Question

Alice invents a key (s₁, s₂, s₃, … , s_k). Bob makes a guess (g₁, g₂, g₃, … , g_k).He is awarded one point for each s_i = g_i.
Each s1 is an integer with the range of 0<=s_i<=11.
Given a q guesses with their scores b_i

(g₁, g₂, g₃, … , g_k) b₁
(g₁, g₂, g₃, … , g_k) b₂
.
.
.
(g₁, g₂, g₃, … , g_k) b_q

Can you state if there is a key possible. Given 0<=si<=11, 1<=k<=11, 1<=q<=8.
For Example
2 2 1 1 2
1 1 2 2 1

For the guess 2 2 1 1 the score is 2
For the guess 1 1 2 2 the score is 1
Because there is a key possible let’s say 2 1 1 3 which gives the desired scores.Hence the answer is yes

Another Example
1 2 3 4 4
4 3 2 1 1

For the guess 1 2 3 4 the score is 4
For the guess 4 3 2 1 the score is 1

This has no key which gives the desired scores hence answer is NO

I tried the brute force approach generating n^k such keys where n is the range of s_i.But it gave Time Limit exceeding error.
Its an interview puzzle. I have seen variants of this question but was not able to solve them.Can you tell me what should I read for such type of questions.

Answer 1

I don’t know the best solution to this problem, but if you did a recursive search of the possible solution space, pruning branches which could not possibly lead to a solution, it would be much faster than trying all (n^k) keys.

Take your example:

1 2 3 4 4 -> 4
4 3 2 1 1 -> 1

The 3 possible values for g1 which could be significant are: 1, 4, and “neither 1 nor 4”. Choose one of them, and then recursively look at the possible values for g2. Choose one, and recursively look at the possible values for g3, etc.

As you search, keep track of a cumulative score for each of the guesses from b1 to bq. Whenever you choose a value for a digit, increment the cumulative scores for all the guesses which have the same number in that position. Keep these cumulative scores on a stack (so you can back up).

When you reach a point where no solution is possible, back up and continue searching a different path. If you back all the way up to g1 and no more paths are left to search, then the answer is NO. If you find a solution, then the answer is YES.

When to stop searching a path and back up:

• If the cumulative score of one of the guesses exceeds the given score
• If the cumulative score of one of the guesses is less than the given score minus the number of levels left in the search tree (before you hit the bottom)

This approach could still be very slow, especially if “k” was large. But again, it will be far faster than generating (n^k) keys.