Shady spots you are going on a very long run of r

Shady spots You are going on a very long run of R miles. Luckily, there are some shady spots along the
way at which you can stop to rest, at distances d
1
< d
2
< · · · < d
n
from the starting point. Some of
these have more shade than others: the benefit you get from stopping at the i th spot is b
i
.
You would like to plan your stops so that you get as much total benefit as possible, subject to the
constraint that you run at least D miles between stops (and also D miles before your first stop, and
also D miles between your last stop and the final destination).
For example, suppose R = 10, D = 3, and there are n = 6 stops at distances d [1 : 6] = [2 , 3 , 4 , 6 , 7 , 8]
with benefits b [1 : 6] = [10 , 10 , 30 , 5 , 30 , 40]. Then the optimal solution is to stop at locations 3 , 5, with
total benefit 60.
(a) [2 pts] Write down a formal specification of the problem, by writing down the following:
1. Instance
2. Solution format
3. Constraints
4. Objective function
Solution:
(b) [2 pts] Here’s a subproblem that can be used for a dynamic programming solution: for 1 ≤ i ≤ n ,
define
B [ i ] = optimal benefit if you start at position d [ i ] .
For instance, in the example above, B [5] = 0 since if you begin at position d [5] = 7, then you are not
allowed to make any stops before the run is over.
For convenience, also define B [0] to be the optimal benefit starting at position 0.
Give the full array B [0 : 6] for the example above.
Solution:
(c) [1 pt] For a general instance with n resting spots, in what order should the subproblems
B [0] , B [1] , . . . , B [ n ] be solved?
Solution:
(d) [4 pts] Give a rule by which the answer to any subproblem B [ i ] can be determined once answers
are known for ”smaller” subproblems (in terms of the ordering from part (c)). Write down a dynamic
programming algorithm that implements this rule and returns the optimal score. (You do not need to
return the chosen locations.)
Solution:
(e) [1 pt] What is the running time of your algorithm? (No need to justify.)
Page 9

Drink breaks You are going on a bike ride that is a total of M miles. You can ride at most S miles without
taking a snack break. Luckily, there are some snack shops along the way, at distances m
1
< m
2
<
· · · < m
n
from the starting point. Buying a drink at the i th shop costs d
i
.
At which snack shops should you stop so as to complete the ride while paying as little as possible?
You may assume that there is a solution: that is, the distance between consecutive shops is at most S
and m
1
≤ S and M - m
n
≤ S .
For example, suppose M = 100, S = 30, and there are n = 6 snack shops at distances m [1 : 6] =
[20 , 30 , 40 , 60 , 70 , 80] with costs d [1 : 6] = [10 , 50 , 15 , 25 , 30 , 50]. Then the optimal solution is to stop at
locations 1 , 3 , 5, with total cost 55.
(a) [2 pts] Write down a formal specification of the problem, by writing down the following:
1. Instance
2. Solution format
3. Constraints
4. Objective function
Solution: