I have been trying this dp problem . But couldn't figure out the recurrence relation. The problem description is given below:
Problem Statement:
A long, linear field has N (1 <= N <= 1,000) clumps of grass at unique integer locations on what will be treated as a number line.Think of the clumps as points on the number line.
Bessie starts at some specified integer location L on the number line (1 <= L <= 1,000,000) and traverses the number line in the two possible directions (sometimes reversing her direction) in order to reach and eat all the clumps. She moves at a constant speed (one unit of distance in one unit of time), and eats a clump instantly when she encounters it.
Clumps that aren't eaten for a while get stale. We say the "staleness" of a clump is the amount of time that elapses from when Bessie starts moving until she eats a clump. Bessie wants to minimize the total staleness of all the clumps she eats.
Find the minimum total staleness that Bessie can achieve while eating all the clumps.
Input:
- Line 1 : Two space-separated integers: N and L.
- Lines 2..N+1: Each line contains a single integer giving the position P of a clump (1 <= P <= 1,000,000).
Output:
- Line 1: A single integer: the minimum total staleness Bessie can achieve while eating all the clumps.
Sample Input:
4 10
1
9
11
19
Sample output:
44
Hint:
INPUT DETAILS: Four clumps: at 1, 9, 11, and 19. Bessie starts at location 10.
OUTPUT DETAILS: Bessie can follow this route:
- start at position 10 at time 0
- move to position 9, arriving at time 1
- move to position 11, arriving at time 3
- move to position 19, arriving at time 11
- move to position 1, arriving at time 29
First of all, it should be clear that Bessie should eat a clump of grass as soon as she reaches it. Now assume Bessie starts between clumps i and i + 1. Then we can create a dp state dp[l][r][k] where l is the index of the leftmost clump of grass she hasn't eaten, and r is similarly defined. The k represents Bessie's current location (0 if on the leftmost clump of grass, 1 otherwise). Transitions are clearly O(1) so this solution works.
The main idea is correct, though there's still a very crucial (and hard to spot) detail missing here.
The answer is the sum of all times you eat the clumps, so you can think of it this way: Every minute that passes, values of all uneaten clumps increase by one, and the answer is the final value of all the clumps. So actually, every minute that passes contributes u to the final answer, where u is the amount of uneaten clumps. If we are at state l, r, k, then u = N - (r - l + 1).
The transitions should be clear.
can you please write out the transition equation?
Edit: I have understood the solution. Thank you very much :)