Let G = (V , E) be an undirected weighted graph on |V | = n vertices and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V , E S ), E S ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well-motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43-year old girth lower bound conjecture made independently by Erdős, Bollobás, and Bondy & Simonovits.Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information, which involves growing either breadth first search trees up to θ(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach ON COMPUTING SPARSE SPANNERS IN WEIGHTED GRAPHS 533of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory.

We present the first linear time (1 + ε)-approximation algorithm for the k-means problem for fixed k and ε. Our algorithm runs in O(nd) time, which is linear in the size of the input. Another feature of our algorithm is its simplicity -the only technique involved is random sampling.

We present an algorithm for maintaining a maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized O(log n) time for each edge update where n is the number of vertices in the graph. Moreover, for any sequence of t edge updates, the total time taken by the algorithm is O(t log n + n log 2 n) with high probability. Note:The previous version of this result appeared in SIAM J. Comp., 44(1): 88-113, 2015. However, the analysis presented there for the algorithm was erroneous. This version rectifies this deficiency without any changes in the algorithm while preserving the performance bounds of the original algorithm.