From 29b60e7f52be3814f1b6b9547ee5dad9a30dc92f Mon Sep 17 00:00:00 2001
From: Eduardo Santos <eduardo.experimental@gmail.com>
Date: Wed, 10 Jul 2024 01:50:36 -0300
Subject: [PATCH] Improve README.md and 217 in O(n^2)

---
 README.md                               | 11 ++++++++---
 python3/217/217-contains-duplicate-a.py | 12 ++++++++----
 python3/217/217-contains-duplicate-b.py |  2 +-
 python3/217/217-contains-duplicate-c.py |  2 +-
 python3/217/217-contains-duplicate-d.py |  2 +-
 5 files changed, 19 insertions(+), 10 deletions(-)

diff --git a/README.md b/README.md
index 51322c3..aa39a28 100644
--- a/README.md
+++ b/README.md
@@ -6,12 +6,17 @@
   <br>
 </h1>
 
-<h4 align="center">Solutions to LeetCode problems written in Python 3.</h4>
+<h4 align="center">Solutions to LeetCode problems written in Python 3</h4>
 
-The algorithm or technique used, time and space complexities, and optimality are given as comments at the top of each file. Optimality always favors time.
+The algorithm or technique used, time complexity, auxiliary space, and
+optimality are given as comments at the top of each file. Optimality always
+favors time.
 
 > [!NOTE]
-> [Binary logarithms](https://en.wikipedia.org/wiki/Binary_logarithm) are notated $\mathop{{}^{}\mathrm{lb}} x$ (e.g., `n lb n`) in compliance with [ISO 80000-2:2019](https://www.iso.org/standard/64973.html).
+> [Binary logarithms](https://en.wikipedia.org/wiki/Binary_logarithm), as in
+> [logarithmic time](https://en.wikipedia.org/wiki/Time_complexity#Logarithmic_time),
+> are notated $\mathop{{}^{}\mathrm{lb}} x$ (e.g., `lb n`) in compliance with
+> [ISO 80000-2:2019](https://www.iso.org/standard/64973.html).
 
 ## 🧩 Solved
 
diff --git a/python3/217/217-contains-duplicate-a.py b/python3/217/217-contains-duplicate-a.py
index de09d10..2cf40aa 100644
--- a/python3/217/217-contains-duplicate-a.py
+++ b/python3/217/217-contains-duplicate-a.py
@@ -1,15 +1,19 @@
 # Programming language: Python 3
 # Algorithm/technique:  nested iteration
 # Time complexity:      O(n^2)
-# Space complexity:     O(1)
+# Auxiliary space:      O(1)
 # Optimal:              no
 # Notes:
 
 class Solution:
     def containsDuplicate(self, nums: list[int]) -> bool:
-        for i, num in enumerate(nums):
-            for j in range(i + 1, len(nums)):
-                if num == nums[j]:
+        n = len(nums)
+
+        # enumerate does not allow the last item to be skipped without slicing,
+        # which would require O(n) auxiliary space.
+        for i in range(n - 1):
+            for j in range(i + 1, n):
+                if nums[i] == nums[j]:
                     return True
 
         return False
diff --git a/python3/217/217-contains-duplicate-b.py b/python3/217/217-contains-duplicate-b.py
index f0a1ae1..05b1af6 100644
--- a/python3/217/217-contains-duplicate-b.py
+++ b/python3/217/217-contains-duplicate-b.py
@@ -1,7 +1,7 @@
 # Programming language: Python 3
 # Algorithm/technique:  sorting
 # Time complexity:      O(n lb n)
-# Space complexity:     O(1)
+# Auxiliary space:      O(1)
 # Optimal:              no
 # Notes:
 
diff --git a/python3/217/217-contains-duplicate-c.py b/python3/217/217-contains-duplicate-c.py
index d02b7aa..ec10908 100644
--- a/python3/217/217-contains-duplicate-c.py
+++ b/python3/217/217-contains-duplicate-c.py
@@ -1,7 +1,7 @@
 # Programming language: Python 3
 # Algorithm/technique:  set
 # Time complexity:      O(n)
-# Space complexity:     O(n)
+# Auxiliary space:      O(n)
 # Optimal:              yes
 # Notes:
 
diff --git a/python3/217/217-contains-duplicate-d.py b/python3/217/217-contains-duplicate-d.py
index 94ea39f..750d8a7 100644
--- a/python3/217/217-contains-duplicate-d.py
+++ b/python3/217/217-contains-duplicate-d.py
@@ -1,7 +1,7 @@
 # Programming language: Python 3
 # Algorithm/technique:  set
 # Time complexity:      O(n)
-# Space complexity:     O(n)
+# Auxiliary space:      O(n)
 # Optimal:              yes
 # Notes: