Optimal Networks from Error Correcting Codes

To address growth challenges facing large Data Centers and supercomputing clusters a new construction is presented for scalable, high throughput, low latency networks. The resulting networks require 1.5-5 times fewer switches, 2-6 times fewer cables, have 1.2-2 times lower latency and correspondingly lower congestion and packet losses than the best present or proposed networks providing the same number of ports at the same total bisection. These advantage ratios increase with network size. The key new ingredient is the exact equivalence discovered between the problem of maximizing network bisection for large classes of practically interesting Cayley graphs and the problem of maximizing codeword distance for linear error correcting codes. Resulting translation recipe converts existent optimal error correcting codes into optimal throughput networks.Comment: 14 pages, accepted at ANCS 2013 conferenc

Quantized Indexing: Beyond Arithmetic Coding

Quantized Indexing is a fast and space-efficient form of enumerative (combinatorial) coding, the strongest among asymptotically optimal universal entropy coding algorithms. The present advance in enumerative coding is similar to that made by arithmetic coding with respect to its unlimited precision predecessor, Elias coding. The arithmetic precision, execution time, table sizes and coding delay are all reduced by a factor O(n) at a redundancy below 2*log(e)/2^g bits/symbol (for n input symbols and g-bit QI precision). Due to its tighter enumeration, QI output redundancy is below that of arithmetic coding (which can be derived as a lower accuracy approximation of QI). The relative compression gain vanishes in large n and in high entropy limits and increases for shorter outputs and for less predictable data. QI is significantly faster than the fastest arithmetic coders, from factor 6 in high entropy limit to over 100 in low entropy limit (`typically' 10-20 times faster). These speedups are result of using only 3 adds, 1 shift and 2 array lookups (all in 32 bit precision) per less probable symbol and no coding operations for the most probable symbol . Further, the exact enumeration algorithm is sharpened and its lattice walks formulation is generalized. A new numeric type with a broader applicability, sliding window integer, is introduced.Comment: Submitted to DCC-200

Cabbage and fermented vegetables : From death rate heterogeneity in countries to candidates for mitigation strategies of severe COVID-19

Large differences in COVID-19 death rates exist between countries and between regions of the same country. Some very low death rate countries such as Eastern Asia, Central Europe, or the Balkans have a common feature of eating large quantities of fermented foods. Although biases exist when examining ecological studies, fermented vegetables or cabbage have been associated with low death rates in European countries. SARS-CoV-2 binds to its receptor, the angiotensin-converting enzyme 2 (ACE2). As a result of SARS-CoV-2 binding, ACE2 downregulation enhances the angiotensin II receptor type 1 (AT(1)R) axis associated with oxidative stress. This leads to insulin resistance as well as lung and endothelial damage, two severe outcomes of COVID-19. The nuclear factor (erythroid-derived 2)-like 2 (Nrf2) is the most potent antioxidant in humans and can block in particular the AT(1)R axis. Cabbage contains precursors of sulforaphane, the most active natural activator of Nrf2. Fermented vegetables contain many lactobacilli, which are also potent Nrf2 activators. Three examples are: kimchi in Korea, westernized foods, and the slum paradox. It is proposed that fermented cabbage is a proof-of-concept of dietary manipulations that may enhance Nrf2-associated antioxidant effects, helpful in mitigating COVID-19 severity.Peer reviewe

Nrf2-interacting nutrients and COVID-19 : time for research to develop adaptation strategies

There are large between- and within-country variations in COVID-19 death rates. Some very low death rate settings such as Eastern Asia, Central Europe, the Balkans and Africa have a common feature of eating large quantities of fermented foods whose intake is associated with the activation of the Nrf2 (Nuclear factor (erythroid-derived 2)-like 2) anti-oxidant transcription factor. There are many Nrf2-interacting nutrients (berberine, curcumin, epigallocatechin gallate, genistein, quercetin, resveratrol, sulforaphane) that all act similarly to reduce insulin resistance, endothelial damage, lung injury and cytokine storm. They also act on the same mechanisms (mTOR: Mammalian target of rapamycin, PPAR gamma:Peroxisome proliferator-activated receptor, NF kappa B: Nuclear factor kappa B, ERK: Extracellular signal-regulated kinases and eIF2 alpha:Elongation initiation factor 2 alpha). They may as a result be important in mitigating the severity of COVID-19, acting through the endoplasmic reticulum stress or ACE-Angiotensin-II-AT(1)R axis (AT(1)R) pathway. Many Nrf2-interacting nutrients are also interacting with TRPA1 and/or TRPV1. Interestingly, geographical areas with very low COVID-19 mortality are those with the lowest prevalence of obesity (Sub-Saharan Africa and Asia). It is tempting to propose that Nrf2-interacting foods and nutrients can re-balance insulin resistance and have a significant effect on COVID-19 severity. It is therefore possible that the intake of these foods may restore an optimal natural balance for the Nrf2 pathway and may be of interest in the mitigation of COVID-19 severity

FAST, OPTIMAL ENTROPY CODER *

Quantized Indexing ‡ (QI) is a fast and space-efficient form of enumerative coding 1, the most “desirable ” 12,30 among asymptotically optimal universal source coding algorithms. The present advance in enumerative coding is similar to that made by arithmetic coding with respect to its unlimited precision predecessor, Elias coding. The arithmetic precision, execution time, table sizes and coding delay are all reduced by a factor ≈ n/g at a redundancy below log(e)/2 g-1 bits/symbol (for n input symbols and precision g ~ log(n)). Due to its tighter enumeration, QI output redundancy is below that of arithmetic coding (which can be derived as a reduced accuracy approximation of QI). The relative compression gain vanishes in large n and in high entropy limits and increases for shorter outputs and for less predictable data. QI is significantly faster than the fastest arithmetic coders, from factor 6 in high entropy limit to over 200 in low entropy limit (‘typically ’ 10-15 times faster). These speedups are result of using only 3 adds, 1 shift and 2 simple array lookups (all in 32 bit precision; α=2) per less probable symbol and no coding operations for the most probable symbol §. Further, the exact enumeration algorithm is sharpened and its lattice walks formulation is generalized. A new numeric type with a broader applicability, sliding window integer, is introduced

Quantized Indexing: Beyond Arithmetic Coding

Quantized Indexing is a fast and space-efficient form of enumerative coding,\ud the most â\u80\u9cdesirableâ\u80\u9d among asymptotically optimal universal source coding\ud algorithms. The present advance in enumerative coding is similar to that made by\ud arithmetic coding with respect to its unlimited precision predecessor, Elias\ud coding. The arithmetic precision, execution time, table sizes and coding delay\ud are all reduced by a factor O(n) at a redundancy below log(e)/2^(g-1) bits/symbol (for\ud n input symbols and a g-bit QI precision). Due to its tighter enumeration, the QI\ud redundancy is below that of arithmetic coding (which can be derived as a lower\ud accuracy approximation of QI). The relative compression gain vanishes in large n\ud and in high entropy limits and increases for shorter outputs and for less\ud predictable data. QI is significantly faster than the fastest arithmetic coders22,\ud from factor 6 in high entropy to over 100 in low entropy limit (â\u80\u98typicallyâ\u80\u99 10-20\ud times faster). These speedups are result of using only 3 adds, 1 shift and 2 array\ud lookups (all in 32 bit precision; α=2) per less probable symbol and no coding\ud operations for the most probable symbol. Further, the exact enumeration\ud algorithm is sharpened and its lattice walks formulation is generalized. A new\ud numeric type with a broader applicability, sliding window integer, is introduced

Quantized Indexing: Background Information *

This report presents background material ‡ for the Quantized Indexing § (QI) form of enumerative coding. Following the introduction to conventional enumerative coding and its reformulation as lattice walks, the relations between arithmetic and enumerative coding are explored. In addition to examining the fundamental origins of the performance differences (in speed and output size), special emphasis was on the distinctions in the two approaches to modeling. General modeling pattern for enumerative coding (including QI) is described, along with the special cases for finite order Markov sources. This approach is compared to the arithmetic coder adaptive modeling as well as to the grammer and dictionary based modeling methods