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add more notes in data structures and algorithms
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@kekeandzeyu
kekeandzeyu committed Oct 6, 2024
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79 changes: 45 additions & 34 deletions Writerside/topics/Data-Structures-and-Algorithms-1.md
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Expand Up @@ -918,7 +918,7 @@ class ArrayDeque:
</list>

<tip>
<p><format color="BlueViolet">Defect:</format> </p>
<p><format color="BlueViolet">Defect</format></p>
<list type="bullet">
<li>
<p>Union too expensive (<math>N</math> array accesses).</p>
Expand Down Expand Up @@ -950,7 +950,7 @@ class ArrayDeque:
</list>

<note>
<p><format color="BlueViolet">Defect:</format> </p>
<p><format color="BlueViolet">Defect</format></p>
<list type="bullet">
<li>
<p>Trees can get tall.</p>
Expand Down Expand Up @@ -1211,7 +1211,7 @@ class UnionFind:

### 4.1 Stacks

<p><format color="BlueViolet">Defintions:</format> </p>
<p><format color="BlueViolet">Defintions</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -1349,7 +1349,7 @@ print(len(stack) == 0)
<img src="../images_data/d4-1-2.png" alt="Stack push"/>
</procedure>

<p><format color="BlueViolet">Properties:</format> </p>
<p><format color="BlueViolet">Properties</format></p>

<list>
<li>
Expand Down Expand Up @@ -2235,7 +2235,7 @@ public class Bag&lt;Item&gt; implements Iterable&lt;Item&gt; {
</step>
</procedure>

<p><format color="BlueViolet">Properties:</format> </p>
<p><format color="BlueViolet">Properties</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -2341,7 +2341,7 @@ def selection_sort(arr):
</step>
</procedure>

<p><format color="BlueViolet">Definitions:</format> </p>
<p><format color="BlueViolet">Definitions</format></p>

<list type="bullet">
<li>
Expand All @@ -2365,7 +2365,7 @@ arrays, insertion sort runs in linear time.</p>
equals the number of inversions (number of compares = exchanges + (N
– 1)).</p>

<p><format color="BlueViolet">Running Time Analysis:</format> </p>
<p><format color="BlueViolet">Running Time Analysis</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -2467,7 +2467,7 @@ def insertion_sort(arr):
</step>
</procedure>

<p><format color="BlueViolet">Increment Sequence:</format> </p>
<p><format color="BlueViolet">Increment Sequence</format></p>

<list type="decimal">
<li>
Expand Down Expand Up @@ -2644,15 +2644,15 @@ array[i], array[r] = array[r], array[i]
</tab>
</tabs>

### 5.5 Convex Hull
### 5.5 Convex Hull {id="convex-hull"}

<p><format color="DarkOrange">Convex Hull:</format> A convex hull of
a set of <math>N</math> points is the smallest perimeter fence
enclosing the points.</p>

<img src="../images_data/d5-5-1.png" alt="Convex Hull"/>

<p><format color="BlueViolet">Equivalent definitions:</format> </p>
<p><format color="BlueViolet">Equivalent definitions</format></p>

<list type="bullet">
<li>
Expand All @@ -2667,7 +2667,7 @@ enclosing the points.</p>
</li>
</list>

<p><format color="BlueViolet">Geometric properties:</format> </p>
<p><format color="BlueViolet">Geometric properties</format></p>

<list type="bullet">
<li>
Expand All @@ -2681,7 +2681,7 @@ enclosing the points.</p>
</li>
</list>

<img src="../images_data/d5-5-2.png" alt="Geometric Properties"/>
<img src="../images_data/d5-5-2.png" alt="Geometric Properties"/>

<procedure title="Graham Scan">
<step>
Expand All @@ -2693,8 +2693,7 @@ enclosing the points.</p>
</step>
<step>
<p>Consider points in order; discard unless it create a ccw
(counterclockwise) turn.
</p>
(counterclockwise) turn.</p>
</step>
</procedure>

Expand Down Expand Up @@ -2722,18 +2721,17 @@ turn?</p>
</step>
</procedure>

<p><format color="LawnGreen">Proof:</format> </p>
<p><format color="LawnGreen">Proof</format></p>

<code-block lang="tex">
\begin{equation}
2 \times \text{Area}(\triangle abc) = \begin{vmatrix} x_a & y_a & 1 \\ x_b & y_b & 1 \\ x_c & y_c & 1 \end{vmatrix} = (x_b - x_a)(y_c - y_a) - (y_b - y_a)(x_c - x_a)
\end{equation}
</code-block>

<img src="../images_data/d5-5-3.png" alt="Determinant and
Positions"/>
<img src="../images_data/d5-5-3.png" alt="Determinant and Positions"/>

<p><format color="BlueViolet">Applications:</format> </p>
<p><format color="BlueViolet">Applications</format></p>

<list type="bullet">
<li>
Expand All @@ -2755,7 +2753,20 @@ Positions"/>
</li>
</list>

<p><format color="BlueViolet">Point2D:</format> </p>
<p><format color="BlueViolet">Cost of Convex Hull:</format> <math>N \log N + N</math></p>

