Serial and parallel algorithms for the medial axis transform

An O(n²) time serial algorithm is developed for obtaining the medial axis transform (MAT) of an n×n image. An O(log n) time CREW PRAM algorithm and an O(log² n) time SIMD hypercube parallel algorithm for the MAT are also developed. Both of these use O(n²) processors. Two problems associated with the MAT, the area and perimeter reporting problem, are studied. An O(log n) time hypercube algorithm is developed for both of them, where n is the number of squares in the MAT, and the algorithms use O(n²) processors     

Optimal parallel initialization algorithms for a class of priorityqueues

An adaptive parallel algorithm for inducing a priority queue structure on an n-element array is presented. The algorithm is extended to provide optimal parallel construction algorithms for three other heap-like structures useful in implementing double-ended priority queues, namely min-max heaps, deeps, and min-max-pair heaps. It is shown that an n-element array can be made into a heap, a deap, a min-max heap, or a min-max-pair heap in O(log n+(n /p)) time using no more than n/log n processors, in the exclusive-read-exclusive-write parallel random-access machine model     

An efficient distributed knot detection algorithm

A distributed knot detection algorithm for general graphs is presented. The knot detection algorithm uses at most O(n log n+m) messages and O(m+n log n) bits of memory to detect all knots' nodes in the network (where n is the number of nodes and m is the number of links). This is compared to O(n²) messages needed in the best algorithm previously published. The knot detection algorithm makes use of efficient cycle detection and clustering techniques. Various applications for the knot detection algorithms are presented. In particular, its importance to deadlock detection in store and forward communication networks and in transaction systems is demonstrated     

A novel discrete relaxation architecture

The discrete relaxation algorithm (DRA) is a computational technique that enforces arc consistency (AC) in a constraint satisfaction problem (CSP). The original sequential AC-1 algorithm suffers from O(n³m³) time complexity, and even the optimal sequential AC-4 algorithm is O (n²m²) for an n-object and m-label DRA problem. Sample problem runs show that these algorithms are all too slow to meet the need for any useful, real-time CSP applications. A parallel DRA5 algorithm that reaches a lower bound of O(nm) (where the number of processors is polynomial in the problem size) is given. A fine-grained, massively parallel hardware computer architecture has been designed for the DRA5 algorithm. For practical problems, many orders of magnitude of efficiency improvement can be reached on such a hardware architecture     

Two-dimensional convolution on a pyramid computer

An algorithm for convolving a k×k window of weighting coefficients with an n×n image matrix on a pyramid computer of O(n²) processors in time O(logn+k²), excluding the time to load the image matrix, is presented. If k=Ω (√log n), which is typical in practice, the algorithm has a processor-time product O(n ² k²) which is optimal with respect to the usual sequential algorithm. A feature of the algorithm is that the mechanism for controlling the transmission and distribution of data in each processor is finite state, independent of the values of n and k. Thus, for convolving two {0, 1}-valued matrices using Boolean operations rather than the typical sum and product operations, the processors of the pyramid computer are finite-state     

Efficient parallel processing of image contours

Describes two parallel algorithms for ranking the pixels on a curve in O (log N) time using either an EREW or CREW PRAM model. The algorithms accomplish this with N processors for a √N×√N image. After applying such an algorithm to an image, it is possible to move the pixels from a curve into processors having consecutive addresses. This is important because one can subsequently apply many algorithms to the curve (such as piecewise linear approximation algorithms or point in polygon tests) using segmented scan operations (i.e. parallel prefix operations). Scan operations can be executed in logarithmic time on many interconnection networks, such as hypercube, tree, butterfly, and shuffle exchange machines as well as on the EREW PRAM. The algorithms were implemented on the hypercube structured Connection Machine, and various performance tests were conducted     

Output feedback eigenstructure assignment using two Sylvesterequations

The eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p+m> n. The main tool used is the concept of (C, A, B)-invariance and two coupled Sylvester equations, the solution of which leads to the computation of an output stabilizing feedback. A computationally efficient algorithm for the solution of these two coupled equations, which leads to the computation of a desired output feedback, is presented     

