid in elective laparoscopic cholecystectomy