A phone company divides a city into n cells C, C2,..., C. In each cell it...

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A phone company divides a city into n cells C₁, C2,..., C₁. In each cell it has a tower. When a call comes to a mobile user the company has a set of probabilities P₁, P2, ..., Pn such that the user is now at cell Ci with Pi probability. The company wants to activate as few towers as possible on average to find the user. When it activates a tower in a cell where the user is, the search stops. Search time is divided into d slots. By the end of d slots the user must be found. The question the company faces is what are the towers (cells) to activate in slot 1, 2, ..., d as to minimize the expected number of activations. More specifically, the company wants a policy which is a collection of d sets S₁, S2, ..., Sa where S; contains towers (cells) to be activated at slot i. = For example, let n = 5 and d = 3, a policy might be S₁ {₁}, S₂ = {C₂, C4), S3 = {C3, C5}. The expected number of activation for this policy is 1+2 (1-p₁) +2 (1-P₁-P2-P4) (a) What is the optimal policy if p; = for all i and d = 2. Prove it to be optimal. [5 pts] n (b) Prove that in an optimal policy, S₁ is a set of cells which is a prefix of the non-increasing sorted order of the cells according to their probability. [5 pts] (c) Design a polynomial time algorithm to calculate the optimal policy in general. Justify the correctness of your algorithm and it's time complexity. [10 pts] Activate Windows Go to Settings to activate Windows. A phone company divides a city into n cells C₁, C2,..., C₁. In each cell it has a tower. When a call comes to a mobile user the company has a set of probabilities P₁, P2, ..., Pn such that the user is now at cell Ci with Pi probability. The company wants to activate as few towers as possible on average to find the user. When it activates a tower in a cell where the user is, the search stops. Search time is divided into d slots. By the end of d slots the user must be found. The question the company faces is what are the towers (cells) to activate in slot 1, 2, ..., d as to minimize the expected number of activations. More specifically, the company wants a policy which is a collection of d sets S₁, S2, ..., Sa where S; contains towers (cells) to be activated at slot i. = For example, let n = 5 and d = 3, a policy might be S₁ {₁}, S₂ = {C₂, C4), S3 = {C3, C5}. The expected number of activation for this policy is 1+2 (1-p₁) +2 (1-P₁-P2-P4) (a) What is the optimal policy if p; = for all i and d = 2. Prove it to be optimal. [5 pts] n (b) Prove that in an optimal policy, S₁ is a set of cells which is a prefix of the non-increasing sorted order of the cells according to their probability. [5 pts] (c) Design a polynomial time algorithm to calculate the optimal policy in general. Justify the correctness of your algorithm and it's time complexity. [10 pts] Activate Windows Go to Settings to activate Windows.