Problem 2: Optimal Milking [Brian Dean, 2013]
Farmer John has recently purchased a new barn containing N milking machines (1 <= N <= 40,000), conveniently numbered 1..N and arranged in a row.
Milking machine i is capable of extracting M(i) units of milk per day (1 <= M(i) <= 100,000). Unfortunately, the machines were installed so close together that if a machine i is in use on a particular day, its two neighboring machines cannot be used that day (endpoint machines have only one neighbor, of course). Farmer John is free to select different subsets of machines to operate on different days.
Farmer John is interested in computing the maximum amount of milk he can extract over a series of D days (1 <= D <= 50,000). At the beginning of each day, he has enough time to perform maintenance on one selected milking machine i, thereby changing its daily milk output M(i) from that day forward. Given a list of these daily modifications, please tell Farmer John how much milk he can produce over D days (note that this number might not fit into a 32-bit integer).
The first line gives the values of N and D.
The next N lines each contain one integer, the initial value of M(i).
The next D lines each contain two integers i and m, indicating that Farmer John updates the value of M(i) to m at the beginning of day d.
There are 5 machines, with initial milk outputs 1,2,3,4,5. On day 1, machine 5 is updated to output 2 unit of milk, and so on.
Output a single integer, the maximum total amount of milk FJ can produce over D days.
On day one, the optimal amount of milk is 2+4 = 6 (also achievable as 1+3+2). On day two, the optimal amount is 7+4 = 11. On day three, the optimal amount is 10+3+2=15.
Problem obtained from USACO December Contest 2013, Gold.