Spatial Search Algorithms
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An interesting problem in the area of algorithms is the spatial search problem, which aims to find one or more marked points in a finite physical region that can be modeled by a graph, for instance, a two-dimensional finite lattice, so that the vertices of the graph are the places one can search and the edges are the pathways one can use to move from one vertex to an adjacent one. The quantum version of this problem was analyzed by Benioff in a very concrete way. He imagined a quantum robot that moves to adjacent vertices in a time unit. The position of the robot can be a superposition of a finite number of places (vertices). How many steps does the robot need to take in order to find a marked vertex with high probability? In this problem, we suppose that the robot only finds the marked vertex by stepping on it and the robot has no hint about the direction of the marked vertex and no compass and no memory.