310 updates

Author: localhost433

notes/courses/CSCI-UA-310/08-09-hashing.md

@@ -51,7 +51,7 @@ The table size $m$ is much smaller than $|U|$.
 
 **Collision**: Two keys $k_1 \neq k_2$ with $h(k_1) = h(k_2)$ is a **collision**. Collisions are unavoidable if $n > m$.
 
-### Hash Function Design
+### 3.1 Hash Function Design
 
 A good hash function satisfies the **simple uniform hashing assumption**: each key is equally likely to hash to any of the $m$ slots, independently of all other keys.
 
@@ -67,7 +67,7 @@ A good hash function satisfies the **simple uniform hashing assumption**: each k
 
 * Slot $j$ contains a pointer to the head of the list of all elements with $h(k) = j$.
 
-### Pseudocode
+### 4.1 Pseudocode
 
 ```text
 CHAINED-HASH-INSERT(T, x)
@@ -131,7 +131,7 @@ HASH-SEARCH(T, k)
 
 **Deletion** is tricky: cannot just set to `NIL` (would break search). Use a special `DELETED` sentinel.
 
-### Probing Strategies
+### 6.1 Probing Strategies
 
 **Linear Probing**: $h(k, i) = (h'(k) + i) \bmod m$.
 

notes/courses/CSCI-UA-310/10-binary-search-trees.md

@@ -48,7 +48,7 @@ Similarly, **pre-order** (root, left, right) and **post-order** (left, right, ro
 
 ## 3. Searching
 
-### SEARCH
+### 3.1 SEARCH
 
 ```text
 TREE-SEARCH(x, k)
@@ -61,7 +61,7 @@ TREE-SEARCH(x, k)
 
 Running time: $O(h)$.
 
-### MINIMUM and MAXIMUM
+### 3.2 MINIMUM and MAXIMUM
 
 ```text
 TREE-MINIMUM(x)
@@ -79,7 +79,7 @@ TREE-MAXIMUM(x)
 
 Both run in $O(h)$.
 
-### SUCCESSOR
+### 3.3 SUCCESSOR
 
 The **successor** of node $x$ is the node with the smallest key greater than $x.\text{key}$.
 
@@ -100,7 +100,7 @@ Running time: $O(h)$.
 
 ## 4. Modification
 
-### INSERT
+### 4.1 INSERT
 
 ```text
 TREE-INSERT(T, z)
@@ -121,7 +121,7 @@ TREE-INSERT(T, z)
 
 Running time: $O(h)$.
 
-### DELETE
+### 4.2 DELETE
 
 Three cases when deleting node $z$:
 

notes/courses/CSCI-UA-310/11-red-black-trees.md

@@ -47,7 +47,7 @@ The **black-height** $\text{bh}(x)$ of a node $x$ is the number of black nodes o
 
 **Rotations** are local restructuring operations that preserve the BST property and run in $O(1)$ time.
 
-### Left Rotation on node $x$
+### 3.1 Left Rotation on node $x$
 
 ```text
 LEFT-ROTATE(T, x)
@@ -84,7 +84,7 @@ RB-INSERT(T, z)
 2. RB-INSERT-FIXUP(T, z)
 ```
 
-### Fixup Cases
+### 4.1 Fixup Cases
 
 Let $z$ be the newly inserted red node, $z.\text{parent}$ be red (violation of Property 4), and $y$ be $z$'s **uncle** (sibling of $z$'s parent).
 

notes/courses/CSCI-UA-310/12-13-14-dynamic-programming.md

@@ -53,7 +53,7 @@ date: 2026-03-02/04/09
 |---|---|---|---|---|---|---|---|---|---|---|
 | Price $p[i]$ | 1 | 5 | 8 | 9 | 10 | 17 | 17 | 20 | 24 | 30 |
 
-### Naive Recursive Solution
+### 3.1 Naive Recursive Solution
 
 $$r_n = \max_{1 \leq i \leq n}(p[i] + r_{n-i})$$
 
@@ -71,7 +71,7 @@ CUT-ROD(p, n)
 
 **Running time**: $T(n) = 1 + \sum_{j=0}^{n-1} T(j)$, giving $T(n) = 2^n$ — exponential due to redundant subproblems.
 
-### Bottom-Up Dynamic Programming
+### 3.2 Bottom-Up Dynamic Programming
 
 Solve subproblems in order of increasing size, filling a table $r[0 \dots n]$.
 
