Efficient Deadlock-Free Multi-dimensional Interval Routing in Interconnection Networks

Abst rac t : We present deadlock-free packet (wormhole) routing algorithms based on multi-dimensional interval schemes for cert~n multiprocessor interconnection networks and give their analysis in terms of the compactness and the size (the maximum number of buffers per node (per link)). The issue of a simultaneous reduction of the compactness and the size is fundamental, worth to investigate and of practical importance, as interval routing and wormhole routing have been realized in INMOS Transputer C104 Router chips. In this paper we give an evidence that for some well-known interconnection networks there are efficient deadlock-free multidimensional interval routing schemes (DFMIRS) despite of a provable nonexistence of efficient deterministic shortest path interval routing schemes (IRS). For d-dimensional butterflies we give a d-dimensional DFMIRS with constant compactness and size, while each shortest path IRS is of the compactness at least 2 d/2. For d-dimensional cube connected cycles we show a d-dimensional DFMIRS with compactness and size polynomial in d, while each shortest path IRS needs compactness at least 2 ~/2. For d-dimensional hypercubes (tori) we present a d-dimensional DFMIRS of compactness 1 and size 2 (4), while for shortest path IRS we can achieve the reduction to 2 (5) buffers with compactness 2 d-1 (O(,~-1)). We also present a nonconstant lower bound (in the form vfd) on the size of deadlock-free packet routing (based on acyclic orientation covering) for a special set of routing paths on d-dimensional hypercubes.