Low-Port Tree Representations

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Consider an n-node undirected graph G(V,E) with a pre-assigned port numbering for the outgoing edges of each node. The port numbers assigned to a node u of degree \(\deg(u)\) are \(\{0,1,\ldots,\deg(u)-1\}\). In certain contexts it is necessary to maintain a directed spanning tree of G, in which case each node needs to remember the port number leading to its parent. Hence the cost of a spanning tree T is the total number of bits the nodes need to store in order to remember T. This paper addresses the question of asymptotically bounding the cost of the optimal tree, as a function of the graph size. A tight upper bound of O(n) is established on this cost, thus improving on the best previously known bound of O(nloglogn) [6] and proving the conjecture raised therein. This is achieved by presenting a polynomial time algorithm for constructing a spanning tree T of cost O(n) for a given general graph G with an arbitrary port labeling.