Optimal parallel algorithms for problems modeled by a family ofintervals

A family of intervals on the real line provides a natural model for a vast number of scheduling and VLSI problems. Recently, a number of parallel algorithms to solve a variety of practical problems on such a family of intervals have been proposed in the literature. The authors develop computational tools and show how they can be used for the purpose of devising cost-optimal parallel algorithms for a number of interval-related problems, including finding a largest subset of pairwise nonoverlapping intervals, a minimum dominating subset of intervals, along with algorithms to compute the shortest path between a pair of intervals and, based on the shortest path, a parallel algorithm to find the center of the family of intervals. More precisely, with an arbitrary family of n intervals as input, all the algorithms run in O(log n) time using O(n) processors in the EREW-PRAM model of computation     

An efficient distributed protocol for finding shortest paths innetworks with negative weights

The design is discussed of distributed algorithms for the single-source shortest-path problem to run on an asynchronous directed network in which some of the edges may be associated with negative weights, and thus in which a cycle of negative total weight may also exist. The only existing solution in the literature for this problem is due to K.M. Chandy and J. Misra (1982), and it has, in the worst case, an unbounded message complexity. A synchronous version of the Chandy-Misra algorithm is described and studied, and it is proved that for a network with m edges and n nodes, the worst case message and time complexities of this algorithm are O(mn ) and O(n), respectively. This algorithm is then combined with an efficient synchronizer to yield an asynchronous protocol that retains the same message and time complexities     

A comment on `An inequality for the trace of matrix product' byJ.K. Baksalary and S. Puntanen

An example is provided which illustrates that the new bounds for the trace of the product of an arbitrary n×n real matrix A and an n×n nonnegative definite real symmetric matrix B derived in the above-titled paper (ibid., vol.37, no.2, pp.239-240, Feb. 1992) are not valid     

Optimal distributed t-resilient election in completenetworks

The problem of distributed leader election in an asynchronous complete network, in the presence of faults that occurred prior to the execution of the election algorithm, is discussed. Failures of this type are encountered, for example, during a recovery from a crash in the network. For a network with n processors, k of which start the algorithm that uses at most O(n log k +n+kt) messages is presented and shown to be optimal. An optimal algorithm for the case where the identities of the neighbors are known is also presented. It is noted that the order of the message complexity of a t-resilient algorithm is not always higher than that of a nonresilient one. The t-resilient algorithm is a systematic modification of an existing algorithm for a fault-free network     

Job scheduling in a partitionable mesh using a two-dimensionalbuddy system partitioning scheme

The job scheduling problem in a partitionable mesh-connected system in which jobs require square meshes and the system is a square mesh whose size is a power of two is discussed. A heuristic algorithm of time complexity O(n(log n+log p)), in which n is the number of jobs to be scheduled and p is the size of the system is presented. The algorithm adopts the largest-job-first scheduling policy and uses a two-dimensional buddy system as the system partitioning scheme. It is shown that, in the worst case, the algorithm produces a schedule four times longer than an optimal schedule, and, on the average, schedules generated by the algorithm are twice as long as optimal schedules     

Constant time algorithms for the transitive closure and somerelated graph problems on processor arrays with reconfigurable bussystems

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O (1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n×n×n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n²×n² processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs     

A new method of image compression using irreducible covers ofmaximal rectangles

The binary-image-compression problem is analyzed using irreducible cover of maximal rectangles. A bound on the minimum-rectangular-cover problem for image compression is given under certain conditions that previously have not been analyzed. It is demonstrated that for a simply connected image, the irreducible cover proposed uses less than four times the number of the rectangles in a minimum cover. With n pixels in a square, the parallel algorithm for obtaining the irreducible cover uses (n/log n) concurrent-read-exclusive write (CREW) processors in O(log n) time     

