Selection from read-only memory with limited workspace

Given an unordered array of $N$ elements drawn from a totally ordered set and an integer $k$ in the range from $1$ to $N$, in the classic selection problem the task is to find the $k$-th smallest element in the array. We study the complexity of this problem in the space-restricted random-access model: The input array is stored on read-only memory, and the algorithm has access to a limited amount of workspace. We prove that the linear-time prune-and-search algorithm---presented in most textbooks on algorithms---can be modified to use $\Theta(N)$ bits instead of $\Theta(N)$ words of extra space. Prior to our work, the best known algorithm by Frederickson could perform the task with $\Theta(N)$ bits of extra space in $O(N \lg^{*} N)$ time. Our result separates the space-restricted random-access model and the multi-pass streaming model, since we can surpass the $\Omega(N \lg^{*} N)$ lower bound known for the latter model. We also generalize our algorithm for the case when the size of the workspace is $\Theta(S)$ bits, where $\lg^3{N} \leq S \leq N$. The running time of our generalized algorithm is $O(N \lg^{*}(N/S) + N (\lg N) / \lg{} S)$, slightly improving over the $O(N \lg^{*}(N (\lg N)/S) + N (\lg N) / \lg{} S)$ bound of Frederickson's algorithm. To obtain the improvements mentioned above, we developed a new data structure, called the wavelet stack, that we use for repeated pruning. We expect the wavelet stack to be a useful tool in other applications as well.Comment: 16 pages, 1 figure, Preliminary version appeared in COCOON-201

Ergot disease of sorghum

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Relations among sorghum ergot strains from the United States, Mexico, Puerto Rico, Bolivia, India and Australia

Sorghum ergot, initially restricted to Asia and Africa, was recently found in the Americas and Australia. Three species causing the disease have been reported: Claviceps sorghi in India, C. sorghicola in Japan, and C. africana in all ergot-positive countries. The objective of our study was to study the intraspecific variation in C. africana isolates in the Americas, Africa, India, and Australia. We confirmed C. africana, C. sorghi, and C. sorghicola as different species using differences in nucleotide sequences of internal transcribed spacer 1 and 5.8S rDNA regions. Sequences of this region obtained from the representative American, Indian, and Australian isolates of C. africana were identical. In addition, random amplified polymorphic DNA (RAPD) banding patterns of sorghum ergot pathogen isolates from the United States, Mexico, Puerto Rico, Bolivia, Australia, and India were evaluated with nearly 100 primers. A total of 65 primers gave identical patterns for all isolates, which confirmed that all were C. africana. The identity of RAPD pattern and rDNA sequence of Indian isolates with those of C. africana confirmed that the species is now present in India. Only 20 primers gave small pattern differences and 7 of them were used for routine testing. All of the American isolates were identical and three isolates of the same type were also found in South Africa, suggesting Africa as the origin of the invasion clone in the Americas. Australian and Indian isolates were distinguishable by a single band difference; therefore, migration from the Asian region to Australia is suspected. Another distinct group was found in Africa. Cluster analysis of the informative bands revealed that the American and African group are on the same moderately (69%) supported clade. Isolates from Australia and India belonged to another clade

Improved Bounds for Finger Search on a RAM

Abstract. We present a new finger search tree with O(1) worst-case update time and O(log log d) expected search time with high probability in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. For the need of the analysis we model the updates by a “balls and bins ” combinatorial game that is interesting in its own right as it involves insertions and deletions of balls according to an unknown distribution.