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Algorithms/Complexity.md @@ -0,0 +1,77 @@ +--- +date: 02.09.2024 +type: theoretical +--- + + +$\mathcal{O}$ + +## Complexity +- Time Complexity: amount of time an algorithm takes to complete as a function of the size of its input. +- Space Complexity: amount of memory it uses as a function of the size of its input. + +### Big O +- **Formal Definition:** + + Big O notation, $\mathcal{O}(f(n))$, describes the upper bound of the time or space complexity of an algorithm in terms of the input size $n$. It characterizes the worst-case scenario of an algorithm's growth rate. + + Formally, a function $T(n)$ is said to be $\mathcal{O}(f(n))$ if there exist positive constants $c$ and $n_0$ such that for all $n \geq n_0$: + + $$ + 0 \leq T(n) \leq c \cdot f(n). + $$ + + This means that $T(n)$ will grow at most as fast as $f(n)$, up to a constant multiple $c$, for sufficiently large $n$. + + +![Big O cheatsheet](Big%20o.png) + + + +### Informally: +Big-O notation provides an upper limit on: +- **Time**: How long an algorithm will take. +- **Space**: How much memory it will require. +It helps compare algorithms by focusing on their growth rate and ignoring constant factors. + + +### Other Notations for Growth Rates + +1. **Big-O ($\mathcal{O}(f(n))$)**: + - Describes the **upper bound** (worst-case) of $T(n)$. + - Informally: $T(n)$ grows no faster than $f(n)$. + - Formal Definition: + $$ + T(n) \text{ is } \mathcal{O}(f(n)) \iff \exists c > 0, n_0 > 0 \text{ such that } T(n) \leq c \cdot f(n) \text{ for all } n \geq n_0. + $$ + +2. **Big-Omega ($\Omega(f(n))$)**: + - Describes the **lower bound** (best-case) of $T(n)$. + - Informally: $T(n)$ grows at least as fast as $f(n)$. + - Formal Definition: + $$ + T(n) \text{ is } \Omega(f(n)) \iff \exists c > 0, n_0 > 0 \text{ such that } T(n) \geq c \cdot f(n) \text{ for all } n \geq n_0. + $$ + +3. **Big-Theta ($\Theta(f(n))$)**: + - Describes the **exact bound** (tight bound) of $T(n)$. + - Informally: $T(n)$ grows exactly as fast as $f(n)$. + - Formal Definition: + $$ + T(n) \text{ is } \Theta(f(n)) \iff T(n) \text{ is } \mathcal{O}(f(n)) \text{ and } T(n) \text{ is } \Omega(f(n)). + $$ + +4. **Little-O ($o(f(n))$)**: + - Describes a **loose upper bound** of $T(n)$. + - Informally: $T(n)$ grows strictly slower than $f(n)$. + - Formal Definition: + $$ + T(n) \text{ is } o(f(n)) \iff \forall c > 0, \exists n_0 > 0 \text{ such that } T(n) < c \cdot f(n) \text{ for all } n \geq n_0. + $$ + +Think of growth rates as vehicles on a highway: +- $\mathcal{O}(f(n))$: $T(n)$ cannot go faster than $f(n)$. +- $\Omega(f(n))$: $T(n)$ cannot go slower than $f(n)$. +- $\Theta(f(n))$: $T(n)$ moves at the exact same speed as $f(n)$. +- $o(f(n))$: $T(n)$ moves slower than $f(n)$ and never catches up. + diff --git a/Advanced Algorithms/Divide and Conquer.md b/Advanced Algorithms/Divide and Conquer.md new file mode 100644 index 0000000..aa200ad --- /dev/null +++ b/Advanced Algorithms/Divide and Conquer.md @@ -0,0 +1,41 @@ +--- +date: 09.09.2024 +type: theoretical +--- + + +Divide and Conquer is a powerful algorithmic paradigm used to solve complex problems by breaking them down into smaller, more manageable sub-problems. + +## Steps in Divide and Conquer + +1. **Divide:** + - The problem is divided into smaller subproblems that are similar to the original but smaller in size. + - This division continues until the subproblems become simple enough to be solved directly. + +2. **Conquer:** + - Solve the smaller subproblems recursively. If the subproblems are small enough (base cases), solve them directly without further recursion. + +3. **Combine:** + - Combine the solutions of the subproblems to get the solution to the original problem. + +### Example: Merge Sort + +**Merge Sort** is a classic example of the Divide and Conquer approach: + +1. **Divide:** + - Split the unsorted array into two halves recursively until each subarray contains a single element. + +2. **Conquer:** + - Merge two sorted halves by comparing elements