On a compaction theorem of ragde

Ragde demonstrated that in constant time a PRAM with $n$ processors can move at most $k$ items, stored in distinct cells of an array of size $n$, to distinct cells in an array of size at most $k^4$. We show that the exponent of 4 in the preceding sentence can be replaced by any constant greater than~2

Parallel Merging and Sorting on Linked List

We study linked list sorting and merging on the PRAM model. In this paper we show that n real numbers can be sorted into a linked list in constant time with n2+e processors or in ) time with n2 processors. We also show that two sorted linked lists of n integers in {0, 1, …, m}  can be merged into one sorted linked list in O(log(c)n(loglogm)1/2) time using n/(log(c)n(loglogm)1/2)  processors, where c is an arbitrarily large constant

A lower bound for linear approximate compaction

The {\em $\lambda$-approximate compaction} problem is: given an input array of $n$ values, each either 0 or 1, place each value in an output array so that all the 1's are in the first $(1+\lambda)k$ array locations, where $k$ is the number of 1's in the input. $\lambda$ is an accuracy parameter. This problem is of fundamental importance in parallel computation because of its applications to processor allocation and approximate counting. When $\lambda$ is a constant, the problem is called {\em Linear Approximate Compaction} (LAC). On the CRCW PRAM model, %there is an algorithm that solves approximate compaction in \order{(\log\log n)^3} time for $\lambda = \frac{1}{\log\log n}$, using $\frac{n}{(\log\log n)^3}$ processors. Our main result shows that this is close to the best possible. Specifically, we prove that LAC requires %$\Omega(\log\log n)$ time using \order{n} processors. We also give a tradeoff between $\lambda$ and the processing time. For $\epsilon < 1$, and $\lambda = n^{\epsilon}$, the time required is $\Omega(\log \frac{1}{\epsilon})$

Sorting Real Numbers in Constant Time Using n^2/log^cn Processors

We study the sorting of real numbers into a linked list on the PRAM (Parallel Random-Access Machine) model. We show that n real numbers can be sorted into a linked list in constant time using n2 processors. Previously n numbers can be sorted into a linked list using n2 processors in O(loglogn) time. We also study the time processor trade-off for sorting real numbers into a linked list on the PRAM (Parallel Random Access Machine) model. We show that n real numbers can be sorted into a linked list with n2/t processors in O(logt) time. Previously n real numbers can be sorted into a linked list using n3 processors in constant time and n2 processors in O(loglogn). And then we show that input array of n real numbers  can be sorted into linked list in constant time using n2/logcn  processors for any positive constant c. We believe that further reduction on the number of processors for sorting real numbers in constant time will be very difficult if not impossible

Sorting Real Numbers into a Linked List on the PRAM Model

Title from PDF of title page, viewed June 2, 2022Thesis advisor: Yijie HanVitaIncludes bibliographical references (pages 10-11)Thesis (M.S.)--School of Computing and Engineering. University of Missouri--Kansas City, 2022We study the sorting of real numbers into a linked list on the PRAM (Parallel Random Access Machine) model. The research work consists of two parts. First part talks about the various techniques involved in sorting the real numbers on the linked list in terms of number of processors and time complexity. We have examined on how to sort the real numbers in the linked list using n^3, n^2 processors which has the time complexity of constant time and O(loglogn) time respectively. We have done good research in that area to come up with an algorithm to sort n real numbers into the linked list using n^2 processors in constant time. In second part, we talk about the time processor trade off for sorting the real numbers in the linked list.Introduction -- Methodology -- Theorem -- Conclusion

An Arbitrary CRCW PRAM Algorithm for Sorting Integers Into a LinkedList and Chaining on a Trie

Title from PDF of title page viewed June 1, 2020Thesis advisor: Yijie HanVitaIncludes bibliographical references (pages 22-23)Thesis (M.S.)--School of Computing and Engineering. University of Missouri--Kansas City, 2020The research work comprises of two parts. Part one is using an Arbitrary CRCW PRAM algorithm for sorting integers into a linked list. There are various algorithms and techniques to sort the integers in LinkedList. Arbitrary CRCW PRAM model, being the weakest model is able to sort n integers in a LinkedList in “constant time” using nlogm processors and if we use nt processors, then it can be sorted in O(loglogm/logt) time by converting Arbitrary CRCW PRAM model to Priority CRCW PRAM model. Part two is Chaining on a Trie. This research paper solves the problem of chaining on a Trie by providing more efficient complexity. This Algorithm takes “constant time” using n(logm+1) processors to chain the nodes on a Trie for n input integers on the Arbitrary CRCW PRAM model.Introduction -- Sort integers into a linked list -- Chaining on a Trie --Conclusio

Fast deterministic processor allocation

Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given $n$ nonnegative integers $x_1,\ldots,x_n$, to allocate $n$ nonoverlapping subarrays of sizes $x_1,\ldots,x_n$ from within a base array of $O(\sum_{j=1}^n x_j)$ cells. We show that interval allocation problems of size $n$ can be solved in $O((\log\log n)^3)$ time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of $\Theta({{\log n}/{\log\log n}})$. We also describe an application to the problem of computing the connected components of an undirected graph

Constant Time Sorting and Searching

Title from PDF of title page, viewed January 4, 2023Thesis advisor: Yijie HanVitaIncludes bibliographical references (pages 25-28)Thesis (M.S.)--Department of Computer Science and Electrical Engineering. University of Missouri--Kansas City, 2022To study the sorting of real numbers into a linked list on Parallel Random Access Machine model. To show that input array of n real numbers can be sorted into a linked list in constant time using n²/logᶜn processors for any positive constant c. The searching problem studied is locating the interval of n sorted real numbers for inserting a query real number. Taking into account an input of n real numbers and organize them in the sorted order to facilitate searching. Initially, sorting the n input real numbers and then convert these real numbers into integers such that their relative order is preserved. Convert the query input real number to a query integer and then search the interval among these n integers for the insertion point of this query real number in constant time.Introduction -- Sorting in constant time -- Searching in constant time -- Theorem -- Conclusion

Optimal parallel string algorithms: sorting, merching and computing the minimum

We study fundamental comparison problems on strings of characters, equipped with the usual lexicographical ordering. For each problem studied, we give a parallel algorithm that is optimal with respect to at least one criterion for which no optimal algorithm was previously known. Specifically, our main results are: % \begin{itemize} \item Two sorted sequences of strings, containing altogether $n$~characters, can be merged in $O(\log n)$ time using $O(n)$ operations on an EREW PRAM. This is optimal as regards both the running time and the number of operations. \item A sequence of strings, containing altogether $n$~characters represented by integers of size polynomial in~$n$, can be sorted in $O({{\log n}/{\log\log n}})$ time using $O(n\log\log n)$ operations on a CRCW PRAM. The running time is optimal for any polynomial number of processors. \item The minimum string in a sequence of strings containing altogether $n$ characters can be found using (expected) $O(n)$ operations in constant expected time on a randomized CRCW PRAM, in $O(\log\log n)$ time on a deterministic CRCW PRAM with a program depending on~$n$, in $O((\log\log n)^3)$ time on a deterministic CRCW PRAM with a program not depending on~$n$, in $O(\log n)$ expected time on a randomized EREW PRAM, and in $O(\log n\log\log n)$ time on a deterministic EREW PRAM. The number of operations is optimal, and the running time is optimal for the randomized algorithms and, if the number of processors is limited to~$n$, for the nonuniform deterministic CRCW PRAM algorithm as we