Space-Efficient Approximation Scheme for Maximum Matching in Sparse Graphs

We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, using only logarithmic space. This deviates from the well known Baker\u27s approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any "recursively sparse" graph class which contains a linear size matching and also for certain other classes like bounded genus graphs. The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker\u27s method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. The bounded degree assumption allows us to obtain a Logspace algorithm

Mechanomodulation of Lymphatic Vessel Contractility Using Oscillatory Pressure Waveforms

The lymphatic system is a network of vessels and nodes transporting and clearing interstitial fluid, orchestrating the immune response, and facilitating lipid transport. An important component of the lymphatic system are the collecting lymphatic vessels which pump lymph through the body by virtue of their intrinsic contractility. The collecting lymphatic vessels are known to be sensitive to their mechanical microenvironment which dictates their contractility. However, relatively little is known about how collecting lymphatic vessel contractility is modulated by their oscillatory mechanical microenvironment and how this mechanosensitivity is affected by lymphatic injury. It is important to know the limits of the mechanomodulation of lymphatic vessels in both physiological and pathological circumstances, since an aberrant mechanical microenvironment is frequently associated with lymphatic dysfunction, such as in the case of lymphedema. The present work investigates the role of the oscillatory mechanical microenvironment in the modulation of collecting lymphatic contractility. The mechanomodulation of isolated collecting lymphatic vessels by oscillatory shear stress was investigated and optimal parameters of stimulation were identified for maximizing lymphatic function. The modulation of lymphatic vessels was also investigated in vivo in response to oscillatory pressure gradients mimicking pressure waveforms during massage. Massage-like pressure waveforms modulated collecting lymphatic vessel contractility, and this modulation was altered by lymphatic injury. Thus, the oscillatory microenvironment is shown to be an important regulator of lymphatic contractility and the present work provides clues on how the mechanomodulation of lymphatics can be harnessed to understand and improve therapeutic approaches to lymphedema.Ph.D

Approximating Edit Distance in the Fully Dynamic Model

The edit distance is a fundamental measure of sequence similarity, defined as the minimum number of character insertions, deletions, and substitutions needed to transform one string into the other. Given two strings of length at most $n$, simple dynamic programming computes their edit distance exactly in $O(n^2)$ time, which is also the best possible (up to subpolynomial factors) assuming the Strong Exponential Time Hypothesis (SETH). The last few decades have seen tremendous progress in edit distance approximation, where the runtime has been brought down to subquadratic, near-linear, and even sublinear at the cost of approximation. In this paper, we study the dynamic edit distance problem, where the strings change dynamically as the characters are substituted, inserted, or deleted over time. Each change may happen at any location of either of the two strings. The goal is to maintain the (exact or approximate) edit distance of such dynamic strings while minimizing the update time. The exact edit distance can be maintained in $\tilde{O}(n)$ time per update (Charalampopoulos, Kociumaka, Mozes; 2020), which is again tight assuming SETH. Unfortunately, even with the unprecedented progress in edit distance approximation in the static setting, strikingly little is known regarding dynamic edit distance approximation. Utilizing the off-the-shelf tools, it is possible to achieve an $O(n^{c})$-approximation in $n^{0.5-c+o(1)}$ update time for any constant $c\in [0,\frac16]$. Improving upon this trade-off remains open. The contribution of this work is a dynamic $n^{o(1)}$-approximation algorithm with amortized expected update time of $n^{o(1)}$. In other words, we bring the approximation-ratio and update-time product down to $n^{o(1)}$. Our solution utilizes an elegant framework of precision sampling tree for edit distance approximation (Andoni, Krauthgamer, Onak; 2010).Comment: Accepted to FOCS 202