A sub-linear time distributed algorithm for minimum-weight spanning trees

TL;DR: This paper presents a lower bound of $\Omega(D+\sqrt n/\log n)$ on the time required for the distributed construction of a minimum-weight spanning tree (MST) in weighted n-vertex networks of diameter D=Omega(\ log n) in the bounded message model.

TL;DR: This work introduces a self organizing network structure called a spine and proposes a spine-based routing infrastructure for routing in ad hoc networks and proposes two spine routing algorithms: Optimal Spine Routing (OSR), which uses full and up-to-date knowledge of the network topology, and (b) Partial-knowledge SpineRouting (PSR, which uses partialknowledge of thenetwork topology.

TL;DR: This work analytically prove that the node degree of the IMRG is at most 6, it is connected and planar, and more importantly, the total edge length of theIMRG is within a constant factor of that of the minimum spanning tree.

TL;DR: This work presents a distributed algorithm that constructs a minimum-weight spanning tree in O(log log n) communication rounds, where in each round any process can send a message to each other process.

Abstract: Lecture 1 The basic statement of extremal graph theory is Mantel's theorem, proved in 1907, which states that any graph on n vertices with no triangle contains at most n 2 /4 edges. This is clearly best possible, as one may partition the set of n vertices into two sets of size n/2 and n/2 and form the complete bipartite graph between them. This graph has no triangles and n 2 /4 edges. As a warm-up, we will give a number of different proofs of this simple and fundamental theorem. Theorem 1 (Mantel's theorem) If a graph G on n vertices contains no triangle then it contains at most n 2 4 edges. First proof Suppose that G has m edges. Let x and y be two vertices in G which are joined by an edge. If d(v) is the degree of a vertex v, we see that d(x) + d(y) ≤ n. This is because every vertex in the graph G is connected to at most one of x and y. Note now that x d 2 (x) = xy∈E (d(x) + d(y)) ≤ mn. On the other hand, since x d(x) = 2m, the Cauchy-Schwarz inequality implies that x d 2 (x) ≥ (x d(x)) 2 n ≥ 4m 2 n. Therefore 4m 2 n ≤ mn, and the result follows. 2 Second proof We proceed by induction on n. For n = 1 and n = 2, the result is trivial, so assume that we know it to be true for n − 1 and let G be a graph on n vertices. Let x and y be two adjacent vertices in G. As above, we know that d(x)+d(y) ≤ n. The complement H of x and y has n−2 vertices and since it contains no triangles must, by induction, have at most (n − 2) 2 /4 edges. Therefore, the total number of edges in G is at most e(H) + d(x) + d(y) − 1 ≤ (n − 2) 2 4 + n − 1 = n 2 4 , where the −1 comes from the fact that we count the edge between x and y twice. 2

TL;DR: A distributed algorithm is presented that constructs the minimum weight spanning tree in a connected undirected graph with distinct edge weights that can be initiated spontaneously at any node or at any subset of nodes.

TL;DR: In this paper, a distributed algorithm is presented that constructs the minimum weight spanning tree in a connected undirected graph with distinct edge weights, where a processor exists at each node of the graph, knowing initially only the weights of the adjacent edges.

TL;DR: The new ARPANET routing algorithm is an improvement over the old procedure in that it uses fewer network resources, operates on more realistic estimates of network conditions, reacts faster to important network changes, and does not suffer from long-term loops or oscillations.