Direct routing is the special case of bufferless routing where N packets, once injected into the network, must be routed along specific paths to their destinations without conflicts. We give a general treatment of three facets of direct routing: (i) Algorithms. We present a polynomial time greedy algorithm for arbitrary direct routing problems which is worst-case optimal, i.e., there exist instances for which no direct routing algorithm is better than the greedy. We apply variants of this algorithm to commonly used network topologies. In particular, we obtain near-optimal routing time for the tree and d-dimensional mesh, given arbitrary sources and destinations; for the butterfly and the hypercube, the same result holds for random destinations. (ii) Complexity. By a reduction from Vertex Coloring, we show that Direct Routing is inapproximable, unless P=NP. (iii) Lower Bounds for Buffering. We show that certain direct routing problems cannot be solved efficiently; to solve these problems, any routing algorithm needs buffers. We give non-trivial lower bounds on such buffering requirements for general routing algorithms.
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