Simple Optimal Algorithm Research Articles

Optimal admission control in cellular DS-CDMA systems with multimedia traffic

The problem of jointly controlling the data rates and transmit powers of users, so as to maximize throughput in cellular direct-sequence code-division multiple-access (CDMA) networks is addressed. The multicode (MC)-CDMA system, where the processing gain of each code is fixed and high-rate users use multiple codes for transmission, and the variable gain (VG)-CDMA architectures are considered. The throughout maximization problem is formulated as a classical optimization problem, modeling the constraints arising from the data rate requirements and power budgets. Optimal strategies to maximize throughput in both systems are derived. While the MC-CDMA system has a simple optimal rate-power allocation algorithm, the VG-CDMA system has a more complex solution.

An optimal parallel algorithm for the multiselection problem

Given a set S of n elements drawn from a linearly ordered set, and a set K={ k 1, k 2,…, k r } of positive integers between 1 and n, the multiselection problem is to select the k i th smallest element for all values of i,1⩽ i⩽ r. We present a simple optimal algorithm to solve this problem that runs in O( n ϵ log r) time on the EREW PRAM with n 1− ϵ processors, 0< ϵ<1.

L∞-Approximation via Subdominants

Given a vector u and a certain subset K of a real vector space E, the problem of l∞-approximation involves determining an element u in K nearest to u in the sense of the l∞-error norm. The subdominant u∗ of u is the upper bound (if it exists) of the set {x∈K:x≺u} (we let x≺y if all coordinates of x are smaller than or equal to the corresponding coordinates of y). We present general conditions on K under which a simple relationship between the subdominant of u and a best l∞-approximation holds. We specify this result by taking as K the cone of isotonic functions defined on a poset (X, ≺), the cone of convex functions defined on a subset of RN, the cone of ultrametrics on a set X, and the cone of tree metrics on a set X with fixed distances to a given vertex. This leads to simple optimal algorithms for the problem of best l∞-fitting of distances by ultrametrics and by tree metrics preserving the distances to a fixed vertex (the latter provides a 3-approximation algorithm for the problem of fitting a distance by a tree metric). This simplifies the recent results of Farach, Kannan, and Warnow (1995) and of Agarwala et al. (1996).

Polynomial complexity optimal detection of certain multiple-access systems

It is demonstrated that optimal multiple-user detection in a linear multiple-access system with identical crosscorrelations requires only /spl Oscr/(KlogK) operations instead of the worst case O(2/sup K/). A simple optimal detection algorithm is given, which when generalized has a complexity that is exponential not in the number of users but in the number of unique crosscorrelation values.

A simple optimal algorithm for scheduling variable-sized requests

In this paper, an optimal algorithm is found for the following scheduling problem: given N parallel channels that can each accommodate one request at a time and a set of requests that require a channel at a specific start time and for a particular holding time, what is the best way to schedule these requests in order to accommodate the most requests? A proof of optimality is given. The algorithm is conceptually and computationally simple. This scheduling problem has application in resource allocation for communication networks.

The queue-read queue-write asynchronous PRAM model

This paper presents results for the queue-read, queue-write asynchronous parallel random access machine ( qrqw asynchronous pram) model, which is the asynchronous variant of the qrqw pram model. The qrqw pram family of models, which was introduced earlier by the authors, permit concurrent reading and writing to shared memory locations, but each memory location is viewed as having a queue which can service at most one request at a time. In the basic qrqw pram model each processor executes a series of reads to shared memory locations, a series of local computation steps, and a series of writes to shared memory locations, and then synchronizes with all other processors; thus this can be viewed as a bulk-synchronous model. In contrast, in the qrqw asynchronous pram model discussed in this paper, there is no imposed bulk-synchronization between processors, and each processor proceeds at its own pace. Thus, the qrqw asynchronous pram serves as a better model for designing and analyzing truly asynchronous parallel algorithms than the original qrqw pram. In this paper we elaborate on the qrqw asynchronous pram model, and we demonstrate the power of asynchrony over bulk-synchrony by presenting a work and time optimal deterministic algorithm on the qrqw asynchronous pram for the leader election problem and a simple randomized work and time optimal algorithm on the qrqw asynchronous pram for sorting. In contrast, no tight bounds are known on the qrqw pram for either deterministic or randomized parallel algorithms for leader election and the only work and time optimal algorithms for sorting known on the qrqw pram are those inherited from the erew pram, which are considerably more complicated. Our sorting algorithm is an asynchronous version of an earlier sorting algorithm we developed for the qrqw pram, for which we use an interesting analysis to bound the running time to be O( log n). We also present a randomized algorithm to simulate one step of a crcw pram on a qrqw asynchronous pram in sublogarithmic time if the maximum contention in the step is relatively small. © 1998—Elsevier Science B.V. All rights reserved

