Linear Complement Research Articles

Oblivious routing for LC permutations on hypercubes

We propose an oblivious algorithm to route linear-complement (LC) permutations on hypercubes in circuit switched and wormhole routing. The algorithm guarantees that N independent paths can be set up simultaneously for any LC permutation with only a comparison of two bits in one routing step for any path. An LC permutation is determined by a transformation matrix T and a constant modifier C. For all the LC permutations with the same transformation matrix T (we call them a type of permutations), an algorithm is executed to find an ordered sequence of dimensions without knowing a particular permutation. When the sequence of dimensions is used in the routing process for all the packets of a permutation of this type, a comparison of two bits is carried out in each routing step in the packet transmission process. It is guaranteed that no contention will occur between any two paths on the use of the dimensional links, thus N independent paths can be set up simultaneously for the N packets of an LC permutation. Time complexity of the algorithm for finding an ordered sequence for the use of the n dimensions is O( n 3) for any type of LC permutations (rather than one particular LC permutation), and it can be carried out off-line. The routing process itself is distributed, and oblivious.

Hierarchical classification of permutation classes in multistage interconnection networks

This paper explores a new hierarchy among different permutation classes, that has many applications in multistage interconnection networks. The well-known LC (linear-complement) class is shown to be merely a subset of the closure set of the BP (bit-permute) class, known as the BPCL (bit-permute-closure) class; the closure is obtained by applying certain group-transformation rules on the BP-permutations. It indicates that for every permutation P of the LC class, there exists a permutation PI in the BP class, such that the conflict graphs of P and P* are isomorphic, for n-stage MIN's. This obviates the practice of treating the LC class as a special case; the existing algorithm for optimal routing of BPC class in an n-stage MIN can take care of optimal routing of the LC class as well. Finally, the relationships of BPCL with other classes of permutations, e.g., LIE (linear-input-equivalence), BPIE (bit-permute-input-equivalence), BPOE (bit-permute-output-equivalence) are also exposed. Apart from lending better understanding and an integral view of the universe of permutations, these results are found to be useful in accelerating routability in n-stage MIN's as well as in (2n-1)-stage Benes and shuffle-exchange networks. >