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Distance oracles and graph spanners are excerpts of a graph that allow to compute approximate shortest paths. Here, we consider the situation where it is possible to access the original graph in addition to the graph excerpt while computing paths. This allows for asymptotically much smaller excerpts than distance oracles or spanners. The quality of an algorithm in this setting is measured by the size of the excerpt (in bits), by how much of the original graph is accessed (in number of edges), and the stretch of the computed path (as the ratio between the length of the path and the distance between its end points). Because these three objectives are conflicting goals, we are interested in a good trade-off. We measure the number of accesses to the graph relative to the number of edges in the computed path. We present a parametrized construction that, for constant stretches, achieves excerpt sizes and number of accessed edges that are both sublinear in the number of graph vertices. We also show that within these limits, a stretch smaller than 5 cannot be guaranteed. [ABSTRACT FROM AUTHOR]

Let G = (V,E) denote an undirected weighted graph of n nodes and m edges, and let U ⊆ V. The relative eccentricity of a node v ∈ U is the maximum distance in G between v and any other node of U, while the radius of U in G is the minimum relative eccentricity of all the nodes in U. Several facility location problems ask for partitioning the nodes of G so as to minimize some global optimization function of the radii of the subsets of the partition. Here, we focus on the problem of partitioning the nodes of G into exactly p ≥ 2 non-empty subsets, so as to minimize the sum of the subset radii, called the total radius of the partition. This problem can be easily seen to be NP-hard when p is part of the input, but when p is fixed it can be solved in polynomial time by reducing it to a similar partitioning problem. In this paper, we first present an efficient O(n3) time algorithm for the notable case p = 2, which improves the O(mn2 + n3 logn) running time obtainable by applying the aforementioned reduction. Then, in an effort of characterizing meaningful polynomial-time solvable instances of the problem when p is part of the input, we show that (i) when G is a tree, then the problem can be solved in O(n3p3) time, and (ii) when G has bounded treewidth h, then the problem can be solved in O(n4h + 4p3) time. [ABSTRACT FROM AUTHOR]