return is_reachable_dfs build_graph (team_a team_b the time complexity and space complexity are both ( )where is the number of outcomes it' natural to use graph when the problem involves spatially connected objectse road segments between cities more generallyconsider using graph when you need to analyze any binary relationshipbetween objectssuch as interlinked webpagesfollowers in social graphetc in such casesquite often the problem can be reduced to well-known graph problem some graph problems entail analyzing structuree looking for cycles or connected components dfs works particularly well for these applications some graph problems are related to optimizatione find the shortest path from one vertex to another bfsdijkstra' shortest path algorithmand minimum spanning tree are examples of graph algorithms appropriate for optimization problems table top tips for graphs graph search computing vertices which are reachable from other vertices is fundamental operation which can be performed in one of two idiomatic waysnamely depth-first search (dfsand breadth-first search (bfsboth have linear time complexity-- (| | |to be precise in the worst-case there is path from the initial vertex covering all vertices without any repeatsand the dfs edges selected correspond to this pathso the space complexity of dfs is (| |(this space is implicitly allocated on the function call stackthe space complexity of bfs is also (| |)since in worst-case there is an edge from the initial vertex to all remaining verticesimplying that they will all be in the bfs queue simultaneously at some point dfs and bfs differ from each other in terms of the additional information they providee bfs can be used to compute distances from the start vertex and dfs can be used to check for the presence of cycles key notions in dfs include the concept of discovery time and finishing time for vertices paint boolean matrix let be boolean array encoding black-and-white image the entry (abcan be viewed as encoding the color at entry (abcall two entries adjacent if one is to the leftrightabove or below the other note that the definition implies that an entry can be adjacent to at most four other entriesand that adjacency is symmetrici if is adjacent to entry then is adjacent to define path from entry to entry to be sequence of adjacent entriesstarting at ending at with successive entries being adjacent define the region associated with point (ijto be all points ( such that there exists path from (ijto ( in which all entries are the same color in particular this implies (ijand ( must be the same color implement routine that takes an boolean array together with an entry (xyand flips the color of the region associated with (xysee figure for an example of flipping