Update README.md

searching/README.md

@@ -1,19 +1,104 @@
-## searching
+## binary search
 
 <br>
 
+* a binary search operates on a contiguous sequence with a specified left and right index (this is called the **search space**).
   
-### binary search
-
-<br>
-
-* a binary search operates on a contiguous sequence with a specified left and right index. this is called the search space.
 * binary searching is composed of 3 sections:
-    * pre-processing: sort if collection is unsorted
+    * **pre-processing**: sort if collection is unsorted
-    * binary search: using a loop or recursion to divide search sapce in half after each comparison
+    * **binary search**: using a loop or recursion to divide search space in half after each comparison (`O(log(N)`)
-    * post-processing: determine viable candidates in the remaining space
+    * **post-processing`: determine viable candidates in the remaining space
 
 * there are 3 "templates" when writing a binary search:
     * `while left < right`, with `left = mid + 1` and `right = mid - 1`
     * `while left < right`, with `left = mid + 1` and `right = mid`, and `left` is returned
-    * `while left + 1 < right`, with `left = 1` and `right = mid`, and `left` and `right` are returned
+    * `while left + 1 < right`, with `left = mid` and `right = mid`, and `left` and `right` are returned
+
+
+<br>
+
+----
+
+### iterative
+
+<br>
+
+```python
+    if lens(nums) == 0:
+        return False
+        
+    lower, higher = 0, len(array)
+
+    while lower < higher:
+        mid = (higher + lower) // 2
+        
+        if array[mid] == item:
+            return mid 
+        elif array[mid] > item:
+            higher = mid - 1
+        else:
+            lower = mid + 1
+            
+    return False
+```
+
+<br>
+
+----
+
+### recursive
+
+<br>
+
+```python
+def binary_search_recursive(array, item, higher=None, lower=0):
+
+    higher = higher or len(array)
+    
+    if higher < lower:
+        return False
+    
+    mid = (higher + lower) // 2
+    
+    if  item == array[mid]:
+        return mid
+        
+    elif item < array[mid]:
+        return binary_search_recursive(array, item, mid - 1, lower)
+        
+    else:
+        return binary_search_recursive(array, item, higher, mid + 1)
+```
+
+<br>
+
+---
+
+### in a matrix
+
+<br>
+
+```python
+def binary_search_matrix(matrix, item, lower=0, higher=None):
+
+    if not matrix:
+        return False
+    
+    rows = len(matrix)
+    cols = len(matrix[0])
+    higher = higher or rows * cols
+
+    if higher > lower:
+        mid = (higher + lower) // 2
+        row = mid // cols
+        col = mid % cols
+
+        if item == matrix[row][col]:
+            return row, col
+        elif item < matrix[row][col]:
+            return binary_search_matrix(matrix, item, lower, mid - 1)
+        else:
+            return binary_search_matrix(matrix, item, mid + 1, higher)
+        
+    return False
+```