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Exploring BFS Trees: Properties, Claims, and Edge Types

Learn about BFS trees, layering in graphs, edge types, ancestor relationships, and graph algorithms with practice problems included.

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Exploring BFS Trees: Properties, Claims, and Edge Types

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  1. CS330 Discussion 5 Spring 2017

  2. BFS Tree • In lecture, we discussed a DFS tree. Given a DFS run on graph edge is in the DFS tree if is first arrived at using . • We can define a BFS tree using the exact same definition.

  3. Properties of BFS trees • When we run BFS and get a BFS tree, we can create a layering on the graph. • We start by saying layer 0 consists of the root of the BFS. • Then, layer 1 is all vertices which are adjacent to the root. • Layer 2 is all vertices adjacent to vertices in layer 1, except the root. • Continuing the pattern, layer is all vertices adjacent to vertices in layer , that do not appear in any previous layers.

  4. Properties of BFS trees • Claim: Let be an edge in a directed graph. If we run a BFS and split the vertices into layers, the layer of cannot exceed the layer of by more than 1. • Why? • What’s the equivalent claim for an undirected graph?

  5. Properties of DFS trees • is an ancestor of in a DFS/BFS tree if is on the path in the tree from the root of the DFS/BFS to . • Claim: If is an edge in an undirected graph, then in any DFS either is an ancestor of , or is an ancestor of . • Why? • Is there an equivalent claim for a directed graph?

  6. Edge types in a DFS/BFS tree • Given a DFS/BFS tree, we can define different types of edges. • First, all edges which are a part of the DFS/BFS tree are referred to as tree edges.

  7. Edge types in a DFS/BFS tree • Edges may exist in the graph but not be included in the DFS/BFS tree (i.e. not be tree edges). These edges are split into three types. • is a forward edge if is an ancestor of • is a back edge if is an ancestor of • is a cross edge if is not an ancestor of and is not an ancestor of

  8. Edge types in a DFS/BFS tree • Consider a directed graph, and a DFS/BFS for the graph. • For both DFS and BFS, if the graph has edges then clearly we will always have tree edges. Whether the other types will exist is not so apparent. • In a DFS, can we have a cross edge? A forward edge? A back edge? • In a BFS, can we have a cross edge? A forward edge? A back edge?

  9. A note on undirected graphs • We can also define tree, forward, back, and cross edges for DFS/BFS on a directed graph. • However, the definition of forward/back becomes a bit less clear since the edge does not have a direction. • To resolve this, we’ll assign each edge whichever direction the DFS/BFS first explores that edge in. • So if we try to visit from before visiting from in the DFS/BFS, we’ll treat the edge as being directed from to .

  10. Graph Algorithm Practice Problem 1 • You are given a directed graph with unweighted edges, as well as two vertices and a subset of vertices which does not include either or . • Write an algorithm to find the shortest path from to that passes through at least one vertex in .

  11. Graph Algorithm Practice Problem 2 • You are given a graph with unweighted edges, and two disjoint subsets of , and • Write an algorithm to determine the shortest path from any vertex in to any vertex in . Also provide its runtime.

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