Treasure hunt is the task of finding an inert target by a mobile agent in an unknown environment. We consider treasure hunt in geometric terrains with obstacles. Both the terrain and the obstacles are modeled as polygons and both the agent and the treasure are modeled as points. The agent navigates in the terrain, avoiding obstacles, and finds the treasure when there is a segment of length at most 1 between them, unobstructed by the boundary of the terrain or by the obstacles. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the amount of information that the agent needs a priori in order to find the treasure at cost $O(L)$, where $L$ is the length of the shortest path in the terrain from the initial position of the agent to the treasure, avoiding obstacles. Following the paradigm of algorithms with advice, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the whole environment: the terrain, the position of the treasure and the initial position of the agent. Advice complexity of treasure hunt is the minimum length of the advice string (up to multiplicative constants) that enables the agent to find the treasure at cost $O(L)$. We first consider treasure hunt in regular terrains which are defined as convex polygons with convex $c$-fat obstacles, for some constant $c>1$. A polygon is $c$-fat if the ratio of the radius of the smallest disc containing it to the radius of the largest disc contained in it is at most $c$. For the class of regular terrains, we establish the exact advice complexity of treasure hunt. We then show that advice complexity of treasure hunt for the class of arbitrary terrains (even for non-convex polygons without obstacles, and even for those with only horizontal or vertical sides) is exponentially larger than for regular terrains.

中文翻译：

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几何地形中寻宝的建议复杂性

寻宝是移动代理在未知环境中寻找惰性目标的任务。我们考虑在有障碍物的几何地形中寻宝。地形和障碍物都被建模为多边形，代理和宝藏都被建模为点。智能体在地形中导航，避开障碍物，当它们之间最多有一段长度为1，不受地形边界或障碍物的阻挡时，找到宝藏。寻宝的代价就是代理的轨迹长度。我们调查了代理需要先验的信息量，以便以 $O(L)$ 的成本找到宝藏，其中 $L$ 是从代理的初始位置到地形中最短路径的长度。珍惜，避障。遵循带有建议的算法范式，这些信息作为二进制字符串预先提供给代理，由与代理合作并了解整个环境的预言机：地形、宝藏位置和代理的初始位置. 寻宝的建议复杂度是使代理能够以 $O(L)$ 的成本找到宝藏的建议字符串的最小长度（最多为乘法常数）。我们首先考虑在规则地形中寻宝，这些地形被定义为具有凸 $c$-fat 障碍物的凸多边形，对于某些常数 $c>1$。如果包含多边形的最小圆盘的半径与包含在其中的最大圆盘的半径之比至多为 $c$，则多边形为 $c$-fat。对于常规地形类，我们建立了寻宝的确切建议复杂度。