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We give faster algorithms for producing sparse approximations of the transition matrices of $k$-step random walks on undirected, weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with $n$ vertices and $m$ edges, our algorithm produces a graph with about $n\log{n}$ edges that approximates the $k$-step random walk graph in about $m + n \log^4{n}$ time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices.

We consider the problem of commutative rank computation of a given matrix space. A matrix space is a (linear) subspace of the (linear) space of n x n matrices over a given field. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient deterministic computation of the commutative rank is a major open problem, although there is a simple and efficient randomized algorithm for it. Recently, there has been a series of results on computing the non-commutative rank of matrix spaces in deterministic polynomial time. Since the non-commutative rank of any matrix space is at most twice the commutative rank, one immediately gets a deterministic 1/2-approximation algorithm for the computation of the commutative rank. This leads to a natural question of whether this approximation ratio can be improved. In this paper, we answer this question affirmatively. We present a deterministic Polynomial-time approximation scheme (PTAS) for computing the commutative rank of a given matrix space B. More specifically, given a matrix space and a rational number e > 0, we give an algorithm, that runs in time O(n^(4 + 3/e)) and computes a matrix A in the given matrix space B such that the rank of A is at least (1-e) times the commutative rank of B. The algorithm is the natural greedy algorithm. It always takes the first set of k matrices that will increase the rank of the matrix constructed so far until it does not find any improvement, where the size of the set k depends on e.

For "large" class $\mathcal{C}$ of continuous probability density functions (p.d.f.), we demonstrate that for every $w\in\mathcal{C}$ there is mixture of discrete Binomial distributions (MDBD) with $T\geq N\sqrt{\phi_{w}/\delta}$ distinct Binomial distributions $B(\cdot,N)$ that $\delta$-approximates a \emph{discretized} p.d.f. $\hat{w}(i/N)\triangleq w(i/N)/[\sum_{\ell=0}^{N}w(\ell/N)]$ for all $i\in[3:N-3]$, where $\phi_{w}\geq\max_{x\in[0,1]}|w(x)|$. Also, we give two efficient parallel algorithms to find such MDBD. Moreover, we propose a sequential algorithm that on input MDBD with $N=2^k$ for $k\in\mathbb{N}_{+}$ that induces a discretized p.d.f. $\beta$, $B=D-M$ that is either Laplacian or SDDM matrix and parameter $\epsilon\in(0,1)$, outputs in $\hat{O}(\epsilon^{-2}m + \epsilon^{-4}nT)$ time a spectral sparsifier $D-\hat{M}_{N} \approx_{\epsilon} D-D\sum_{i=0}^{N}\beta_{i}(D^{-1} M)^i$ of a matrix-polynomial, where $\hat{O}(\cdot)$ notation hides $\mathrm{poly}(\log n,\log N)$ factors. This improves the Cheng et al.'s\cite{CCLPT15} algorithm whose run time is $\hat{O}(\epsilon^{-2} m N^2 + NT)$. Furthermore, our algorithm is parallelizable and runs in work $\hat{O}(\epsilon^{-2}m + \epsilon^{-4}nT)$ and depth $O(\log N\cdot\mathrm{poly}(\log n)+\log T)$. Our main algorithmic contribution is to propose the first efficient parallel algorithm that on input continuous p.d.f. $w\in\mathcal{C}$, matrix $B=D-M$ as above, outputs a spectral sparsifier of matrix-polynomial whose coefficients approximate component-wise the discretized p.d.f. $\hat{w}$. Our results yield the first efficient and parallel algorithm that runs in nearly linear work and poly-logarithmic depth and analyzes the long term behaviour of Markov chains in non-trivial settings. In addition, we strengthen the Spielman and Peng's\cite{PS14} parallel SDD solver.

Impedance plethysmograms were recorded from thoracic region in 254 normal subjects, 183 patients with coronary artery disease, 391 patients with valvular heart disease and 107 patients with congenital septal disorder. The data in 18 normal subjects and 55 patients showed that basal impedance decreases markedly during exercise in patients with ischaemic heart disease. Estimation of cardiac index by this technique in a group of 99 normal subjects has been observed to be more consistent than that of the stroke volume. Estimation of systolic time index from impedance plethysmograms in 34 normal subjects has been shown to be as reliable as that from electrocardiogram, phonocardiogram and carotid pulse tracing. Changes in the shape of plethysmographic waveform produced by valvular and congenital heart diseases are briefly described and the role of this technique in screening cardiac patients has been highlighted.

