Succinct Representations of Permutations and Functions

We investigate the problem of succinctly representing an arbitrary permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for any i and any (positive or negative) integer power k. A representation taking (1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant \epsilon <= 1. A representation taking the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time. We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed quickly for any i and any integer power k. We give a representation that takes (1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and computes arbitrary positive powers in constant time. It can also be used to compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time. We place emphasis on the redundancy, or the space beyond the information-theoretic lower bound that the data structure uses in order to support operations efficiently. A number of lower bounds have recently been shown on the redundancy of data structures. These lower bounds confirm the space-time optimality of some of our solutions. Furthermore, the redundancy of one of our structures "surpasses" a recent lower bound by Golynski [Golynski, SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the Proceedings of ICALP 2003 and 2004. However, all results in this version are improved over the earlier conference versio

Succinct Indexable Dictionaries with Applications to Encoding kk-ary Trees, Prefix Sums and Multisets

We consider the {\it indexable dictionary} problem, which consists of storing a set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting the operations of \Rank(x), which returns the number of elements in $S$ that are less than $x$ if $x \in S$, and -1 otherwise; and \Select(i) which returns the $i$-th smallest element in $S$. We give a data structure that supports both operations in O(1) time on the RAM model and requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits to store a set of size $n$, where {\cal B}(n,m) = \ceil{\lg {m \choose n}} is the minimum number of bits required to store any $n$-element subset from a universe of size $m$. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the $O(\lg \lg m)$ additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: An information-theoretically optimal representation of a $k$-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\cal B}(n,m+n) + o(n)$ bits that supports (appropriate generalizations of) \Rank and \Select operations in constant time, and A representation of a sequence of $n$ non-negative integers summing up to $m$ in ${\cal B}(n,m+n) + o(n)$ bits that supports prefix sum queries in constant time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report 2002/1

Organophosphorous intoxication and hyperthyroidism

It is well established that transient thyroid dysfunction can occur in many non-thyroid illness. But abnormal thyroid function test in the range of hyperthyroidism is not reported many a times. Here we present two cases of chlorpyrifos intoxication presenting with hyperthyroidism and its implications in the management of the patient in acute illness.Role of atropine in antagonizing muscarinic effects of organophosphorus intoxication is well documented. Interplay between atropine, hyperthyroid state and tachycardia discussed

A mechanical engineer cannot open his fist - myotonia congenita

Myotonia congenita is a rare congenital neurological disorder, not encountered by many physicians in their clinical practice. Two types are described Thomsen and Becker. The former autosomal dominant, presents at early age and less aggressive as in our patient. Backer is autosomal recessive more aggressive and with muscle weakness. In these disorders group of skeletal muscles contract longer and slow to relax due to mutation in CLCN1 gene leading to chloride channelopathy. They present with difficulty to initiate an action but can do it normally after repeated attempts. Hand and facial muscles are involved in Thomsen disease; they are not weak or atrophic but might have cramps. Diagnosis of Myotonia congenita is clinical and confirmed by electromyography. Many other clinical conditions have myotonia, dystrophic and non-dystrophic. Physiotherapy and membrane stabilizing drugs like anti-epileptics are of choice but with varying success. Genetic counseling is an important part of management of these cases

Diffraction of light by a transparent lamina

The present Paper embodies an attempt to consider the problem of diffraction of light by a very thin plate of transparent material, bounded by a straight edge, with greater exactness than is attained in the usual elementary treatment on the Fresnel-Huygen principle. The method adopted, though not completely rigorous, bases itself on the electromagnetic theory of light, and seeks to express the disturbance in the field in the form of functions which are solutions of the equations of wave-propagation. The formulae obtained indicate that the light diffracted by the edge should exhibit colour and polarization effects varying in a remarkable manner with the thickness of the plate and the direction of observation. Effects having the general character of those indicated by the theory have actually been observed in experiment. The theory, however, requires modification in the case of thicker laminae, where further complications arise which are not here taken account of