Smallest Sequence With Given Primes
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.graphs;
import java.util.*;
/
* ============================================================================================
* PROBLEM: Smallest sequence with given Primes
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given three prime numbers A, B, and C, and an integer D.
* Find the first D integers (in ascending order) whose prime factors are limited
* to the set {A, B, C}.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* We need to generate a sequence \(S\) where each element \(S[i]\) is of the form:
* $\(S[i] = A^x \cdot B^y \cdot C^z\)$
* where \(x, y, z \ge 0\) (and at least one is \(> 0\) as we typically start from 1
* but the problem asks for the first D such integers, usually excluding 1).
*
* Algorithm (Three-Pointer Approach):
* 1. Initialization:
* - Create a result array res of size D.
* - Maintain three pointers: pA, pB, pC, all initially at -1.
* - These pointers track which element in res will be multiplied by A, B, or C next.
*
* 2. Iterative Generation:
* - For each step \(i\) from 0 to D-1:
* a. Calculate candidates:
* - \(vA = (pA == -1 ? A : res[pA] * A)\)
* - \(vB = (pB == -1 ? B : res[pB] * B)\)
* - \(vC = (pC == -1 ? C : res[pC] * C)\)
* b. The next smallest number is \(min(vA, vB, vC)\).
* c. Store this minimum in res[i].
* d. Crucial Pointer Update*:
* - If \(res[i] == vA\), increment pA.
* - If \(res[i] == vB\), increment pB.
* - If \(res[i] == vC\), increment pC.
* - (Using if instead of else if handles duplicates like \(A \cdot B = B \cdot A\)).
*
* ---------------------
* 3. Complexity Analysis
* ---------------------
* Time Complexity: O(D)
* - We perform constant time work to generate each of the D numbers.
* Space Complexity: O(D)
* - To store the resulting sequence.
*
* ============================================================================================
/
public class SmallestSequenceWithGivenPrimes {
public int solve(int A) {
// TODO: Implement solution
return 0;
}
public int[] solve(int i, int i1, int i2, int i3) {
return new int[0];
}
}