Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics
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We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on n points. We first consider the o(n)-space regime and show that, when the input is a stream of all n2 entries of the metric, for any α≥ 2, both MST and TSP cost can be α-approximated using Õ(n/α) space, and that Ω(n/α^2) space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed p passes over a metric stream, it still requires Ω̃(√(n/α p^2)) space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than 2. We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any ε > 0, any one-pass (2-ε)-approximation of TSP cost requires Ω(ε^2 n^2) space; on the other hand, there is an Õ(n) space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than 2, when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a (1+ε)-approximation is achievable with Õ(n/ε^O(1)) queries. We design an algorithm that performs Õ(n^1.5) distance queries and achieves a strictly better than 2-approximation when either the metric is known to contain a spanning tree supported on weight-1 edges or the algorithm is given access to a minimum spanning tree of the graph.
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