Selection, Routing and Sorting on the Star Graph

We consider the problems of selection, routing and sorting on an n-star graph (with n! nodes), an interconnection network which has been proven to possess many special properties. We identify a tree like subgraph (which we call as a \u27(k, l, k) chain network\u27) of the star graph which enables us to design efficient algorithms for the above mentioned problems. We present an algorithm that performs a sequence of n prefix computations in O(n2) time. This algorithm is used as a subroutine in our other algorithms. In addition we offer an efficient deterministic sorting algorithm that runs in O(n3lg n) steps. Though an algorithm with the same time bound has been proposed before, our algorithm is very simple and is based on a different approach. We also show that sorting can be performed on the n star graph in time O(n3) and that selection of a set of uniformly distributed n keys can be performed in O(n2) time with high probability. Finally, we also present a deterministic (non oblivious) routing algorithm that realizes any permutation in O(n3) steps on the n-star graph. There exists an algorithm in the literature that can perform a single prefix computation in O(n lg n) time. The best known previous algorithm for sorting has a run time of O(n3 lg n) and is deterministic. To our knowledge, the problem of selection has not been considered before on the star graph

Aspects of k-k-Routing in Meshes and OTIS Networks

Aspects of k-k Routing in Meshes and OTIS-Networks Abstract Efficient data transport in parallel computers build on sparse interconnection networks is crucial for their performance. A basic transport problem in such a computer is the k-k routing problem. In this thesis, aspects of the k-k routing problem on r-dimensional meshes and OTIS-G networks are discussed. The first oblivious routing algorithms for these networks are presented that solve the k-k routing problem in an asymptotically optimal running time and a constant buffer size. Furthermore, other aspects of the k-k routing problem for OTIS-G networks are analysed. In particular, lower bounds for the problem based on the diameter and bisection width of OTIS-G networks are given, and the k-k sorting problem on the OTIS-Mesh is considered. Based on OTIS-G networks, a new class of networks, called Extended OTIS-G networks, is introduced, which have smaller diameters than OTIS-G networks.Für die Leistungfähigkeit von Parallelrechnern, die über ein Verbindungsnetzwerk kommunizieren, ist ein effizienter Datentransport entscheidend. Ein grundlegendes Transportproblem in einem solchen Rechner ist das k-k Routing Problem. In dieser Arbeit werden Aspekte dieses Problems in r-dimensionalen Gittern und OTIS-G Netzwerken untersucht. Es wird der erste vergessliche (oblivious) Routing Algorithmus vorgestellt, der das k-k Routing Problem in diesen Netzwerken in einer asymptotisch optimalen Laufzeit bei konstanter Puffergröße löst. Für OTIS-G Netzwerke werden untere Laufzeitschranken für das untersuchte Problem angegeben, die auf dem Durchmesser und der Bisektionsweite der Netzwerke basieren. Weiterhin wird ein Algorithmus vorgestellt, der das k-k Sorting Problem mit einer Laufzeit löst, die nahe an der Bisektions- und Durchmesserschranke liegt. Basierend auf den OTIS-G Netzwerken, wird eine neue Klasse von Netzwerken eingeführt, die sogenannten Extended OTIS-G Netzwerke, die sich durch einen kleineren Durchmesser von OTIS-G Netzwerken unterscheiden

Hamiltonian Decompositions of Regular Topology Networks with Convergence Routing

This paper introduces new methods to construct multiple virtual rings for loss-free routing of non-reserved bursty data in high-speed environments such as ATM LANs. The routing algorithm on multiple virtual rings is convergence routing which combines the actual routing decision with the internal flow control state. Multiple virtual rings are obtained on the hypercube and the circulant networks such that each virtual ring is hamiltonian, and are mutually edge-disjoint. It is shown that multiple virtual rings improve (i) the bound on the length of routing, and (ii) the fault tolerance. On the circulant graphs, necessary and sufficient conditions for hamiltonian decomposition is established. On the hypercube, three algorithms are designed for an N-node hypercube with even dimension: (i) an O(N) time algorithm to find two edge-disjoint hamiltonian circuits, (ii) an O(N log N) time algorithm to find I01N hamiltonian circuits with only E ~ 0.1 common edges, and (iii) a recursive algorithm for the hamiltonian decomposition of the hypercube with dimension power of two. It is shown analytically, and verified by simulations on the circulants that with the d virtual ring embeddings, a bound of O( N / d) is established on the maximum length of routing

Optimal Permutation Routing for Low-dimensional Hypercubes

We consider the offline problem of routing a permutation of tokens on the nodes of a d-dimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d-dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting side-result, that the number of possible communication steps in a d-dimensional hypercube is the same as the number of perfect matchings in a (d + 1)-dimensional hypercube, a combinatorial quantity for which there is no closed-form expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d. 2

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Due to the decoherence of the state-of-the-art physical implementations of quantum computers, it is essential to parallelize the quantum circuits to reduce their depth. Two decades ago, Moore et al. demonstrated that additional qubits (or ancillae) could be used to design "shallow" parallel circuits for quantum operators. They proved that any $n$-qubit CNOT circuit could be parallelized to $O(\log n)$ depth, with $O(n^2)$ ancillae. However, the near-term quantum technologies can only support limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for the design of CNOT circuits. We prove that for any $m\geq0$, any $n$-qubit CNOT circuit can be parallelized to $O\left(\max \left\{\log n, \frac{n^{2}}{(n+m)\log (n+m)}\right\} \right)$ depth, with $O(m)$ ancillae. We show that this bound is tight by a counting argument, and further show that even with arbitrary two-qubit quantum gates to approximate CNOT circuits, the depth lower bound still meets our construction, illustrating the robustness of our result. Our work improves upon two previous results, one by Moore et al. for $O(\log n)$-depth quantum synthesis, and one by Patel et al. for $m = 0$: for the former, we reduce the need of ancillae by a factor of $\log^2 n$ by showing that $m=O(n^2/\log^2 n)$ additional qubits suffice to build $O(\log n)$-depth, $O(n^2/\log n)$ size --- which is asymptotically optimal --- CNOT circuits; for the later, we reduce the depth by a factor of $n$ to the asymptotically optimal bound $O(n/\log n)$. Our results can be directly extended to stabilizer circuits using an earlier result by Aaronson et al. In addition, we provide relevant hardness evidences for synthesis optimization of CNOT circuits in term of both size and depth.Comment: 25 pages, 5 figures. Fixed several minor typos and a mistake about CNOT+Rz circui