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Graph Algorithms

Breadth-First Search (BFS)

     

Use the breadth first strategy to traverse the graph below

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List the vertices in the order in which breadth first search traverses them.

  • Start from the node \(1\).
  • With the choice of two children, visit first the one of higher value.

Given a complete binary tree \(T\) with \(n\) nodes, which of the following best describes the time it takes to find a path using breadth first search from a root \(V\) to another node \(s\) (\(s \in T\))?

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Consider the above tree. Using breadth first search, we can trace the path \(P\) from any node \(S\) to any other node \(V\). For example the path from \(S=1\) to \(V=15\) is \(\{1,2,15\}\).

Let us define a function \(sP(S,V)\) that returns the sum of the nodes in the path \(P\). So for \(S=1\) and \(V=15\), \(sP(1,15)=sum(\{1,2,15\})=18\).

Find the value of \(\sum_{i=1}^{12} sP(1,v_i)\) where \(v_i\) is the \(i\)th node in the tree and \(v_i \neq 1\).

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