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Goaoc, Xavier ; Kerber, Michael (Ed.)We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NPhard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group atmore »

Dictionaries remain the most well studied class of data structures. A dictionary supports insertions, deletions, membership queries, and usually successor, predecessor, and extractmin. In a RAM, all such operations take O(log n) time on n elements. Dictionaries are often crossreferenced as follows. Consider a set of tuples {〈ai,bi,ci…〉}. A database might include more than one dictionary on such a set, for example, one indexed on the a ‘s, another on the b‘s, and so on. Once again, in a RAM, inserting into a set of L crossreferenced dictionaries takes O(L log n) time, as does deleting. The situation is more interesting in external memory. On a Disk Access Machine (DAM), Btrees achieve O(logB N) I/Os for insertions and deletions on a single dictionary and Kelement range queries take optimal O(logB N + K/B) I/Os. These bounds are also achievable by a Btree on crossreferenced dictionaries, with a slowdown of an L factor on insertion and deletions. In recent years, both the theory and practice of external memory dictionaries has been revolutionized by write optimization techniques. A dictionary is write optimized if it is close to a Btree for query time while beating Btrees on insertions. The best (and optimal) dictionariesmore »