GE330_ps_DP_poisson - Problem 1(Dynamic Programming The...

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Problem 1. (Dynamic Programming) The residents of the underground city of Zion defend themselves through a combination of kung fu, heavy artillery, and efficient algorithms. Recently they have become interested in automated methods that can help fend off attacks by swarms of robots. Here’s what one of these robot attacks looks like. A swarm of robots arrives over the course of n seconds; in the i th second, x i robots arrive. Based on remote sensing data, you know this sequence x 1 , x 2 , · · · , x n in advance. You have at your disposal an electromagnetic pulse (EMP), which can destroy some of the robots as they arrive; the EMP’s power depends on how long it’s been allowed to charge up. To make this precise, there is a function f ( · ) so that if j seconds have passed since the EMP was last used, then it is capable of destroying up to f ( j ) robots. So specifically, if it is used in the k th second, and it has been j seconds since it was previously used, then it will destroy min( x k , f ( j )) robots. (After this use, it will be completely drained.) We will also assume that the EMP starts off completely drained, so if it is used for the first time in the j th second, then it is capable of destroying up to f ( j ) robots. The problem. Given the data on robot arrivals x 1 , x 2 , · · · , x n , and given the recharging function f ( · ),
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