from each half in sorted order. + +3. **Combine:** + - Combine the sorted halves to form a single sorted array. + +The time complexity of Merge Sort is $\mathcal{O}(n \log n)$ because the array is repeatedly divided in half ($\log n$ divisions) and each division requires a linear amount of work ($\mathcal{O}(n)$) to merge. + +### Other Examples + +- **Quick Sort:** Uses Divide and Conquer by selecting a "pivot" element and partitioning the array into elements less than and greater than the pivot. +- **Binary Search:** Recursively divides a sorted array in half to search for an element, reducing the search space by half each time. +- **Strassen's Matrix Multiplication:** An algorithm that reduces the time complexity of matrix multiplication from $\mathcal{O}(n^3)$ to approximately $\mathcal{O}(n^{2.81})$ by recursively breaking down matrices. + diff --git a/Advanced Algorithms/Dynamic Programming.md b/Advanced Algorithms/Dynamic Programming.md new file mode 100644 index 0000000..e9de82e --- /dev/null +++ b/Advanced Algorithms/Dynamic Programming.md @@ -0,0 +1,139 @@ +--- +date: 11.09.2024 +type: theoretical +--- + + +**travelling salesperson** + +## Definition +- Approach to design algorithms +- Optimal solution in polynomial time +- Usually used alongside [Divide and Conquer](Divide%20and%20Conquer.md) +- Used to better the time [complexity](Complexity.md), but usually worsens the space complexity + + +### How do we create a DP soluton? +- Characterize optimal solution +- Recursively define the value of an optimal solution +- Compute the value +- Construct an optimal solution **from the computed information** + +## Problem discussion + +### Calculating Fibonacci numbers + +- Recursion is extremely bad -> PF was wrong + - Because the branches are too many + - We compute f(x) where f(x) has already been computed + - Memoization! + + +#### Approaches +- Linear time $\mathcal{O}(n)$ - **Bottom to top** +```pseudo +F[0] = 1 +F[1] = 1 +for i in {2, .., n} do + F[i] = F[i − 1] + F[i − 2] +end for +return F[n] +``` + +Compute the entire array and just return the result. + +- Memoization $\mathcal{O}(n)$ - **Top to bottom** +```javascript +function fibMemo(index, cache) { + cache = cache || []; + if (cache[index]) return cache[index]; + else { + if (index < 3) return 1; + else { + cache[index] = fibMemo(index - 1, cache) + fibMemo(index - 2, cache); + } + } + + return cache[index]; +} +``` + +### [Rod cutting](https://www.geeksforgeeks.org/problems/rod-cutting0840/1) +Given a rod of length N inches and an array of prices, price[]. price[i] denotes the value of a piece of length i. Determine the maximum value obtainable by cutting up the rod and selling the pieces. + +#### Optimal solution +- Let the left part have a length $l$ +- Then $R[n] = P[l] + R[n-l]$ +- Where $P[l]$ is the price +- First cut of length should be the maximal one + + +### Knight Collecting Rewards +>Knight Collecting rewards + Input: Integers $n, m$, table of rewards of size $n \times m$ + Question: What is the maximum reward that the knight can get for its journey. + + + +#### Strategy +- We look at the bounds defined by the problem statement + - We know that the knight can get into the cell $(n-1, m-1)$ either from $(n-2,m-3)$ or $(n-3,m-2)$ + - Hence, we have to know the most rewarding path to these two points from $(0,0)$ + +#### Approaches + +- $\mathcal{O}(n*m)$ +``` +Reward(int n, int m, matrix A) +Create matrix R of size n × m with −∞ values +R[0, 0] = A[0, 0] +for i in {0, .., n − 1} do + for j in {0, .., m − 1} do + R[i, j] = A[i, j] + max R[i − 1, j − 2], R[i − 2, j − 1] + end for +end for +return R[n-1, m-1] +``` + +This doesn't tell us how we got there, but it does give us max rewards. +We can easily include path memory by making elements be a linked list or create a separate table blah blah. + + + +### [Longest common non-contiguous sequence](https://www.geeksforgeeks.org/longest-common-subsequence-dp-4/)[^1] + +> LCS + Input: Two sequences X = [x1, x2, . . . xm], Y = [y1, y2, . . . , Yn]. + Question: Longest common subsequence + +#### Strategy +![[LCS.png]] + +Or this formula: +$$ + +c[i,j] = \begin{cases} + 0, & \text{if } i=0 \text{ or } j=0\\ + c[i-1,j-1]+1, & \text{if } i,j>0 \text{ and } x_i = y_i\\ + max\{c[i,j-1], c[i-1], j\}, & \text{if } i,j>0 \text{ and } x_i \neq y_i\\ + +\end{cases} +$$ + + + +### [Dominating set](https://www.geeksforgeeks.org/dominant-set-of-a-graph/)[^2] in a path + +>Input: A path $P$ with a specified positive cost for each vertex. + Output: Choose a subset of vertices $S$ with a minimum cost such that for each $v \in P$: either $v \in S$ or there is u such that $u \in S$ and $u, v$ are neighbors. + + +#### Strategy +- Identify some subproblems + - $A[i]$ cost of the cheapest set $S_i \subset P_i$ which dominates all vertices in $P_i$ <- not great + - $A[i,0]$ equals the cost of the cheapest $S_i$ which dominates $P_i$ and $v_i \notin S_i$ + - $A[i,1]$ equals the cost of the cheapest + + +[^1]: **contiguous** - next or near in time or sequence +[^2]: [Subset of the vertices of the graph $G$, such that any vertex of $G$ is in it, or has a neighbor in it](https://en.wikipedia.org/wiki/Dominating_set) diff --git a/Advanced Algorithms/Graph Algorithms.md b/Advanced Algorithms/Graph Algorithms.md new file mode 100644 index 0000000..35c6eba --- /dev/null +++ b/Advanced Algorithms/Graph Algorithms.md @@ -0,0 +1,157 @@ +--- +type: theoretical +--- + + +Can be utilized in algorithms. Here's a rundown of a couple of popular graph algorithms along with their use case and [Complexity](Complexity.md). TSP is [NP](P%20vs.%20NP.md), but it can be approximated/solved using a graph algorithm! + +![](graph-algorithms-infographic.gif) +# Graph Algorithms + +## Shortest Path Algorithms + +### Dijkstra's Algorithm + +Finds the shortest path from a source vertex to all other vertices in a graph with non-negative edge weights. + + 1. Initialize distances: + - Set the distance to the source vertex as 0 and all others as infinity. + 2. Use a priority queue (min-heap) to extract the vertex with the smallest distance. + 3. For each neighboring vertex, calculate the tentative distance via the current vertex: + - If the tentative distance is smaller than the known distance, update it. + 4. Repeat until all vertices are processed. + + - Using a priority queue: $O((V + E) \log V)$, where $V$ is the number of vertices and $E$ is the number of edges. + +![](Dijkstra01.gif) +### Floyd-Warshall Algorithm + +Computes shortest paths between all pairs of nodes in a weighted graph. + + 1. Initialize distances with edge weights. + 2. Iteratively update distances by considering each node as an intermediate point. + 3. Handle both positive and negative edge weights (no negative cycles allowed). +- **Complexity**: $O(V^3)$, where $V$ is the number of vertices. + +![Floyd-Warshall Algorithm](218610439-933d939c-a9e1-489d-a0a2-da8cf7bf51c1.gif) + +--- + +## Minimum Spanning Tree Algorithms + +### Kruskal's Algorithm + +Finds the minimum spanning tree (MST) of a graph. + + 1. Sort all edges by weight. + 2. Add edges to the MST in increasing order of weight. + 3. Skip edges that form a cycle (using Union-Find data structure :) ). + +- **Complexity**: $O(E \log E)$, where $E$ is the number of edges. +![](Animation%20of%20Kruskal's%20Algorithm.gif) + + +### Prim's Algorithm + +Constructs the minimum spanning tree (MST) of a graph. + + 1. Start with an arbitrary node. + 2. Repeatedly add the smallest edge that connects a node in the MST to a node outside it. + 3. Continue until all nodes are included. + + - $O(E + V \log V)$ using a priority queue, where $V$ is the number of vertices and $E$ is the number of edges. + +![Prim's Algorithm](4e486f5e-8437-4e2b-8964-5d5860208502_1650942396.093046.gif) + +--- + +## Graph Traversal Algorithms + +### Depth-First Search (DFS) + +Explores as far as possible along each branch before backtracking. + + - Detects cycles. + - Topological sorting. + - Pathfinding in mazes. + +- **Complexity**: $O(V + E)$. + +![](1_WR4AtjT_nhwSOtAW99Yd5g.gif) + + +--- + +### Breadth-First Search (BFS) + +explores all neighbors at the current depth before moving to the next level. + + - Finding shortest paths in unweighted graphs. + - Level-order traversal of trees. + +- **Complexity**: $O(V + E)$. +![](bfs.gif) + +![](dfs-vs-bfs.gif) + +--- + +## Topological Sorting + +Arranges the vertices of a directed acyclic graph (DAG) in a linear order such that for every directed edge $(u, v)$, $u$ comes before $v$. + + - Perform a DFS and use a stack to store the vertices in reverse order of completion. + +- **Complexity**: $O(V + E)$. +![](anim.gif) + +--- + +## Maximum Flow Algorithms + +### Ford-Fulkerson Algorithm + +**Finds the maximum flow in a flow network. + + 1. Initialize the flow to 0. + 2. While there is an augmenting path, increase the flow along the path. + 3. Repeat until no augmenting paths remain. + +- **Complexity**: $O(E \cdot \text{max\_flow})$. +![](FordFulkerson.gif) +--- + +### Edmonds-Karp Algorithm + +A refinement of the Ford-Fulkerson algorithm that uses BFS to find augmenting paths. + +- **Complexity**: $O(V \cdot E^2)$. + +![](56e7f380-cd33-11eb-90b2-658e8d102a95.gif) + + +--- + +## Grid-Based Algorithms + +### A* Algorithm + +Finds the shortest path in a weighted grid using a heuristic. + + 1. Start from the source node. + 2. Use a priority queue to explore the most promising paths first. + 3. Use a heuristic to guide the search toward the goal. + +- **Complexity**: Depends on the heuristic but generally better than BFS for weighted grids. +![](A.gif) +--- + +## Graph Coloring + +Assigns colors to vertices such that no two adjacent vertices share the same color. + +- **Algorithms**: + - Greedy coloring: $O(V^2)$. + - Backtracking for exact coloring. + +--- diff --git a/Advanced Algorithms/Graphs.md b/Advanced Algorithms/Graphs.md new file mode 100644 index 0000000..85896c6 --- /dev/null +++ b/Advanced Algorithms/Graphs.md @@ -0,0 +1,98 @@ +--- +type: mixed +--- +## Definition of a Graph + +A data structure that consists of: +- A set of vertices (or nodes). +- A set of edges, which connect pairs of vertices. + +### Types of Graphs +1. **Directed Graph**: + - Edges have a direction, going from one vertex to another. + - Example: Websites linked via hyperlinks. + +2. **Undirected Graph**: + - Edges have no direction; they simply connect two vertices. + - Example: Social networks where a connection is mutual. + +3. **Weighted Graph**: + - Each edge is assigned a weight or cost. + - Example: Road networks where edge weights represent distances. + +## Binary Tree + +A hierarchical data structure where each node has at most two children, referred to as the left and right child. This structure is foundational for tasks such as expression parsing and hierarchical data representation. + + - Maximum of two children per node. + - Recursive structure: subtrees are binary trees themselves. + +![](Pasted%20image%2020241203231354.png) + +--- + +## Binary Search Tree (BST) + +A Binary Search Tree is a binary tree that maintains sorted order: +- For any node: + - All values in the left subtree are smaller. + - All values in the right subtree are larger. +- Enables $O(\log n)$ search, insertion, and deletion. + +![](binary-search-tree-sorted-array-animation.gif) + +--- + +## Balanced Tree (AVL Tree) + +A self-balancing binary search tree where: +- The balance factor (difference in heights of left and right subtrees) of any node is at most one. +- Rotations are used to maintain balance during insertions and deletions. + - Guaranteed $O(\log n)$ operations. + - Ensures balanced growth for fast access. +![](YieLsCqeuV-avlbal.gif) + + +--- + +## Grids as Graphs + +Can be modeled as a graph: +- Vertices: Represent grid cells. +- Edges: Represent valid movements between cells (e.g., up, down, left, right). +![](Pasted%20image%2020241203231638.png) +--- + +## Famous Graph Problems +Solved using [Graph Algorithms](Graph%20Algorithms.md). +### Shortest Path Problems +- Find the minimum distance or cost to travel between two nodes. +- **We can use**: + - Dijkstra's Algorithm (if non-negative