Streaming and Fully Dynamic Centralized Algorithms for Constructing and Maintaining Sparse Spanners

Abstract

Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm out-performs significantly the state-of-the-art algorithm for this task [20]. Specifically, the processing time-per-edge of our algorithm is drastically smaller than that of the algorithm of [20], and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters for the state-of-the-art algorithm.We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has a very small incremental update time, and a non-trivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides non-trivial bounds on both incremental and decremental update time for a wide range of stretch parameter t.

Streaming algorithm for graph spanners—single pass and constant processing time per edge

“…Recently and independently Elkin [5] came up with a streaming algorithm for graph spanners. The efficiency parameters of his algorithm are the same as that of our algorithm except a slight difference in the processing time per edge.…”

Fully dynamic randomized algorithms for graph spanners

“…Feigenbaum et al [30] are the first to design an incremental algorithm for (2k − 1)-spanner which takes O(k 2 n 1/(k−1) log n) time per edge insertion and maintain spanners with nearly optimal size. Baswana [4] and Elkin [24,25] independently de-signed incremental algorithms which guarantee (kn 1+1/k ) bound on the expected size of the spanner and achieve O(1) update time per insertion. However, the algorithm of Elkin [24,25] is superior in that it guarantees worst case O(1) time which is better than amortized O(1) time per edge insertion guaranteed by Baswana [4].…”

“…Elkin [25] also designed an efficient fully dynamic algorithm for (2k − 1)-spanners for any k. This algorithm handles any edge insertion in O(1) time. However, for deletion of an edge, the algorithm achieves O(1) time with probability 1 − n −1/k and O(m) time with probability n −1/k .…”

“…However, for deletion of an edge, the algorithm achieves O(1) time with probability 1 − n −1/k and O(m) time with probability n −1/k . Therefore, the expected time per update guaranteed by the algorithm of Elkin [25] is O(m/n 1/k ). Due to this reason, this algorithm is suitable only if the parameter k as well as the number of edge deletions is very small.…”

A near-optimal distributed fully dynamic algorithm for maintaining sparse spanners

“…Related Work: In a companion paper [22] similar techniques are used for devising efficient streaming and fully dynamic centralized algorithms for maintaining sparse spanners. Dynamic distributed algorithms for shortest paths computation were studied in [6,26,27,36,18].…”