<p><format color="LawnGreen">Proof:</format> Convex hull reduces to sorting</p>

<list type="bullet">
<li>
<p><math>N \log N</math> cost of sorting</p>
</li>
<li>
<p><math>N</math> cost of reduction</p>
</li>
</list>

<p><format color="BlueViolet">Point2D</format></p>

<tabs>
<tab title="Java">
Expand Down Expand Up @@ -2967,7 +2978,7 @@ class Point2D:
</tab>
</tabs>

<p><format color="BlueViolet">Graham Scan:</format> </p>
<p><format color="BlueViolet">Graham Scan</format></p>

<tabs>
<tab title="Java">
Expand Down Expand Up @@ -3228,7 +3239,7 @@ is <format color="OrangeRed"> in-place</format> if it uses
most <math>N \lg N</math> compares and <math>6N \lg N</math> array
accesses to sort any array of size <math>N</math>.</p>

<p><format color="LawnGreen">Proof:</format> </p>
<p><format color="LawnGreen">Proof</format></p>

<p>The number of compares <math>C(N)</math> and array accesses
<math>A(N)</math> to mergesort an array of size
Expand Down Expand Up @@ -3525,7 +3536,7 @@ def sort(a):

### 6.3 Computational Complexity

<p><format color="BlueViolet">Definitions:</format> </p>
<p><format color="BlueViolet">Definitions</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -3585,7 +3596,7 @@ def sort(a):
sorting algorithm must use at least <math>\lg(N!) \sim N \lg N</math>
compares in the worst case.</p>

<p><format color="BlueViolet">Proof:</format> </p>
<p><format color="BlueViolet">Proof</format></p>

<list>
<li>
Expand Down Expand Up @@ -3619,7 +3630,7 @@ compares in the worst case.</p>
="OrangeRed">stable</format> sort preserves the relative order of
items with equal keys.</p>

<p><format color="BlueViolet">Methods:</format> </p>
<p><format color="BlueViolet">Methods</format></p>

<list type="decimal">
<li>
Expand All @@ -3633,7 +3644,7 @@ items with equal keys.</p>
</li>
</list>

<p><format color="BlueViolet">Conclusion:</format> </p>
<p><format color="BlueViolet">Conclusion</format></p>

<list>
<li>
Expand Down Expand Up @@ -3693,7 +3704,7 @@ items with equal keys.</p>
</step>
</procedure>

<p><format color="BlueViolet">Runtime Analysis:</format> </p>
<p><format color="BlueViolet">Runtime Analysis</format></p>

<list type="bullet">
<li>
Expand All @@ -3715,7 +3726,7 @@ compares <math>C_{N}</math> to quicksort an array of <math>N</math>
distinct keys is <math>\sim 2N \ln N </math> (and the number of
exchanges is <math>\sim \frac{1}{3} N \ln N </math>).</p>

<p><format color="LawnGreen">Proof:</format> </p>
<p><format color="LawnGreen">Proof</format></p>

<p><math>C_{N}</math> satisfies the recurrence <math>C_0 = C_1 = 0
</math> and for <math>N \leq 2</math>:</p>
Expand Down Expand Up @@ -3804,7 +3815,7 @@ for partitioning probability.</p>
</list>
</li>
<li>
<p><format color="Fuchsia">Random shuffle:</format> </p>
<p><format color="Fuchsia">Random shuffle</format></p>
<list type="bullet">
<li>
<p>Probabilistic guarantee against worst case.</p>
Expand All @@ -3816,7 +3827,7 @@ for partitioning probability.</p>
</list>
</li>
<li>
<p><format color="Fuchsia">Caveat emptor:</format> </p>
<p><format color="Fuchsia">Caveat emptor</format></p>
<p>Many textbook implementations go <format color="OrangeRed">
quadratic</format> if array:</p>
<list type="bullet">
Expand Down Expand Up @@ -3999,7 +4010,7 @@ def quicksort(arr, comparator=lambda x, y: x &lt; y):
<p><format color="BlueViolet">Property:</format> Quick-select takes
<format color="OrangeRed">linear</format> time on average.</p>

<p><format color="LawnGreen">Proof:</format> </p>
<p><format color="LawnGreen">Proof</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -4260,7 +4271,7 @@ def sort_all(arr: list):
</tab>
</tabs>

<p><format color="BlueViolet">Conclusion:</format> </p>
<p><format color="BlueViolet">Conclusion</format></p>

<table style="both" id="conclusion">
<tr>
Expand Down Expand Up @@ -4336,7 +4347,7 @@ def sort_all(arr: list):

### 8.1 API and Elementary Implementations

<p><format color="BlueViolet">Applications:</format> </p>
<p><format color="BlueViolet">Applications</format></p>

<list type="bullet">
<li>
Expand Down Expand Up @@ -5404,7 +5415,7 @@ class IndexedPriorityQueue:
</step>
</procedure>

<p><format color="BlueViolet">Properties:</format> </p>
<p><format color="BlueViolet">Properties</format></p>

<list>
<li>
Expand Down Expand Up @@ -7801,7 +7812,7 @@ following three operations in a certain order: </p>
</li>
</list>

<p><format color="BlueViolet">Three types of traversal:</format> </p>
<p><format color="BlueViolet">Three types of traversal</format></p>

<list type="alpha-lower">
<li>
Expand Down
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