# Dynamic Ring Exploration with \((H,s)\) View +
### Related Works  
To see various settings and exploration algorithms on static networks, there is a good survey [2], for example, a network with distinct node labeling or without node labeling (an anonymous network), exploration with termination or perpetual exploration, and from the point of the number of agents, exploration by a single agent or by multiple agents.  
The literature of dynamic networks are surveyed in [1; 16].  
The recent works about mobile agents (or robots) on dynamic networks are summarized in [17] including exploration; _gathering_ on 1-interval connected rings [13] which requires all the agents to gather at one node or at adjacent two nodes; _dispersion_[18] which stipulates that every node must be occupied by exactly one agent where the number of agents is the same as that of nodes on _permuting rings_ in which the nodes may be permuted at each time step, i.e., the neighbors of a node may change at each time step while the topologies are rings or paths at each time step.  
The following works consider the exploration of dynamic networks by multiple agents (or robots) without the knowledge of a link scheduling (or only with the ability to detect whether the adjacent links are present or not at the moment). In [10], the exploration for 1-interval connected rings is considered. The _perpetual exploration_ (i.e., the exploration without termination) on _connected-over-time rings_ is considered in [8; 9]. In [11], the perpetual exploration on two kinds of temporal networks with arbitrary footprints is considered: _connected-over-time graphs_ and 1-_interval connected graphs with bounded missing links_. The difference between with or without the ability to detect whether the adjacent links are present or not at the moment (called the _link presence detection_) is considered in [12] for an \(n\times m\)_dynamic torus_ which consists of \(n\) horizontal rings and \(m\) vertical rings each of which is a 1-intervalconnected ring. It is shown that the minimum number of agents with the link presence detection to explore the networks is a half of the minimum number of agents without the one to explore.  
The following works consider the exploration of dynamic networks by an agent with the full knowledge of a link scheduling, i.e., the information about when and which links disappear or appear. In [7], the exploration is considered on _\(T\)-interval connected rings_ where at most one link is missing at each time step and for any \(T\) successive time steps, there exists a common spanning connected component. It is shown that the optimal exploration time is \(2n-3\) when \(T=1\) where \(n\) is the network size. In [6], the exploration on _\(1\)-interval connected cactuses_ is considered. They show that the graphs can be explored in \(2^{O(\sqrt{\log n})}\) time which is much less than the known upper bound for the general graph, \(O(n^{2})\). In [3], the authors reveal the existence of the \(1\)-interval connected graphs which have the exploration time \(\Omega(n^{2})\), proving the exploration time of arbitrary \(1\)-interval connected graphs is \(\Theta(n^{2})\). In [5], it is shown that when the maximum degree at each time step is upper-bounded by \(d\), the exploration time is reduced to \(O((dn^{2}\log d)/\log n)\). In [4], the authors prove that the exploration time is reduced to \(O(n^{1.75})\) if an agent can move two hops in each time step.  
Other problems are also considered on dynamic networks; _patrolling_ on \(1\)-interval connected rings [14] which requires the maximum length of the interval between two visits to a node to be minimized; _compacting_ on \(1\)-interval connected rings [15] which stipulates that all the agents in a network must be located in a continuous part of the ring and at each node there exists at most one agent.