Therefore if there is a majority element the algorithm outputs the majority

Therefore if there is a majority element the

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Therefore, if there is a majority element, the algorithm outputs the majority element. 2 (d) (3 pts.) Optional bonus problem: Check my proof (This is an optional bonus challenge problem. Only solve it if you can’t get enough of the algorithms goodness. We reserve the right not to grade bonus problems if necessary.) If G = ( V , E ) is an undirected graph and S V is a set of vertices, call the edge ( u , v ) good if one of u , v is in S and the other is not in S ; and call the set S nice if it makes at least half of the edges of the graph CS 170, Fall 2014, Sol 1 4
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good. We want an algorithm that, given an undirected graph G = ( V , E ) , outputs a nice set S . Consider the following algorithm and proof of correctness: Algorithm M AKE N ICE ( G ) : 1. Set S : = /0. 2. Loop until S is nice: 3. Let w be a vertex whose degree is maximal (resolve ties arbitrarily). 4. Set S : = S [ { w } . 5. Delete all edges that have w as an endpoint. 6. If S is nice (for the original graph), return S . Proof : Imagine keeping track of the number of good edges (from the original graph), as the algorithm executes. We will show that the number of good edges increases in each iteration of the loop. In particular, when we add w to S in step 4, every edge currently in the graph that has w as an endpoint will shift from non-good to good. (Why? Well, the algorithm immediately deletes all edges whenever one of their endpoints enters S , so whenever execution reaches step 3, all remaining edges connect two vertices in V \ S . In particular, all edges that haven’t been deleted yet are non-good. But the ones that have w as an endpoint will immediately become good when w is added to S .) Since we strictly increase the number of good edges each time we execute an iteration of the loop, eventually the number of good edges must exceed | E | / 2. 2 Is this proof of correctness valid? Write “Valid” or “Invalid.” If you write “Valid”, analyze the running time of the algorithm. If you write “Invalid”, explain what is wrong (list the first step in the reasoning that doesn’t follow). Solution: The proof is Invalid. The proof states correctly that: . . . when we add w to S in step 4, every edge currently in the graph that has w as an endpoint will shift from non-good to good. It then uses this to claim that: Since we strictly increase the number of good edges each time we execute an iteration of the loop, eventually the number of good edges must exceed | E | / 2. However, this claim does not follow. The problem is that there is no proof that the number of good edges is strictly increasing, only that in each iteration some edges become good. It ignores the fact that some edges may become non-good again. Indeed, an edge first become good when one of its vertices is added to S and goes back to being non-good when the other vertex is added to S .
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