Efficient deadlock-free multidimensional interval routing in hypercube-like networks

Abstract

We present deadlock-free packet/wormhole routing algorithms ba\-sed on multidimensional interval schemes for certain hypercube related multiprocessor interconnection networks and give their analysis in terms of the compactness (i.e.~the maximum number of intervals per link) and the buffer-size (i.e.~the ma\-xi\-mum number of buffers per node/link). The issue of a simultaneous reduction of the compactness and the buffer-size is fundamental, worth to investigate and of practical importance, since the interval routing and wormhole routing have been industrially 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 non-existence of efficient deterministic shortest path interval routing schemes (IRS). For $d$-dimensional hypercubes (tori) we present a $d$-dimensional DFMIRS of compactness $1$ and size $2$ (of compactness $1$ and size $4$), while for shortest path IRS we can achieve the reduction to $2$ (to at most $5$) buffers per node with compactness $2^{d-1}$ (with compactness $O(n^{d-1})$). For $d$-dimensional generalized butterflies we give a $d$-dimensional DFMIRS with compactness $2$ and size $3$, while each shortest path IRS is of the compactness at least superpolynomial in $d$. 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 $2^{d/2}$.
We also present a nonconstant lower bound (in the form $\sqrt{d}$) on the size of deadlock-free packet routing (based on acyclic orientation covering) for a set of monotone routing paths on $d$-dimensional hypercubes.