Work-time optimal k-merge algorithms on the PRAM

For 2/spl les/k/spl les/n, the k-merge problem is to merge a collection of ksorted sequences of total length n into a new sorted sequence. The k-merge problem is fundamental as it provides a common generalization of both merging and sorting. The main contribution of this work is to give simple and intuitive work-time optimal algorithms for the k-merge problem on three PRAM models, thus settling the status of the k-merge problem. We first prove that /spl Omega/(n log k) work is required to solve the k-merge problem on the PRAM models. We then show that the EREW-PRAM and both the CREW-PRAM and the CRCW require /spl Omega/(log n) time and /spl Omega/(log log n+log k) time, respectively, provided that the amount of work is bounded by O(n log k). Our first k-merge algorithm runs in /spl Theta/(log n) time and performs /spl Theta/(n log k) work on the EREW-PRAM. Finally, we design a work-time optimal CREW-PRAM k-merge algorithm that runs in /spl Theta/(log log n+log k) time and performs /spl Theta/(n log k) work. This latter algorithm is also work-time optimal on the CREW-PRAM model. Our algorithms completely settle the status of the k-merge problem on the three main PRAM models.

Planar stage graphs: Characterizations and applications

We consider combinatorial and algorithmic aspects of the well-known paradigm “killing two birds with one stone”. We define a stage graph as follows: vertices are points from a planar point set, and { u, v} is an edge if and only if the (infinite, straight) line segment joining u to v intersects a given line segment, called a stage. We show that a graph is a stage graph if and only if it is a permutation graph. The characterization results in a compact linear space representation of stage graphs. This has been exploited for designing improved algorithms for maximum matching in permutation graphs, two processor task scheduling for dependency graphs known to be permutation graphs, and dominance-related problems for planar point sets. We show that a maximum matching in permutation graphs can be computed in Ω(n log 2 n) time, where n is the number of vertices. We provide simple optimal sequential and parallel algorithms for several dominance related problems for planar point sets.

A Simple Optimal Parallel Algorithm for Reporting Paths in a Tree

We present optimal parallel solutions to reporting paths between pairs of nodes in an n-node tree. Our algorithms are deterministic and designed to run on an exclusive read exclusive write parallel random-access machine (EREW PRAM). In particular, we provide a simple optimal parallel algorithm for preprocessing the input tree such that the path queries can be answered efficiently. Our algorithm for preprocessing runs in O( log n) time using O(n/ log n) processors. Using the preprocessing, we can report paths between k node pairs in O( log n + log k) time using O(k + (n + S)/ log n) processors on an EREW PRAM, where S is the size of the output. In particular, we can report the path between a single pair of distinct nodes in O( log n) time using O(L/ log n) processors, where L denotes the length of the path.

Optimal algorithms for the single and multiple vertex updating problems of a minimum spanning tree

The vertex updating problem for a minimum spanning tree (MST) is defined as follows: Given a graphG=(V, E G) and an MSTT forG, find a new MST forG to which a new vertexz has been added along with weighted edges that connectz with the vertices ofG. We present a set of rules that produce simple optimal parallel algorithms that run inO(lgn) time usingn/lgn EREW PRAM processors, wheren=¦V¦. These algorithms employ any valid tree-contraction schedule that can be produced within the stated resource bounds. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best-known parallel result was a rather complicated algorithm that usedn processors in the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST whenk new vertices are introduced simultaneously. This problem is solved inO(lgk·lgn) parallel time using (k·n)/(lgk·lgn) EREW PRAM processors. This is optimal for graphs having Ω (kn) edges.

