Kth Smallest Element In A Sorted Matrix
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: Kth Smallest Element in a Sorted Matrix (K-way Merge)
* ============================================================================================
*
* ---------------------
* 1. Logical Breakdown & Mathematical Formulation
* ---------------------
* Given a matrix where A[i][j] <= A[i][j+1] (rows) and A[i][j] <= A[i+1][j] (cols).
* We want the B-th smallest element.
*
* Step 1: Min-Heap Initialization
* - Insert the first element of every row into a Min-Heap: {value, row, col}.
* - The heap size will be exactly N (number of rows).
*
* Step 2: Extract and Advance
* - Repeat B times:
* 1. Extract the minimum element from the heap: {val, r, c}.
* 2. If this is the B-th extraction, return val.
* 3. Otherwise, if the extracted element has a successor in its row (c + 1 < M),
* insert the next element from that row into the heap: {A[r][c+1], r, c+1}.
*
* ---------------------
* 2. Complexity Analysis
* ---------------------
* Time Complexity:
* O(min(B, N) + B log N)
* - Initializing heap: O(N log N) or O(N).
* - B extractions and insertions: O(B log N).
*
* Space Complexity:
* O(N) to store the first element of each row in the Priority Queue.
*
* ---------------------
* 3. Base Case Verification
* ---------------------
* - If B = 1: The heap extracts the very first element A[0][0], which is the absolute minimum.
* - If B = NM: The algorithm will traverse all elements and return the largest at A[N-1][M-1].
* ============================================================================================
/
public class KthSmallestElementInASortedMatrix {
public int solve(int[][] a3, int A) {
// TODO: Implement solution
return 0;
}
}