Backtracking Problem in the Traversal of an Unknown Directed Graph by a Finite Robot
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A covering path in a directed graph is a path passing through all vertices and arcs of the graph, with each arc being traversed only in the direction of its orientation. A covering path exists for any initial vertex only if the graph is strongly connected. The traversal of an unknown graph implies that the topology of the graph is not a priori known, and we learn it only in the course of traversing the graph. This is similar to the problem of traversing a maze by a robot in the case where the plan of the maze is not available. If the robot is a “general-purpose computer” without any limitations on the number of its states, then traversal algorithms with the estimate O(nm) are known, where n is the number of vertices and m is the number of arcs. If the number of states is finite, then this robot is a finite automaton. Such a robot is an analogue of the Turing machine, where the tape is replaced by a graph and the cells are assigned to the graph vertices and arcs. The selection of the arc that has not been traversed yet among those originating from the current vertex is determined by the order of the outgoing arcs, which is a priori specified for each vertex. The best known traversal algorithms for a finite robot are based on constructing the output directed spanning tree of the graph with the root at the initial vertex and traversing it with the aim to find all untraversed arcs. In doing so, we face the backtracking problem, which consists in searching for all vertices of the tree in the order inverse to their natural partial ordering, i.e., from the leaves to the root. Therefore, the upper estimate of the algorithms is different from the optimal estimate O(nm) by the number of steps required for the backtracking along the outgoing tree. The best known estimate O(nm + n2 log log n) has been suggested by the author in the previous paper . In this paper, a finite robot is suggested that performs a backtracking with the estimate O(n2 log*(n)). The function log* is defined as an integer solution of the inequality 1 ≤ log2log*(n) < 2, where log t = log º log º ... º log (the superposition º is applied t – 1 times) is the tth compositional degree of the logarithm. The estimate O(nm + n2 log*(n)) for the covering path length is valid for any strongly connected graph for a certain (unfortunately, not arbitrary) order of the outgoing arcs. Interestingly, such an order of the arcs can be marked by symbols of the finite robot traversing the graph. Hence, there exists a robot that traverses the graph twice: first traversal with the estimate O(nm + n2 log log n) and the second traversal with the estimate O(nm + n2 log*(n)).
KeywordsDirect Graph Turing Machine Integer Solution Finite Automaton Initial Vertex
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