@@ -89,7 +89,7 @@ BOTTOM-UP-CUT-ROD(p, n)
 
 **Running time**: $\Theta(n^2)$ — two nested loops.
 
-### Top-Down Memoization
+### 3.3 Top-Down Memoization
 
 ```text
 MEMOIZED-CUT-ROD(p, n)
@@ -141,7 +141,7 @@ PRINT-CUT-ROD-SOLUTION(p, n)
 
 ## 5. Longest Common Subsequence
 
-### Problem
+### 5.1 Problem
 
 A **subsequence** of a sequence is the sequence with zero or more elements removed (not necessarily contiguous).
 
@@ -152,21 +152,21 @@ A **subsequence** of a sequence is the sequence with zero or more elements remov
 **Example**: $X = \langle A, B, C, B, D, A, B \rangle$, $Y = \langle B, D, C, A, B, A \rangle$.
 LCS: $\langle B, C, B, A \rangle$ (length 4) or $\langle B, D, A, B \rangle$ (length 4).
 
-### Optimal Substructure
+### 5.2 Optimal Substructure
 
 **Theorem**: Let $Z = \langle z_1, \dots, z_k \rangle$ be any LCS of $X$ and $Y$.
 
 1. If $x_m = y_n$, then $z_k = x_m = y_n$ and $Z_{k-1}$ is an LCS of $X_{m-1}$ and $Y_{n-1}$.
 2. If $x_m \neq y_n$ and $z_k \neq x_m$, then $Z$ is an LCS of $X_{m-1}$ and $Y$.
 3. If $x_m \neq y_n$ and $z_k \neq y_n$, then $Z$ is an LCS of $X$ and $Y_{n-1}$.
 
-### Recurrence
+### 5.3 Recurrence
 
 Define $c[i,j]$ = length of LCS of $X_i = \langle x_1,\dots,x_i \rangle$ and $Y_j = \langle y_1,\dots,y_j \rangle$.
 
 $$c[i,j] = \begin{cases} 0 & \text{if } i = 0 \text{ or } j = 0 \\ c[i-1,j-1] + 1 & \text{if } i,j > 0 \text{ and } x_i = y_j \\ \max(c[i-1,j],\; c[i,j-1]) & \text{if } i,j > 0 \text{ and } x_i \neq y_j \end{cases}$$
 
-### Algorithm
+### 5.4 Algorithm
 
 ```text
 LCS-LENGTH(X, Y)
@@ -196,28 +196,28 @@ LCS-LENGTH(X, Y)
 
 ## 6. 0/1 Knapsack
 
-### Problem
+### 6.1 Problem
 
 **Input**: $n$ items where item $i$ has weight $w_i \in \mathbb{Z}^+$ and value $v_i \geq 0$; knapsack capacity $W \in \mathbb{Z}^+$.
 
 **Output**: Choose a subset $S \subseteq \{1, \dots, n\}$ to maximize $\sum_{i \in S} v_i$ subject to $\sum_{i \in S} w_i \leq W$.
 
 The **0/1** constraint means each item is either taken or not (no fractions).
 
-### Optimal Substructure
+### 6.2 Optimal Substructure
 
 For item $n$:
 
 * **Skip item $n$**: optimal value from items $\{1,\dots,n-1\}$ with capacity $W$.
 * **Take item $n$** (if $w_n \leq W$): $v_n$ plus optimal value from items $\{1,\dots,n-1\}$ with capacity $W - w_n$.
 
-### Recurrence
+### 6.3 Recurrence
 
 Define $\text{OPT}(i, w)$ = maximum value using items $\{1,\dots,i\}$ with capacity $w$.
 
 $$\text{OPT}(i, w) = \begin{cases} 0 & \text{if } i = 0 \\ \text{OPT}(i-1, w) & \text{if } w_i > w \\ \max\bigl(\text{OPT}(i-1, w),\; v_i + \text{OPT}(i-1, w - w_i)\bigr) & \text{otherwise} \end{cases}$$
 
