Problem 45470. Count the number of reaction chains achievable in T mins
This problem is related to Problem 45467.
Let's denote a list of N compounds as 1, 2, ..., N. You are then given a list of valid reactions for converting one compound to another (e.g. 1 > 2), as well as the time it takes to complete them ( completion time ). With this information, we can generate reaction chains. A reaction chain is a series of valid reaction steps taken one after the other. Examples are given below:
Given N = 4 and the following valid reactions: Reaction 1: 1 > 2 takes 1.5 mins Reaction 2: 1 > 3 takes 2.5 mins Reaction 3: 2 > 3 takes 0.6 mins Reaction 4: 3 > 4 takes 4.1 mins Reaction 5: 4 > 2 takes 3.2 mins Sample reaction chains: 1 > 3 > 4 takes (2.5 + 4.1) mins 1 > 2 > 3 > 4 takes (1.5 + 0.6 + 4.1) mins 4 > 2 > 3 takes (3.2 + 0.6) mins
Note that conversion reactions can only move forward. But if the list states that converting to and from the same two compounds is possible, then a reaction chain can take only one of these paths.
Your task is this: Given a starting compound S, can you count how many reaction chains are possible whose total completion time does not exceed T mins?
Note that if multiple valid reaction steps are possible between two compounds (e.g. "1 > 2 takes 1.5 mins" and "1 > 2 takes 2.5 mins" both appear in the list), then reaction chains that take either of these paths are distinct. Lastly, for this problem, reaction chains must not contain duplicate compounds; otherwise, they become "cycles" rather than "chains".
The inputs to this problem are R, S, and T. The meanings of S and T are given above. Variable R is a 3column matrix containing the list of valid reaction steps at each row i:
"Reaction i: R( i, 1) > R( i, 2) takes R( i, 3) mins"
Output the required number of reaction chains. You are ensured that:
 N and T are integers satisfying: 2 <= N <= 20 and 1 <= T <= 100.
 S and all elements in the first 2 columns of R are integers within [1, N].
 Completion times are decimal numbers within (0,10].
 Each compound 1, ..., N is mentioned at least once in R. Hence, N can be inferred from matrix R.
The following sample test case is the one illustrated above:
>> R = [1 2 1.5; 1 3 2.5; 2 3 0.6; 3 4 4.1; 4 2 3.2]; >> reaction_chain2(R,1,5) ans = 3 >> reaction_chain2(R,1,10) ans = 6
In this example, all the reaction chains starting from Compound 1 are:
(1.50 mins) 1 > 2 (2.10 mins) 1 > 2 > 3 (6.20 mins) 1 > 2 > 3 > 4 (2.50 mins) 1 > 3 (6.60 mins) 1 > 3 > 4 (9.80 mins) 1 > 3 > 4 > 2
Solution Stats
Problem Comments
Solution Comments
Show commentsProblem Recent Solvers13
Suggested Problems

Remove all the words that end with "ain"
2080 Solvers

2086 Solvers

Selfsimilarity 1  Every other term
53 Solvers

Selfsimilarity 3  Every other pair of terms
44 Solvers

158 Solvers
More from this Author19
Problem Tags
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!