(Almost) Optimal Constructions of UOWHFs from 1-to-1, Regular One-way Functions and Beyond

We revisit the problem of black-box constructions of universal one-way hash functions (UOWHFs) from several (from specific to more general) classes of one-way functions (OWFs), and give respective constructions that either improve or generalize the best previously known. In addition, the parameters we achieve are either optimal or almost optimal simultaneously up to small factors, e.g., arbitrarily small $\omega(1)$. For any 1-to-1 one-way function, we give an optimal construction of UOWHFs with key and output length $\Theta(n)$ by making a single call to the underlying OWF. This improves the constructions of Naor and Yung (STOC 1989) and De Santis and Yung (Eurocrypt 1990) that need key length $O(n*\omega(log n))$. For any known-(almost-)regular one-way function with known hardness, we give an optimal construction of UOWHFs with key and output length $\Theta(n)$ and a single call to the one-way function. For any known-(almost-)regular one-way function, we give a construction of UOWHFs with key and output length $O(n*\omega(1))$ and by making $\omega(1)$ non-adaptive calls to the one-way function. This improves the construction of Barhum and Maurer (Latincrypt 2012) that requires key and output length $O(n*\omega(log n))$ and $\omega(log n)$ calls. For any weakly-regular one-way function introduced by Yu et al. at TCC 2015 (i.e., the set of inputs with maximal number of siblings is of an $n^{-c}$-fraction for some constant $c$), we give a construction of UOWHFs with key length $O(n*log n)$ and output length $\Theta(n)$. This generalizes the construction of Ames et al. (Asiacrypt 2012) which requires an unknown-regular one-way function (i.e., $c=0$). Along the way, we use several techniques that might be of independent interest. We show that almost 1-to-1 (except for a negligible fraction) one-way functions and known (almost-)regular one-way functions are equivalent in the known-hardness (or non-uniform) setting, by giving an optimal construction of the former from the latter. In addition, we show how to transform any one-way function that is far from regular (but only weakly regular on a noticeable fraction of domain) into an almost-regular one-way function

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Large Intragenic Deletion in DSTYK Underlies Autosomal-Recessive Complicated Spastic Paraparesis, SPG23

SPG23 is an autosomal-recessive neurodegenerative subtype of lower limb spastic paraparesis with additional diffuse skin and hair dyspigmentation at birth followed by further patchy pigment loss during childhood. Previously, genome-wide linkage in an Arab-Israeli pedigree mapped the gene to an approximately 25 cM locus on chromosome 1q24–q32. By using whole-exome sequencing in a further Palestinian-Jordanian SPG23 pedigree, we identified a complex homozygous 4-kb deletion/20-bp insertion in DSTYK (dual serine-threonine and tyrosine protein kinase) in all four affected family members. DSTYK is located within the established linkage region and we also found the same mutation in the previously reported pedigree and another Israeli pedigree (total of ten affected individuals from three different families). The mutation removes the last two exons and part of the 3′ UTR of DSTYK. Skin biopsies revealed reduced DSTYK protein levels along with focal loss of melanocytes. Ultrastructurally, swollen mitochondria and cytoplasmic vacuoles were also noted in remaining melanocytes and some keratinocytes and fibroblasts. Cultured keratinocytes and fibroblasts from an affected individual, as well as knockdown of Dstyk in mouse melanocytes, keratinocytes, and fibroblasts, were associated with increased cell death after ultraviolet irradiation. Keratinocytes from an affected individual showed loss of kinase activity upon stimulation with fibroblast growth factor. Previously, dominant mutations in DSTYK were implicated in congenital urological developmental disorders, but our study identifies different phenotypic consequences for a recurrent autosomal-recessive deletion mutation in revealing the genetic basis of SPG23.The Centre for Dermatology and Genetic Medicine is supported by a Wellcome Trust Strategic Award (reference 098439/Z/12/Z). The work was supported by the MRC (MR/M018512/1) and the UK National Institute for Health Research (NIHR) comprehensive Biomedical Research Centre (BRC) award to Guy’s and St. Thomas’ NHS Foundation Trust, in partnership with the King’s College London and King’s College Hospital NHS Foundation Trust. This study was also supported by UK Medical Research Council Project Grant (MR/M00046X/1) and Action Research grant SP3706 as well as medical student grants from the Jean Shanks Foundation and the British Association of Dermatologists

Disjoint path allocation with sublinear advice

We study the disjoint path allocation problem. In this setting, a path P of length L is given, and a sequence of subpaths of P arrives online, one in every time step. Each such path requests a permanent connection between its two end-vertices. An online algorithm can admit or reject such a request; in the former case, none of the involved edges can be part of any other connection. We investigate how much additional binary information (called “advice”) can help to obtain a good solution. It is known that, with roughly log2log2L advice bits, it can be guaranteed that a log2L-competitive solution is computed. In this paper, we prove the surprising result that, with L1−ε advice bits, it is not possible to obtain a solution with a competitive ratio better than (δlog2L)/2, where 0<δ<ε<1. This shows an interesting threshold behavior of the problem. A fairly good competitive ratio, namely log2L, can be obtained with very few advice bits. However, any increase of the advice does not help any further until an almost linear number of advice bits is supplied. Then again, it is also known that linear advice allows for optimality

A Technique to Obtain Hardness Results for Randomized Online Algorithms - A Survey

ISSN:0302-9743ISSN:1611-334

Online minimum spanning tree with advice

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size n, we show an asymptotically tight bound of \u398(n log n) on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., 5(n 12 1) advice bits allow to compute an optimal result if the weight function is the Euclidean distance; if the graph is complete, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal\u2019s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results from the advice complexity finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem

On Energy-Efficient Computations With Advice

ISSN:0302-9743ISSN:1611-334

Advice Complexity of the Online Search Problem

The online search problem is a fundamental problem in finance. The numerous direct applications include searching for optimal prices for commodity trading and trading foreign currencies. In this paper, we analyze the advice complexity of this problem. In particular, we are interested in identifying the minimum amount of information needed in order to achieve a certain competitive ratio. We design an algorithm that reads b bits of advice and achieves a competitive ratio of (M/m)^(1/(2^b+1)) where M and m are the maximum and minimum price in the input. We also give a matching lower bound. Furthermore, we compare the power of advice and randomization for this problem