-### Algorithm
+### 6.4 Algorithm
 
 ```text
 KNAPSACK(n, W, w[1..n], v[1..n])
@@ -234,7 +234,7 @@ KNAPSACK(n, W, w[1..n], v[1..n])
 
 **Note**: $O(nW)$ is **pseudopolynomial** — it is polynomial in $n$ and $W$, but $W$ can be exponential in the number of bits required to represent it. The 0/1 Knapsack problem is NP-hard.
 
-### Worked Example
+### 6.5 Worked Example
 
 Items: $(w_1, v_1) = (2, 6)$, $(w_2, v_2) = (2, 10)$, $(w_3, v_3) = (3, 12)$. Capacity $W = 5$.
 
@@ -272,7 +272,7 @@ FIBONACCI-DP(n)
 
 ## 8. Matrix Chain Multiplication
 
-### Problem
+### 8.1 Problem
 
 **Input**: A chain of $n$ matrices $A_1, A_2, \dots, A_n$ where $A_i$ has dimensions $p_{i-1} \times p_i$.
 
@@ -285,19 +285,19 @@ FIBONACCI-DP(n)
 * $((A_1 A_2) A_3) A_4$: $30 \cdot 35 \cdot 15 + 30 \cdot 15 \cdot 5 + 30 \cdot 5 \cdot 10 = 18{,}000$ multiplications.
 * $(A_1 (A_2 (A_3 A_4)))$: $15 \cdot 5 \cdot 10 + 35 \cdot 15 \cdot 10 + 30 \cdot 35 \cdot 10 = 16{,}250$ multiplications.
 
-### Optimal Substructure
+### 8.2 Optimal Substructure
 
 Any optimal parenthesization of $A_i A_{i+1} \cdots A_j$ must split at some $k$ ($i \leq k < j$) into $(A_i \cdots A_k)(A_{k+1} \cdots A_j)$. Each part must also be optimally parenthesized.
 
-### Recurrence
+### 8.3 Recurrence
 
 Define $m[i,j]$ = minimum number of multiplications to compute $A_i \cdots A_j$.
 
 $$m[i,j] = \begin{cases} 0 & \text{if } i = j \\ \min_{i \leq k < j} \bigl( m[i,k] + m[k+1,j] + p_{i-1} \cdot p_k \cdot p_j \bigr) & \text{if } i < j \end{cases}$$
 
 **Guess**: the split position $k$.
 
-### Algorithm
+### 8.4 Algorithm
 
 Solve by increasing **chain length** $\ell = j - i$.
 
@@ -325,21 +325,21 @@ MATRIX-CHAIN-ORDER(p)
 
 ## 9. Subset Sum
 
-### Problem
+### 9.1 Problem
 
 **Input**: A set $S = \{a_1, a_2, \dots, a_n\}$ of positive integers and a target $W$.
 
 **Output**: Does there exist a subset $T \subseteq S$ such that $\sum_{a \in T} a = W$?
 
-### Recurrence
+### 9.2 Recurrence
 
 Define $\text{DP}[i][w] = \mathtt{true}$ if some subset of $\{a_1, \dots, a_i\}$ sums to exactly $w$.
 
 $$\text{DP}[i][w] = \text{DP}[i-1][w] \;\lor\; \bigl(w \geq a_i \;\land\; \text{DP}[i-1][w - a_i]\bigr)$$
 
 Base cases: $\text{DP}[i][0] = \mathtt{true}$ for all $i$; $\text{DP}[0][w] = \mathtt{false}$ for $w > 0$.
 
-### Algorithm
+### 9.3 Algorithm
 
 ```text
 SUBSET-SUM(a[1..n], W)

notes/courses/CSCI-UA-310/15-16-greedy.md

@@ -33,13 +33,13 @@ A greedy algorithm is correct when the **greedy choice property** holds: a globa
 
 ## 2. Interval Scheduling (Activity Selection)
 
-### Problem
+### 2.1 Problem
 
 **Input**: $n$ activities, each with start time $s_i$ and finish time $f_i$. Two activities $i$ and $j$ are **compatible** if they do not overlap (i.e., $[s_i, f_i)$ and $[s_j, f_j)$ are disjoint).
 
 **Output**: A maximum-size set of mutually compatible activities.
 
-### Greedy Algorithm
+### 2.2 Greedy Algorithm
 
 **Greedy rule**: Always select the compatible activity with the **earliest finish time**.
 
@@ -57,7 +57,7 @@ GREEDY-ACTIVITY-SELECTOR(s, f)
 
 **Running time**: $O(n \log n)$ for sorting; $O(n)$ for selection. Total: $O(n \log n)$.
 
-### Correctness (Exchange Argument)
+### 2.3 Correctness (Exchange Argument)
 
 **Theorem**: `GREEDY-ACTIVITY-SELECTOR` produces an optimal solution.
 
@@ -75,15 +75,15 @@ Since the greedy finishes no later than optimal at each step, if $O$ has $m$ act
 
 ## 3. Fractional Knapsack
 
-### Problem
+### 3.1 Problem
 
 Same as 0/1 Knapsack but items can be split: take a **fraction** $x_i \in [0,1]$ of item $i$.
 
