Juggling Networks

Switching networks of various kinds have come to occupy a prominent position in computer science as well as communication engineering. The classical switching network technology has been spacedivision-multiplex switching, in which each switching function is performed by a spatially separate switching component (such as a crossbar switch). A recent trend in switching network technology has been the advent of time-division-multiplex switching, wherein a single switching component performs the function of many switches at successive moments of time according to a periodic schedule. This technology has the advantage that nearly all of the cost of the network is in inertial memory (such as delay lines), with the cost of switching elements growing much more slowly as a function of the capacity of the network. In order for a classical space-division-multiplex network to be adaptable to time-division-multiplex technology, its interconnection pattern must satisfy stringent requirements. For example, networks based on randomized interconnections (an important tool in determining the asymptotic complexity of optimal networks) are not suitable for time-division-multiplex implementation. Indeed, time-division-multiplex implementations have been presented for only a few of the simplest classical space-division-multiplex constructions, such as rearrangeable connection networks. This paper shows how interconnection patterns based on explicit constructions for expanding graphs can be implemented in time-division-multiplex networks. This provides time-division-multiplex implementations for switching networks that are within constant factors of optimal in memory cost, and that have asymptotically more slowly growing switching costs. These constructions are based on a metaphor involving teams of jugglers whose throwing, catching and passing patterns result in intricate permutations of the balls. This metaphor affords a convenient visualization of time-division-multiplex activities that should be of value in devising networks for a variety of switching tasks