Shorter Labels for Routing in Trees
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A routing labeling scheme assigns a binary string, called a label, to each node in a network, and chooses a distinct port number from {1,...,d} for every edge outgoing from a node of degree d. Then, given the labels of u and w and no other information about the network, it should be possible to determine the port number corresponding to the first edge on the shortest path from u to w. In their seminal paper, Thorup and Zwick [SPAA 2001] designed several routing methods for general weighted networks. An important technical ingredient in their paper that according to the authors “may be of independent practical and theoretical interest” is a routing labeling scheme for trees of arbitrary degrees. For a tree on n nodes, their scheme constructs labels consisting of (1+o(1))log n bits such that the sought port number can be computed in constant time. Looking closer at their construction, the labels consist of log n + O(log n·logloglog n / loglog n) bits. Given that the only known lower bound is log n+Ω(loglog n), a natural question that has been asked for other labeling problems in trees is to determine the asymptotics of the smaller-order term. We make the first (and significant) progress in 19 years on determining the correct second-order term for the length of a label in a routing labeling scheme for trees on n nodes. We design such a scheme with labels of length log n+O((loglog n)^2). Furthermore, we modify the scheme to allow for computing the port number in constant time at the expense of slightly increasing the length to log n+O((loglog n)^3).
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