 **Input**: $n$ items with weights $w_i$ and values $v_i$; capacity $W$.
 
 **Output**: Fractions $x_1, \dots, x_n \in [0,1]$ maximizing $\sum_i x_i v_i$ subject to $\sum_i x_i w_i \leq W$.
 
-### Greedy Algorithm
+### 3.2 Greedy Algorithm
 
 **Key insight**: Take as much as possible of the item with the highest **value density** $v_i / w_i$.
 
@@ -123,7 +123,7 @@ The exchange argument is the canonical way to prove greedy correctness:
 
 ## 5. Interval Partitioning (Scheduling on Multiple Machines)
 
-### Problem
+### 5.1 Problem
 
 **Input**: $n$ jobs with start times $s_j$ and finish times $f_j$.
 
@@ -133,7 +133,7 @@ The exchange argument is the canonical way to prove greedy correctness:
 
 **Lower bound**: We need at least depth($\mathcal{I}$) machines.
 
-### Greedy Algorithm
+### 5.2 Greedy Algorithm
 
 Sort jobs by start time. Maintain a set of machines. For each job, assign it to any machine that is free; if none is free, open a new machine.
 
@@ -145,7 +145,7 @@ Sort jobs by start time. Maintain a set of machines. For each job, assign it to
 
 ## 6. Huffman Coding
 
-### Problem
+### 6.1 Problem
 
 **Input**: An alphabet $C = \{c_1, \dots, c_n\}$ where character $c_i$ has frequency $f_i$.
 
@@ -182,7 +182,7 @@ HUFFMAN(C)
 
 **Running time**: $O(n \log n)$ using a binary min-heap.
 
-### Worked Example
+### 7.1 Worked Example
 
 Frequencies: $a:45, b:13, c:12, d:16, e:9, f:5$.
 
@@ -202,13 +202,13 @@ $B(T) = 45 \cdot 1 + 13 \cdot 3 + 12 \cdot 3 + 16 \cdot 3 + 9 \cdot 4 + 5 \cdot
 
 ## 8. Correctness Proof
 
-### Lemma 1 (Greedy Choice Property)
+### 8.1 Lemma 1 (Greedy Choice Property)
 
 Let $x$ and $y$ be the two characters with the smallest frequencies. There exists an optimal prefix-free code in which $x$ and $y$ have codewords of the same length that differ only in the last bit (i.e., they are siblings in the code tree).
 
 **Proof**: Take any optimal tree $T$. Let $b$ and $c$ be siblings at maximum depth. WLOG $f_b \leq f_c$ and $f_x \leq f_y$. Swapping $x \leftrightarrow b$ and $y \leftrightarrow c$ does not increase $B(T)$ because $x, y$ have the smallest frequencies and move to the deepest positions.
 
-### Lemma 2 (Optimal Substructure)
+### 8.2 Lemma 2 (Optimal Substructure)
 
 Let $T$ be an optimal code tree for $C$, and suppose $x$ and $y$ are sibling leaves. Let $T'$ be $T$ with $x$ and $y$ replaced by their parent $z$ with $f_z = f_x + f_y$. Then $T'$ is optimal for $C' = C \setminus \{x, y\} \cup \{z\}$.
 

notes/courses/CSCI-UA-310/17-graphs.md

@@ -37,7 +37,7 @@ A **graph** $G = (V, E)$ consists of a set of **vertices** $V$ and a set of **ed
 
 ## 2. Representations
 
-### Adjacency List
+### 2.1 Adjacency List
 
 An array $\text{Adj}[1 \dots n]$ where $\text{Adj}[u]$ is a linked list of all vertices $v$ such that $(u, v) \in E$.
 
@@ -46,7 +46,7 @@ An array $\text{Adj}[1 \dots n]$ where $\text{Adj}[u]$ is a linked list of all v
 * **Time to iterate** over all neighbors of $u$: $\Theta(\deg(u))$.
 * **Preferred** for sparse graphs.
 
-### Adjacency Matrix
+### 2.2 Adjacency Matrix
 
 An $n \times n$ matrix $A$ where $A[u][v] = 1$ if $(u,v) \in E$, else $0$.