Competitive searching in a generalized street

We consider the problem of a robot which has to find a path in an unknown simple polygon from one point <italic>s</italic> to another point <italic>t</italic>, based only on what it has seen so far. A <italic>Street</italic> is a polygon for which the two boundary chains from <italic>s</italic> to <italic>t</italic> are mutually weakly visible, and the set of streets was the only class of polygons for which a competitive search algorithm was known. We define a new, strictly larger class of polygons, called <italic>generalized streets</italic> or <inline-equation> <f> <sc>G</sc></f> </inline-equation>-streets which are characterized by the property that every point on the boundary of a <inline-equation> <f> <sc>G</sc></f> </inline-equation>-street is visible from a point on a horizontal line segment connecting the two boundary chains from <italic>s</italic> to <italic>t</italic>. We present an on-line strategy for a robot placed at <italic>s</italic> to find <italic>t</italic> in an unknown rectilinear <inline-equation> <f> <sc>G</sc></f> </inline-equation>-street; the length of the path created is at most 9 times the length of the shortest path in the <italic>L</italic><subscrpt>1</subscrpt> metric. This is optimal since we show that no strategy can achieve a smaller competitive factor for all rectilinear <inline-equation> <f> <sc>G</sc></f> </inline-equation>-streets. Compared to the <italic>L</italic><subscrpt>2</subscrpt>-shortest path, the strategy is 9.06-competitive which leaves only a very small gap to the lower bound of 9.