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

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 $\Omega(\log \frac{1}{\epsilon})$

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

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Work-Efficient Query Evaluation with PRAMs

The paper studies query evaluation in parallel constant time in the PRAM model. While it is well-known that all relational algebra queries can be evaluated in constant time on an appropriate CRCW-PRAM, this paper is interested in the efficiency of evaluation algorithms, that is, in the number of processors or, asymptotically equivalent, in the work. Naive evaluation in the parallel setting results in huge (polynomial) bounds on the work of such algorithms and in presentations of the result sets that can be extremely scattered in memory. The paper first discusses some obstacles for constant time PRAM query evaluation. It presents algorithms for relational operators that are considerably more efficient than the naive approaches. Further it explores three settings, in which efficient sequential query evaluation algorithms exist: acyclic queries, semi-join algebra queries, and join queries - the latter in the worst-case optimal framework. Under natural assumptions on the representation of the database, the work of the given algorithms matches the best sequential algorithms in the case of semi-join queries, and it comes close in the other two settings. An important tool is the compaction technique from Hagerup (1992)

Searching in a Sorted Linked List and Sort Integers into a Linked List

Title from PDF of title page viewed June 12, 2019Thesis advisor: Yijie HanVitaIncludes bibliographical references (pages 25-27)Thesis (M.S.)--School of Computing and Engineering. University of Missouri--Kansas City, 2019The research work consists of two parts. Part one is about Searching for an integer in a sorted Linked list. A tree is constructed in O(nloglogm/p+loglogm) time with p processors based on the trie with all the given integers. Additional nodes (O(nloglogm) of them) are added to the tree. After the tree is constructed, for any given integer we can find the predecessor and successor of the integer, insert or delete the integer in O(loglogm) time. The result demonstrates for the searching purpose we need not to sort the input numbers into a sorted array for this would need at least O(logn/loglogn) time while this algorithm for constructing the tree can run in O(loglogm) time with n processors. Part two is on sorting integers into a linked list. There are various best algorithms for sorting integers. The current research work applies the recent important results of sorting integers in Ω(logn/loglogn) time. This algorithm takes “constant time” to sort integers into a linked list with nlogm processors and O(loglogm/logt) time using nt processors on the Priority CRCW PRAM model.Introduction -- Searching in a sorted linked list -- Sort integers into a linked list -- Conclusio

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum