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We consider an information dissemination problem where the root of anundirected graph constantly updates its information. The goal is to keep everyother node in the graph about the root as freshly informed as possible. Oursynchronous information spreading model uses telephone calls at each time step,in which any node can call at most one neighbor, thus forming a matching overwhich information is transmitted at each step. We introduce two problems inminimizing two natural objectives (Maximum and Average) of the latency of theroot's information at all nodes in the network.  After deriving a simple reduction from the maximum rooted latency problem tothe well-studied minimum broadcast time problem, we focus on the average rootedlatency version. We introduce a natural problem of finding a finite schedulethat minimizes the average broadcast time from a root. We show that any averagerooted latency induces a solution to this average broadcast problem within aconstant factor and conversely, this average broadcast time is within alogarithmic factor of the average rooted latency. Then, by approximating theaverage broadcast time problem via rounding a time-indexed linear programmingrelaxation, we obtain a logarithmic approximation to the average latencyproblem.  Surprisingly, we show that using the average broadcast time for averagerooted latency introduces this necessary logarithmic factor overhead even intrees. 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 treeswhere each vertex receives information from its parent in regular intervals. Onthe other side, we show how such well-behaved schedules approximate the optimalschedule within a constant factor.

Timeliness Through Telephones: Approximating Information Freshness in Vector Clock Models