Maximum independent sets of circular-arc graphs: Simplified algorithm and proofs

We present a simple optimal algorithm for the problem of finding maximum independent sets of circular-arc graphs. Given an intersection model S of a circular-arc graph G, our algorithm computes a maximum independent set of G in O(n) space and O(n) or O(n log n) time, depending on whether the endpoints of arcs in S are sorted or not. The proofs of the correctness and the complexities of the algorithm are straightforward, and the techniques used can be generalized to solve other problems on circular-arc graphs efficiently.

A simple iterative algorithm for the towers of Hanoi problem

The towers of Hanoi game is well known. The basic version, a favorite example for many authors, is often used in introductory textbooks on computer programming to demonstrate the elegance of writing recursive code. Here, a simple iterative optimal algorithm for the towers of Hanoi problem is presented.

Moldable and castable polygons

This paper introduces the concepts of moldability and castability of simple polygons and relates moldability to monotonicity. We detail a Θ( n) algorithm for determining all n forward maximal monotone chains of a simple polygon and apply this algorithm to the problems of determining 2-moldability, 2-castability and the minimum monotone decomposition of a simple polygon (Swaminathan et al., 1992). Our results include a simple optimal algorithm solving the minimum monotone decomposition problem, an optimal algorithm to determine 2-moldability and an O( n log n) algorithm to determine 2-castability.

A simple optimal parallel algorithm for the minimum coloring problem on interval graphs

In this paper, we develop a new parallel algorithm for the minimum coloring problem on interval graphs. Our method is both easy to understand and easy to implement. Given a set of n intervals, our parallel algorithm takes O(log n) time and O( n) processors on the EREW PRAM model. Also, if the intervals are sorted, it takes O(log n) time and O( n/log n) processors on the EREW PRAM model. Both the sorted and unsorted cases are cost optimal.

Rotator graphs: an efficient topology for point-to-point multiprocessor networks

Rotator graphs, a set of directed permutation graphs, are proposed as an alternative to star and pancake graphs. Rotator graphs are defined in a way similar to the recently proposed Faber-Moore graphs. They have smaller diameter, n-1 in a graph with n factorial vertices, than either the star or pancake graphs or the k-ary n-cubes. A simple optimal routing algorithm is presented for rotator graphs. The n-rotator graphs are defined as a subset of all rotator graphs. The distribution of distances of vertices in the n-rotator graphs is presented, and the average distance between vertices is found. The n-rotator graphs are shown to be optimally fault tolerant and maximally one-step fault diagnosable. The n-rotator graphs are shown to be Hamiltonian, and an algorithm for finding a Hamiltonian circuit in the graphs is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal merging and sorting on the EREW PRAM

We describe very simple optimal EREW PRAM algorithms for the tasks of sorting n elements and of merging two sorted sequences of total length n. The running times achieved are O(log n) for merging and O((log n) 2) for sorting. In addition, the number of comparisons performed by our algorithms is within lower-order terms of the minimum achievable, even by a sequential algorithm.

On-line optimization of a rectification column

A simple on-line optimal control algorithm is suggested for a rectification column. Two optimization criteria, energy supply and distillate production, lead to a vector optimization problem. The Pareto compromise set is constructed by the method of weight coefficients or by the method of ε -constraints. Penalty function approach is used to determine a constrained optimum. A simple short-cut model is used to simulate the column repeatedly. Application to the on-line control of two industrial columns is described.

Optimal algorithms theory for robust estimation and prediction

This paper deals with the theory of optimal algorithms for problems which cannot be solved exactly. The theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved, but only a bound on them is known, in particular, the derivation of computationally simple optimal algorithms for these two problems is investigated. The practical effectiveness of the algorithms obtained is illustrated by several numerical examples.

Scheduling Trees in Parallel/Pipelined Processing Environments

Scheduling task trees to be executed in parallel and/or pipelined processing systems are examined under individual situations. Processor structural requirements at task nodes are also included in the model of consideration. While simple techniques can serve as heuristics, counterexamples are constructed in some crucial cases. Simple optimal algorithms are presented in two important cases: 1) unistructure, multipipe, uniform latency, and flush time; and 2) vector loops. Finally, the complexity of the remaining cases is scrutinized with different structural parameter combinations.