Vector impedance cardiograms in horizontal lead configuration [VICG(H)] were recorded in 34 normal subjects, 18 patients with mitral stenosis, 9 patients with mitral regurgitation, 14 patients with aortic regurgitation and 13 patients with non- cyanotic septal disorders. Data in normal subjects revealed that most of the phase reversal points in VICG(H) waveform did not coincide with those of conventional impedance cardiogram. The shape of VICG(H) waveform and values of VICG indices were observed to be markedly affected in patients having significant alteration in the pulmonary circulation and the changes observed were specific of the type of the disorder. The results of the study suggested that right side of the heart had dominant contribution in generation of VICG(H) waveform.

Impedance plethysmographic observations have been compared with arteriographic findings in 216 patients with peripheral arterial occlusive diseases. Impedance plethysmographic diagnosis in these patients was obtained by Parulkar′s method without apriori knowledge of arteriographic diagnosis. But for minor discrepancy in the anatomical location of the block in few patients, impedance plethysmographic observations correlated very well with arteriographic findings. Impedance plethysmographic diagnosis was found to be correct in 312, wrong in 53, false negative in 8 limbs respectively, yielding a sensitivity of 97.5% and specificity of 98.1% of this technique.

Forty patients with symptoms of neuro-vascular compression in the upper extremities were subjected to impedance plethysmographic study using Parulkar′s method. Two patients recorded decreased blood flow (BFI) in supine position and were diagnosed as having partial occlusion at subclavian level. Sixteen of the patients recorded decreased BFI on 90 degrees abduction and hyper-abduction. Twelve of these patients had radiological evidence of anomalous cervicle ribs. In remaining four patients extrinsic impression on the subclavian artery due to fibrous deposits was confirmed by arteriography. Remaining 22 patients recorded normal impedance plethysmograms. Impedance plethysmography thus provided a non-invasive modality for confirmation of vascular compression in thoracic outlet syndrome.

A review of investigations into the origin of impedance plethysmographic waveform is presented in this article. Attempts made by several investigators in the comparison of impedance plethysmographic estimations of peripheral blood flow with that obtained by standard methods are briefly described. Investigations indicating the negligible contribution from contact impedance at body electrode interface are highlighted. Temporal correlation of impedance plethysmographic waveform recorded from thorax with various important events of the cardiac cycle is summarised and various hypotheses on the genesis of this waveform are presented.

Impedance Plethysmography technique has been discussed with explanation of two compartment model and parallel conductor theory for the estimation of peripheral blood flow and stroke volume. Various methods for signal enhancement to facilitate computation of blood flow are briefly described. Source of error in the estimation of peripheral blood flow is identified and the correction has been suggested.

This paper describes the basic methods for measurement of body impedance, electrodes and their configuration, and the measuring instrument with its limitations. A microcomputer assisted impedance plethysmograph system, developed at BARC and different lead configurations for impedance plethysmographic investigation are also described. Typical impedance plethysmographic waveforms recorded from a normal subject and measurement of their amplitude and various time intervals are illustrated.

Seven patients with history of radiation burn were subjected to impedance plethysmographic investigation at Non-invasive Vascular Laboratory, K. E. M. Hospital. Impedance plethysmograms were recorded from various locations in both the upper extremities in supine and digits of both the hands in supine as well as on hyper-abduction in sitting position. Control values of blood flow index (BFI) and differential pulse arrival time (DPAT) were derived from similar data in 15 normal subjects. It was observed that digits having thinning of epidermis of skin recorded marked decrease in values of BFI and significant change in value of DPAT and these observations correlated well with thermography and vascular scintigraphy. Digits recording significant decrease in BFI, which were clinically normal, were found to have changes in the skin during follow up examination. Thus IPG provided a sensitive modality for detecting ischaemia in early stages in patients with history of radiation injury.

Impedance plethysmographic observations have been correlated with aortographic observations in 57 patients suspected of aortic occlusive diseases. Aortic occlusions have been characterised by marked decrease in blood flow index and significant increase in differential pulse arrival time at thigh level bilaterally. Atherosclerotic affection of the aorta has been featured by a bilateral decrease in the value of blood flow index as well as differential pulse arrival time at thigh level. Leriche′s syndrome, however, has been found to decrease the blood flow index moderately at thigh in both the legs without any significant change in differential pulse arrival time. Aortography in all the patients has confirmed the diagnosis made by impedance plethysmography.