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The smallest spanner size, in graph theory, refers to the minimum stretch factor needed for a spanner, which is a subgraph that approximates the distances of the original graph. For a given graph, the smallest spanner size can vary based on the graph's structure and the desired stretch factor. In general, for a graph with ( n ) vertices, a spanner of size ( O(n^{1 + \epsilon}) ) can be achieved for any ( \epsilon > 0 ). However, specific constructions can yield smaller spanners depending on the properties of the graph, such as being a metric space or having certain dimension constraints.

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1w ago

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