Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied during the last decade. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree.Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ − 1 versus n 1+1/κ , for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of log O(1) n in the space.) The only existing solution of the second type uses n 1/2−O(1/κ) passes over the stream (for space O(n 1+1/κ )), and applies only to unweighted graphs.In this paper we show that (1 + )-approximate single-source shortest paths can be computed in this setting with Õ(n 1+1/κ ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs (assuming and κ are constant). Moreover, in fact, the same result applies for multi-source shortest paths, as long as the number of sources is O(n 1/κ ).We achieve these results by devising efficient dynamic streaming constructions of (1 + , β)spanners and hopsets. We believe that these constructions are of independent interest.

We present algorithms for distributed verification and silentstabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of O(n log ∆) bits per node and O(nD) respectively, where ∆ is the highest degree of a node, n is the number of nodes and D is the diameter of the network. In contrast, our algorithm has a space complexity of O(log n) bits per node, which is optimal for silent-stabilizing spanning trees and runs in O(n) time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, we also present a (non-silent) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones.

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Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ − 1 versus n 1+1/κ , for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 +ϵ in the stretch for weighted graphs, and an additional factor of log O (1) n in the space.) The only existing solution of the second type uses n 1/2−O (1/κ ) passes over the stream (for space O (n 1+1/κ )), and applies only to unweighted graphs.We show that (1 + ϵ )-approximate single-source shortest paths can be computed with Õ (n 1+1/κ ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multisource shortest paths, as long as the number of sources is O (n 1/κ ). We achieve these results by devising efficient dynamic streaming constructions of (1 + ϵ, β )-spanners and hopsets. We believe that these constructions are of independent interest.