Depth first: Optimal path discovery between designated nodes in random ring-based graphs

This paper focuses on optimally determining the existence of connected paths between some given nodes in random ring-based graphs. Serving as a fundamental underlying structure in network modeling, ring topology appears as commonplace in many realistic scenarios. Regarding this, we consider graphs composed of rings, with some possible connected paths between them. Without prior knowledge of the exact node permutations on rings, the existence of each edge can be unraveled through edge testing at a unit cost in one step. The problem examined is that of determining whether the given nodes are connected by a path or separated by a cut, with the minimum expected costs involved. Dividing the problem into different cases based on different topologies of the ring-based networks, we propose the corresponding policies that aim to quickly seek the paths between nodes. A common feature shared by all those policies is that we stick to going in the same direction during edge searching, with edge testing in each step only involving the test between the source and the node that has been tested most. The simple searching rule, interestingly, can be interpreted as a delightful property stemming from the neat structure of ring-based networks, which makes the searching process not rely on any sophisticated behaviors. We prove the optimality of the proposed policies by calculating the expected cost incurred and making a comparison with the other class of strategies. The effectiveness of the proposed policies is also verified through extensive simulations, from which we even disclose three extra intriguing findings: i) in a onering network, the cost will grow drastically with the number of designated nodes when the number is small and will grow slightly when that number is large; ii) in ring-based network, Depth First is optimal in detecting the connectivity between designated nodes; iii) the problem of multi-ring networks shares large similarity with that of two-ring networks, and a larger number of ties between rings will not influence the expected cost.