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We show how to preprocess a set S of points in R d into an external memory data structure that efficiently supports linear-constraint queries. Each query is in the form of a linear constraint x d a 0 + d&1 i=1 a i x i ; the data structure must report all the points of S that satisfy the constraint. This problem is called halfspace range searching in the computational geometry literature. Our goal is to minimize the number of disk blocks required to store the data structure and the number of disk accesses (IÂOs) required to answer a query. For d=2, we present the first data structure that uses linear space and answers linear-constraint queries using an optimal number of IÂOs in the worst case. For d=3, we present a near-linear-size data structure that answers queries using an optimal number of IÂOs on the average. We present linear-size data structures that can answer d-dimensional linear-constraint queries (and even more general d-dimensional simplex queries) efficiently in the worst case. For the d=3 case, we also show how to obtain trade-offs between space and query time.

Abstract. We present fully dynamic algorithms for maintaining 3-and 5-spanners of undirected graphs. For unweighted graphs we maintain a 3-or 5-spanner under insertions and deletions of edges in O(n) amortized time per operation over a sequence of Ω(n) updates. The maintained 3-spanner (resp., 5-spanner) has O(n 3/2 ) edges (resp., O(n 4/3 ) edges), which is known to be optimal. On weighted graphs with d different edge cost values, we maintain a 3-or 5-spanner in O(n) amortized time per operation over a sequence of Ω(d · n) updates. The maintained 3-spanner (resp., 5-spanner) has O(d·n 3/2 ) edges (resp., O(d·n 4/3 ) edges). The same approach can be extended to graphs with real-valued edge costs in the range [1, C]. All our algorithms are deterministic and are substantially faster than recomputing a spanner from scratch after each update.

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We show how to maintain a shortest path tree of a general directed graph G with unit edge weights and n vertices, during a sequence of edge deletions or a sequence of edge insertions, in O(n) amortized time per operation using linear space. Distance queries can be answered in constant time, while shortest path queries can be answered in time linear in the length of the retrieved path. These results are extended to the case of integer edge weights in [1, C], with a bound of O(Cn) amortized time per operation. We also show how to maintain a breadth-first search tree of a directed graph G in an incremental or a decremental setting in O(n) amortized time per operation using linear space

We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a weighted graph G. Roughly speaking, we say that S is resilient if all its pointto-point distances are resilient to edge failures. Namely, whenever any edge in G fails, then as a consequence of this failure all distances do not degrade in S substantially more than in G (i.e., the relative distance increases in S are very close to those in the underlying graph G). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently.

The vitality of an arc/node of a graph with respect to the maximum flow between two fixed nodes s and t is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper, we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only 2(n − 1) max flow instances and, in st‐planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in O(n) worst‐case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a “contiguous” set of arcs/nodes in time proportional to the size of the set.