An EREW PRAM algorithm for image component labeling

An important midlevel task for computer vision is addressed. The problem consists of labeling connected components in N^1/2 ×N^2/2 binary images. This task can be solved with parallel computers by using a simple and novel algorithm. The parallel computing model used is a synchronous fine-grained shared-memory model where only one processor can read from or write to the same memory location at a given time. This model is known as the exclusive-read exclusive-write parallel RAM (EREW PRAM). Using this model, the algorithm presented has O(log N) complexity. The algorithm can run on parallel machines other than the EREW PRAM. In particular, it offers an optimal image component labeling algorithm for mesh-connected computers     

A qualitative bound in robustness of stabilization by statefeedback

It is proved that placing the poles of a linear time-invariant system arbitrarily far to the left of the imaginary axis is not possible if small perturbations in the model coefficients are taken into account. Given a nominal controllable system (A₀, B ₀) with one input and at least two states and an open ball around B₀ (no matter how small), there exists a real number γ and a perturbation B within that ball such that for any feedback matrix K placing the eigenvalues of A ₀+B₀K to the left of Res=γ, there is an eigenvalue of A₀+BK with real part not less than γ     

Performance of shared memory in a parallel computer

A method for analyzing the lengths of memory queues when the network is conflict-free is described. An algorithm based on this method is shown to efficiently determine the upper and lower bounds of the queue length. Analysis indicates that the strategy of using hashing to spread data across memory modules is a good one. Results show that if the size of the system is increased while maintaining a constant ratio of numbers of processors to memories, then, asymptotically, the slowdown in performance from conflicts at the memory modules is Θ(log m /log log m). For m and n less than 100000 and λ between 0.25 and 4.0, the graphical data confirm this growth rate     

An efficiently computable metric for comparing polygonal shapes

A method for comparing polygons that is a metric, invariant under translation, rotation, and change of scale, reasonably easy to compute, and intuitive is presented. The method is based on the L₂ distance between the turning functions of the two polygons. It works for both convex and nonconvex polygons and runs in time O(mn log mn), where m is the number of vertices in one polygon and n is the number of vertices in the other. Some examples showing that the method produces answers that are intuitively reasonable are presented     

Fast image labeling using local operators on mesh-connectedcomputers

A new parallel algorithm is proposed for fat image labeling using local operators on image pixels. The algorithm can be implemented on an n×n mesh-connected computer such that, for any integer k in the range [1, log (2n)], the algorithm requires Θ(kn¹k/) bits of local memory per processor and takes Θ(kn) time. Bit-serial processors and communication links can be used without affecting the asymptotic time complexity of the algorithm. The time complexity of the algorithm has very small leading constant factors, which makes it superior to previous mesh computer labeling algorithms for most practical image sizes (e.g. up to 4096×4096 images). Furthermore, the algorithm is based on using stacks that can be realized using very fast shift registers within each processing element     

The external Heapsort

Heapsort is an internal sorting method which sorts an array of n records in place in O(n log n) time. Heapsort is generally considered unsuitable for external random-access sorting. By replacing key comparisons with merge operations on pages, it is shown how to obtain an in-place external sort which requires O(m log m) page references, where m is the number of pages which the file occupies. The new sort method (called Hillsort) has several useful properties for advanced database management systems. Not only does Hillsort operate in place, i.e., no additional external storage space is required assuming that the page table can be kept in core memory, but accesses to adjacent pages in the heap require one seek only if the pages are physically contiguous. The authors define the Hillsort model of computation for external random-access sorting, develop the complete algorithm and then prove it correct. The model is next refined and a buffer management concept is introduced so as to reduce the number of merge operations and page references, and make the method competitive to a basic balanced two-way external merge. Performance characteristics are noted such as the worst-case upper bound, which can be carried over from Heapsort, and the average-case behavior, deduced from experimental findings. It is shown that the refined version of the algorithm which is on a par with the external merge sort