PROBLEM LINK:
Author: vipulvikram
Tester: D Teja Vardhan Reddy
Editorialist: vipulvikram
DIFFICULTY:
MEDIUM
PREREQUISITES:
DP,Geometric progression
PROBLEM:
Given the size of an array and two integers with their index where they are already filled.Find the number of way to fill the remaining array with following conditions:
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The numbers filled should be in range [L,R] (both inclusive).
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consecutive positions should have different numbers.
QUICK EXPLANATION:
Let i and j are the indexes and X and Y are the corresponding integers which are already filled.
Now, The problem can be divided into 3 sub-problems. First is from index 1 to i, second is from i to j, and third is from j to n (size of array).
Now, it can be solved using dynamic programming. Let’s say we have to solve for index i to j. Index i contain X and index j contain Y. So, first we find the number of way to fill index i+1 with Y ( such that it is not equal to number in index i) , then we find the total number of ways to fill index i+2 with Y and so on till we reach index j. So the number of ways to fill up to index j with Y is its answer. But, if you observe, this series forms geometric progression which can be solved in O(logn) time.
Similarly, we find for index 1 to i, and for index j to n. Then, multiply all these three to get final answer.
AUTHOR’S AND TESTER’S SOLUTIONS:
Author’s solution can be found here.
Tester’s solution can be found here.