Question

As the newly-hired commissioner of the Gotham City Police Department, James Gordon’s first act is to immediately fire all of the dirty cops, stamping out Gotham’s widespread police corruption. To do this, Commissioner Gordon must first figure out which officers are honest and which are dirty. There are n officers in the department. The majority (> n/2) of the officers are honest, and every officer knows whether or not each other officer is dirty. He will identify the dirty cops by asking the officers, in pairs, to indicate whether the other is dirty. Honest officers will 3 always answer truthfully, dirty cops may answer arbitrarily. Thus the following responses are possible:

Officer A "B is honest" "A is honest" Either both are honest or both are dirty "B is honest" "A is dirty" At least one is dirty "B is dirty" "A is honest" At least one is dirty "B is dirty" A is dirty" At least one is dirty Officer B Implication 1. A group of n officers is uncorrupted if more than half are honest. Suppose we start with an uncorrupted group of n officers. Describe a method that uses only In/2] pair-wise tests between officers to find a smaller uncor- rupted group of at most [n/21 officers. Prove that your method satisfies each of the three requirements. 2. Using this approach, devise an algorithm that identifies which officers are honest and which are dirty using only (n) pairwise tests. Prove the cor- rectness of your algorithm, and prove that only O(n) tests are used. Prove that a conspiracy of In/2| +1 dirty officers (who may share a plan) can foil any attempt to find a honest officer. L.e, not only will method above not work, but that there is no way at all for Commissioner Gordon to identify 3.

Answer 1

DATE Tering TcHr4.QUe-ウ deigmed to exece al pasible discnt Combinat all tha Co ns 分 NEST All oas n