We consider an information dissemination problem where the root node in an undirected graph constantly updates its information. The goal is to keep every other node in the graph as freshly informed about the root as possible. Our synchronous information spreading model uses telephone calls at each time step, in which any node can communicate with at most one neighbor, thus forming a matching over which information is transmitted at each step. We introduce two problems in minimizing two natural objectives (Maximum and Average) of the latency of the root's information at all nodes in the network. After deriving a simple reduction from the maximum rooted latency problem to the well-studied minimum broadcast time problem, we focus on the average rooted latency version. We introduce a natural problem of finding a finite schedule that minimizes the average broadcast time from a root. We show that any average rooted latency scheme induces a solution to this average broadcast problem within a constant factor and conversely, this average broadcast time is within a logarithmic factor of the average rooted latency. Then, we derive a log-squared approximation algorithm for the average broadcast time problem via rounding a time-indexed linear programming relaxation, resulting in a log-cubed approximation for the average latency problem. Surprisingly, we show that using the average broadcast time for average rooted latency introduces a necessary logarithmic factor overhead even in trees. We overcome this hurdle and give a 40-approximation for trees. For this, we design an algorithm to find near-optimal locally-periodic schedules in trees where each vertex receives information from its parent in regular intervals. On the other side, we show how such well-behaved schedules approximate the optimal schedule within a constant factor. * This material is based upon work supported in part by the U. S. Office of Naval Research under award number N00014-21-1-2243 and the Air Force Office of Scientific Research under award number FA9550-20-1-0080.

We study a fundamental class of two-layer network design problems. A hub layer is configured by establishing hubs at selected nodes at considerable cost so that the routes between hubs can be operated cheaply. The remaining edges in the network are operated at regular cost. The resulting problem is to determine the set of nodes to open hubs and the set of edges to establish in order to find a network of minimum total cost. We consider the case where the network is required to form a Steiner tree spanning a given set of terminal vertices. When edge costs are non-metric, we show logarithmic approximation hardness even for the special case of spanning trees. On the other hand, we show a polynomial-time reduction for Steiner trees to its corresponding node-weighted version thus proving a logarithmic approximation factor. When edge costs are metric, we show the problem is only a constant factor harder to approximate than its original version (with no hub installation) using a similar reduction.

Let $a_1,\ldots,a_n$ be a permutation of $[n]$. Two disjoint order-isomorphic subsequences are called twins. We show that every permutation of $[n]$ contains twins of length $\Omega(n^{3/5})$ improving the trivial bound of $\Omega(n^{1/2})$. We also show that a random permutation contains twins of length $\Omega(n^{2/3})$, which is sharp.

We study rumor spreading in graphs, specifically multicommodity multicast problem under the wireless model: given source-destination pairs in the graph, one needs to find the fastest schedule to transfer information from each source to the corresponding destination. Under the wireless model, nodes can transmit to any subset of their neighbors in synchronous time steps, as long as they either transmit or receive from at most one transmitter during the same time step. We improve approximation ratio for this problem from O~(n^(2/3)) to O~(n^((1/2) + epsilon)) on n-node graphs. We also design an algorithm that satisfies p given demand pairs in O(OPT + p) steps, where OPT is the length of an optimal schedule, by reducing it to the well-studied packet routing problem. In the case where underlying graph is an n-node tree, we improve the previously best-known approximation ratio of O((log n)/(log log n)) to 3. One consequence of our proof is a simple constructive rule for optimal broadcasting in a tree under a widely studied telephone model.