problem: 0378 kth smallest element in a sorted matrix

Author: cychiu8

0378 Kth Smallest Element in a Sorted Matrix/0378 Kth Smallest Element in a Sorted Matrix.md

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+# 0378. Kth Smallest Element in a Sorted Matrix
+
+* Difficulty: medium
+* Link: https://leetcode.com/problems/kth-smallest-element-in-a-sorted-matrix/
+* Topics: Binary-Search, Heap
+
+# Clarification
+
+1. Check the inputs and outputs
+    - INPUT: matrix[int][int]
+    - OUTPUT: int
+2. Check the main goal
+    - time complexity < $O(n^2)$
+
+# Naive Solution
+
+<aside>
+💡 從最簡單的方法開始 easy solution → only speak out
+
+</aside>
+
+### Thought Process
+
+1. 遍歷整個矩陣，找出第 k 小的值 ⇒ $O(n^2)$ 不符合題目要求
+
+# Solution (Binary search)
+
+### Thought Process
+
+1. 定義 (依據 matrix 之特性)
+    - lower = matrix[0][0]
+    - upper = matrxi[n][n]
+- 猜中間值，並算出有多少個數小於該個中間值
+    - 個數 < target ⇒ lower = mid + 1
+    - 個數 ≥ target ⇒ upper = mid
+- 當 lower ≥ upper 時跳出迴圈，並回傳 lower(/upper)都行
+- Implement
+    
+    ```python
+    class Solution:
+        def kthSmallest(self, matrix: List[List[int]], k: int) -> int:
+            lower = matrix[0][0]
+            upper = matrix[-1][-1]
+            while lower < upper:
+                mid = (lower+upper) // 2
+                count = self.numberOfElementsLessOrEqualThan(matrix, mid)
+                if count < k:
+                    lower = mid + 1
+                else:
+                    upper = mid
+            return lower
+        
+        def numberOfElementsLessOrEqualThan(self, matrix: List[List[int]], target: int) -> int:
+            l = len(matrix)
+            i = 0
+            j = l - 1
+            count = 0
+            while j >= 0 and i < l:
+                if target < matrix[i][j]:
+                    j -= 1
+                if target >= matrix[i][j]:
+                    count += (j+1)
+                    i += 1
+            return count
+    ```
+    
+
+### Complexity
+
+- Time complexity: O(log n)
+- Space complexity: O(1)
+
+# Solution (Heap)
+
+- tbd
+
+# Note
+
+- ****[K-th Smallest Element in a Sorted Matrix](https://www.tutorialcup.com/interview/matrix/k-th-smallest-element-in-a-sorted-matrix.htm#Using_min_Heap_data_structure)****
+- ****[LeetCode 378(Find K Pairs with Smallest Sums) 心得(Medium)](https://medium.com/@ChYuan/leetcode-378-find-k-pairs-with-smallest-sums-%E5%BF%83%E5%BE%97-medium-c2430d02f260)****