Ke Qiu

Fundamental algorithms for the star and pancake interconnection networks with applications to computational geometry

The star and pancake networks were recently proposed as attractive alternatives to the hypercube topology for interconnecting processors in a parallel computer. However, few parallel algorithms are known for these networks. In this paper, we present several data communication schemes and basic algorithms for these two networks. These algorithms are then used to develop parallel solutions to various computational geometric problems on both networks. Computational geometry is just one area where the algorithms proposed here can be applied. Indeed, we believe that these algorithms are interesting and important in their own right and are fundamental to the design of solutions on the star and pancake networks to a host of other problems.

Data communication and computational geometry on the star and pancake interconnection networks

The star and pancake networks were recently proposed as attractive alternatives to the hypercube topology for interconnecting processors in a parallel computer. However, little has been done to design parallel algorithms on these networks. The paper presents several data communication algorithms that are fundamental to designing algorithms on these two networks. These algorithms are then used to develop parallel solutions to various computational geometric problems on both networks. Computational geometry is just one area where the data communication algorithms proposed can be applied. It is believed that these algorithms are interesting and important in their own right, and are basic to the design of solutions on the star and pancake networks to a host of other problems.

Decomposing a star graph into disjoint cycles

Abstract The star graph was proposed by S.B. Akers, D. Harel, and B. Krishnamurthy as an attractive alternative to the n -cube for interconnecting processors on a parallel computer. In this paper, we show that an n -star can be decomposed into ( n −2)! vertex disjoint cycles of lenght ( n −1) n . These cycles may be used in designing parallel algorithms on an interconnection network based on the star topology.

A novel routing scheme on the star and pancake networks and its applications

Abstract A new scheme for routing data on the star and pancake networks is described. It unifies data routing on these two networks, and makes them as powerful as the hypercube when solving a host of problems. Consequently, it allows a certain class of algorithms designed for the hypercube to be implemented directly on the star and pancake networks without time loss. The new scheme is used to derive parallel (star and pancake) algorithms for computing minimum spanning forests in both sparse and dense weighted graphs. The time complexities of these algorithms match those of the equivalent hypercube algorithms. These results take added importance when one recalls the many attractive properties that the star and pancake networks possess by comparison with the hypercube, in particular their smaller degree and diameter.

Parallel Routing and Sorting of the Pancake Network

The pancake graph along with the star graph were proposed in 1986 [1] as attractive alternatives to the hypercube topology for interconnecting processors in a parallel computer. In this paper, we study some of their topological properties. We then present parallelrouting schemes for both networks. Finally, we present an efficient algorithm for sorting K numbers on a pancake interconnection network with n! nodes, where K≥n!, and each node holds at most N=[K/n!] numbers;the algorithm runs in O(NlogN(nlogn)+Nn3logn) time, which is O(n3logn) when K=n!.

On Some Properties of the Star Graph

We derive some properties of the star graph in this paper. In particular, we compute the number of nodes at distance i from a fixed node e in a star graph. To this end, a recursive formula is first obtained. This recursive formula is, in general, hard to solve for a closed form solution. We then study the relations among the number of nodes at distance i to node e in star graphs of different dimensions. This study reveals a very interesting relation among these numbers, which leads to a simple homogeneous linear recursive formula whose characteristic equation is easy to solve. Thus, we get a systematic way to obtain a closed form solution with given initial conditions for any fixed i.

Parallel Minimum Spanning Forest Algorithms on the Star and Pancake Interconnection Networks

Parallel algorithms are described for computing minimum spanning forests in both sparse and dense weighted graphs. The algorithms are designed to run on the star and pancake interconnection networks. Their time complexities match those of the equivalent hypercube algorithms, thanks mainly to a novel routing scheme. The latter unifies data routing on the star and pancake networks, and makes these networks as powerful as the hypercube when solving a host of problems, and it allows a certain class of algorithms designed for the hypercube to be implemented directly on the star and pancake networks without time loss. These results take added importance when one recalls the many attractive properties that the star and pancake networks possess by comparison with the hypercube, in particular their smaller degree and diameter.

Load balancing and selection on the star and pancake interconnection networks

Algorithms for load balancing and selection on the star and pancake interconnection networks are presented. The time complexity of the selection algorithm is discussed. A major component of the selection algorithm is a procedure that balances the loads among all the processors in both networks. Previous results that are either related to the result, or that are used by the algorithms are reviewed.<<ETX>>

Finding the maximum subsequence sum on interconnection networks

We develop parallel algorithms for the maximum subsequence sum (or maximum sum for short) problem on several interconnection networks. For the 1-D version of the problem, we find an algorithm that computes the maximum sum of N elements on these networks of size p, where p ≤ N, with a running time of , which is optimal in view of the lower bound. When , our algorithm computes the maximum sum in O(log N) time, resulting in an optimal cost of O(N). This result also matches the performance of two previous algorithms that are designed to run on the more powerful PRAM model. Our 1-D maximum sum algorithm can be used to solve the problem of maximum subarray, the 2-D version of the problem. In particular, for the same interconnection networks studied here, our parallel algorithm finds the maximum subarray of an N × N array in time O(log N) with processors, once again, matching the performance of a previous PRAM algorithm.

A note on diameter of acyclic directed hypercubes

It has been shown by Everett and Gupta [2] that the edges of an n-cube can be acyclically oriented such that the diameter of the cube is at least F,, ,, the (n + 1)st Fibonacci number, where F, = F, = 1. The result is established by showing that (a) any undirected graph with a chordless path of length p (where the length of a path is defined as the number of vertices on the path) can be oriented such that the graph becomes acyclic with diameter at least p, and (b) a chordless path of length F,, , in an n-cube can be constructed recursively. An open question asked in [2] is whether this lower bound is tight. We will show that this is not the case. Consider a hypercube of dimension 5, Hs. Label its vertices with 5-bit binary numbers as usual such that any two neighboring vertices differ in exactly one bit. A path ul, u2,. . . , up is chordless if ui differs from uj in at least two bits for all 1 i -j 1 > 1. A chordless path of length 14 in H, is shown in Fig. 1, where unrelevant hypercube edges and labels are omitted for simplicity. Thus, by the results of [2, Lemma 21 H, can be acyclically oriented such that its diameter is 14 > F6 = 13. In fact, HS is the smallest hypercube such that its acyclic diameter exceeds the bound given in [2].