匿名图中Pebble引导的近优寻宝
Pebble Guided Near Optimal Treasure Hunt in Anonymous Graphs
摘要：
研究了移动Agent在图中的寻宝问题。图中的节点是匿名的，并且度为$deg(v)$的任何节点$v$处的边被任意标记为$0,1,\ldots, deg(v)-1$。移动代理从一个节点开始，必须找到一个静止的对象，称为{ \it宝藏}，它位于一个未知的节点上，距离它的初始位置$D$。代理在到达宝藏所在的节点时找到宝藏。寻宝的{ \it time}定义为代理在找到宝藏之前访问的边数。代理人没有任何关于图或宝藏位置的先验知识。一个知道图、代理的初始位置和宝藏位置的预言家在图的节点上放置一些鹅卵石，每个节点最多一个鹅卵石，以引导代理走向宝藏。我们的目标是回答这个问题：什么是最快的寻宝算法，无论放置多少鹅卵石？我们展示了一个算法，使用$O(D \log Δ)$卵石找到宝藏在图$G$中的时间$O(D \log Δ+ \log^3 Δ)$，其中$Δ$是最大程度的节点在$G$和$D$是从初始位置的代理到宝藏的距离。我们显示了一个几乎匹配的下限$Ω(D \log Δ)$的寻宝时间使用任何数量的鹅卵石。
英文摘要：
We study the problem of treasure hunt in a graph by a mobile agent. The nodes in the graph are anonymous and the edges at any node $v$ of degree $deg(v)$ are labeled arbitrarily as $0,1,\ldots, deg(v)-1$. A mobile agent, starting from a node, must find a stationary object, called {\it treasure} that is located on an unknown node at a distance $D$ from its initial position. The agent finds the treasure when it reaches the node where the treasure is present. The {\it time} of treasure hunt is defined as the number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An Oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure. We target to answer the question: what is the fastest possible treasure hunt algorithm regardless of the number of pebbles are placed? We show an algorithm that uses $O(D \log Δ)$ pebbles to find the treasure in a graph $G$ in time $O(D \log Δ+ \log^3 Δ)$, where $Δ$ is the maximum degree of a node in $G$ and $D$ is the distance from the initial position of the agent to the treasure. We show an almost matching lower bound of $Ω(D \log Δ)$ on time of the treasure hunt using any number of pebbles.
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