Question

Giles has asked Buffy to optimize her procedure for nighttime patrol of Sunnydale. (This takes place sometime in Season 2.) Giles points out that proper slaying technique would allow Buffy to traverse all of the streets of Sunnydale in such a way that she would walk on each side of each street, exactly once, going up the street in one direction and down the street in the other direction. Buffy now has slayer homework: how can it be done? (If you have to assume anything about the layout of the city of Sunnydale, make it clear!).

Answer 1

Assumptions for Sunnydale- Sunnydale is not some kind of multi-level city and any elevations can be flattened out into a 2D road map that can be abstracted as an undirected, unweighted, and connected graph. The roads of the city are the edges of the graph and the intersections (3-way and 4-way) and cul-de-sacs are the nodes of the graph.

Thus, it becomes Buffy’s job to traverse the entire graph and visit every edge exactly once, so that Buffy can take an exactly reversed path to travel the other side for the second time. For any unweighted, connected and undirected graph, there exists a path to every node exactly once because of the existence and operation of the DFS traversal algorithm. So, to start with, one can perform this search with dynamic programming, starting at any node to create most of the journey. While visiting each node, treat the recursive call as walking one side on a street, and treat the eventual return as walking the other side of the street. This provides the bulk of a forward path and a reverse path to traverse every edge. The only issue is that in the normal working of DFS with dynamic programming, the search function is only called on nodes that have not been visited, so there are edges that would typically be skipped so that the nodes are visited once only. However, in our real-world scenario, we can instruct Buffy that, for each node 9intersections in this case) that have been visited already, Buffy needs to walk to it and immediately return, such that although there may be intersections that will be visited multiple times, each street is walked down in both directly exactly one time. In an abstract sense, this forms a path that includes all of the edges that are normally excluded when using DFS to create the DFS tree via the standard creation of an adjacency list or matrix.

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