Deterministic Graph Exploration with Advice

We consider the task of graph exploration. An $n$-node graph has unlabeled nodes, and all ports at any node of degree $d$ are arbitrarily numbered $0,\dots, d-1$. A mobile agent has to visit all nodes and stop. The exploration time is the number of edge traversals. We consider the problem of how much knowledge the agent has to have a priori, in order to explore the graph in a given time, using a deterministic algorithm. This a priori information (advice) is provided to the agent by an oracle, in the form of a binary string, whose length is called the size of advice. We consider two types of oracles. The instance oracle knows the entire instance of the exploration problem, i.e., the port-numbered map of the graph and the starting node of the agent in this map. The map oracle knows the port-numbered map of the graph but does not know the starting node of the agent. We first consider exploration in polynomial time, and determine the exact minimum size of advice to achieve it. This size is $\log\log\log n -\Theta(1)$, for both types of oracles. When advice is large, there are two natural time thresholds: $\Theta(n^2)$ for a map oracle, and $\Theta(n)$ for an instance oracle, that can be achieved with sufficiently large advice. We show that, with a map oracle, time $\Theta(n^2)$ cannot be improved in general, regardless of the size of advice. We also show that the smallest size of advice to achieve this time is larger than $n^\delta$, for any $\delta <1/3$. For an instance oracle, advice of size $O(n\log n)$ is enough to achieve time $O(n)$. We show that, with any advice of size $o(n\log n)$, the time of exploration must be at least $n^\epsilon$, for any $\epsilon <2$, and with any advice of size $O(n)$, the time must be $\Omega(n^2)$. We also investigate minimum advice sufficient for fast exploration of hamiltonian graphs

Pebble guided Treasure Hunt in Plane

We study the problem of treasure hunt in a Euclidean plane by a mobile agent with the guidance of pebbles. The initial position of the agent and position of the treasure are modeled as special points in the Euclidean plane. The treasure is situated at a distance at most $D>0$ from the initial position of the agent. The agent has a perfect compass, but an adversary controls the speed of the agent. Hence, the agent can not measure how much distance it traveled for a given time. The agent can find the treasure only when it reaches the exact position of the treasure. The cost of the treasure hunt is defined as the total distance traveled by the agent before it finds the treasure. The agent has no prior knowledge of the position of the treasure or the value of $D$. An Oracle, which knows the treasure's position and the agent's initial location, places some pebbles to guide the agent towards the treasure. Once decided to move along some specified angular direction, the agent can decide to change its direction only when it encounters a pebble or a special point. We ask the following central question in this paper: ``For given $k \ge 0$, What is cheapest treasure hunt algorithm if at most $k$ pebbles are placed by the Oracle?" We show that for $k=1$, there does not exist any treasure hunt algorithm that finds the treasure with finite cost. We show the existence of an algorithm with cost $O(D)$ for $k=2$. For $k>8$ we have designed an algorithm that uses $k$ many pebbles to find the treasure with cost $O(k^{2}) + D(\sin\theta' + \cos\theta')$, where $\theta'=\frac{\pi}{2^{k-8}}$. The second result shows the existence of an algorithm with cost arbitrarily close to $D$ for sufficiently large values of $D$

The effect of non-sulphide gangue minerals on the flotation of sulphide minerals from Carlin-type gold ores

Flotation pre-concentration of sulphide and gold values from certain Carlin-type deposits characterised as double-refractory gold ores is quite challenging. Numerous studies conducted on these ores in many laboratories globally (including the present study) under a variety of chemical and physical conditions have merely confirmed low recovery (and poor concentrate grades) for sulphide minerals and gold, and poor separation between sulphide minerals and carbonaceous matter, even when the valuable minerals are adequately liberated. None of the traditional reasons based on liberation or the choice of chemical and physical conditions and separation strategies could provide satisfactory explanation for the observed poor separation. In this study, the focus was on the role of non-sulphide gangue (NSG) minerals. It was hypothesised that specific NSG minerals have a detrimental effect on flotation recovery of gold bearing minerals and their separation selectivity. In order to test this hypothesis and delineate the effect of the various gangue minerals, a new approach was taken. This involved first isolating the various mineral components of a double-refractory gold ore from one of the Carlin-type deposits using a gravity separation technique. Then flotation experiments were performed using a mixture design on various mixtures of these isolated components under controlled conditions. The results of these mixture experiments supported the hypothesis and demonstrated, for the first time for these types of ores, that even small amounts of NSG minerals, especially carbonaceous matter and clays, had a large adverse effect on the flotation of sulphides and selectivity of separation. While it is tempting to attribute the observed effects solely to slime coating, there is no basis to do so at this stage; it is more reasonable to propose that multiple contributions exist. The results of this study provide the much-needed context and direction for further fundamental studies and for developing processing strategies

Moving and Computing Models: Agents

International audienc