Bth Smallest Prime Fraction
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.heaps;
/* * ============================================================================================ * PROBLEM: B-th Smallest Prime Fraction * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a sorted array A of primes. Consider all fractions p/q where p < q and * p, q are in A. Find the B-th smallest such fraction. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * For each index i (numerator), the fractions are: * A[i]/A[N-1], A[i]/A[N-2], ..., A[i]/A[i+1] * Since A is sorted, for a fixed A[i], the smallest fraction is always A[i]/A[N-1]. * * Algorithm (Min-Heap approach): * 1. Initialization: * - Use a Min-Heap to store triples: {numerator_index, denominator_index, value}. * - Initially, for every possible numerator index i (from 0 to N-2), push the * smallest possible fraction into the heap: {i, N-1, A[i]/A[N-1]}. * * 2. Iterative Extraction: * - Extract the minimum element from the heap B-1 times. * - Every time we extract {i, j, value}, the next smallest fraction for that * specific numerator A[i] is A[i]/A[j-1]. * - Push {i, j-1, A[i]/A[j-1]} back into the heap (if j-1 > i). * * 3. Result: * - The B-th element extracted from the heap is the answer. * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Time Complexity: * O((N + B) log N) * - Initial heap build: O(N log N). * - B extractions and insertions: O(B log N). * * Space Complexity: * O(N) to store the heap elements. * ============================================================================================ / public class BthSmallestPrimeFraction { public int[] solve(int[] a1, int A) { // TODO: Implement solution return new int[0]; } }