weights). + - Bellman-Ford Algorithm (handles negative weights). + - Floyd-Warshall Algorithm (if we want all-pairs shortest paths). + +### Minimum Spanning Tree (MST) +- Find a subset of edges that connects all vertices with the minimum total weight. +- **We can use**: + - Kruskal's Algorithm. + - Prim's Algorithm. + +### Topological Sorting +- Arrange the vertices of a DAG in linear order such that for every directed edge $(u, v)$, $u$ appears before $v$. + +### Max Flow / Min Cut +- Find the maximum flow possible in a network from a source to a sink. + - Ford-Fulkerson Algorithm. + - Edmonds-Karp Algorithm. + +### Traveling Salesman Problem (TSP) +- Find the shortest tour that visits each vertex exactly once and returns to the starting point. + +### Pathfinding in Grids +- Find the shortest path in a grid-based graph. +- **We can use**: + - A* Algorithm (heuristic-based). + - Breadth-First Search (BFS) for unweighted grids. + +--- \ No newline at end of file diff --git a/Advanced Algorithms/P vs. NP.md b/Advanced Algorithms/P vs. NP.md new file mode 100644 index 0000000..36c5f41 --- /dev/null +++ b/Advanced Algorithms/P vs. NP.md @@ -0,0 +1,82 @@ +--- +type: theoretical +--- + +![Video](https://www.youtube.com/watch?v=mdQzAp7gSns) +![Video 2](https://youtu.be/pQsdygaYcE4) +## Definition + +- Class P + - The set of decision problems that can be solved by a deterministic Turing machine in polynomial time[^1]. These are problems considered to be efficiently solvable. +- $O(\log n)$ + - The set of decision problems for which a given solution can be verified in polynomial time by a deterministic Turing machine[^2]. NP stands for "nondeterministic polynomial time." +- :LiLoaderPinwheel: Does $\mathbf{P = NP}$? + - This is one of the most important open questions in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved. + +## Understanding P and NP +![](Pasted%20image%2020241203234032.png) +### Class P Problems + + - Solvable in polynomial time. + - Algorithms exist that can find a solution efficiently as the input size grows. +- **As seen in**: + - Prime Testing: Determining if a number is prime. + - Shortest Path: Finding the shortest path in a graph (e.g., Dijkstra's algorithm). + - Sorting Algorithms: Such as Quick Sort and Merge Sort. + +### Class NP Problems + + - Solutions can be verified in polynomial time. + - No known polynomial-time algorithms to solve all NP problems. +- **As in**: + - Subset Sum: Determining if a subset of a given set of integers sums up to a target integer. + - 3-SAT: Determining if a Boolean formula in conjunctive normal form with at most three literals per clause is satisfiable. + - Hamiltonian Cycle: Determining if a Hamiltonian cycle exists in a graph. + +## NP-Complete Problems + +- The hardest problems in NP. A problem is NP-Complete if: + - It is in NP. + - Every problem in NP can be reduced to it in polynomial time[^3]. +- Implications: If any NP-Complete problem is solvable in polynomial time, then $P = NP$. + - Traveling Salesperson Problem (Decision Version): Determining if there's a tour shorter than a given length. + - Clique Problem: Finding a complete subgraph (clique) of a certain size in a graph. + - Vertex Cover: Determining if there exists a set of vertices covering all edges. + +## Problems + +### Subset Sum Problem + +- **Input**: A set of integers and a target sum. +- **Question**: Is there a subset whose sum equals the target? +- **Approach**: + - **Exponential Time**: Checking all possible subsets. + - **Dynamic Programming**: Pseudo-polynomial time algorithm when numbers are small. + +### 3-SAT Problem + +- **Input**: A Boolean formula in 3-CNF (Conjunctive Normal Form). +- **Question**: Is there a truth assignment that satisfies the formula? +- **Importance**: The first problem proven to be NP-Complete (Cook-Levin theorem). + +## Strategies +- Find solutions close to optimal in polynomial time. +- Practical methods that find good-enough solutions without guaranteeing optimality (heuristics). +- Restricting the problem to a subset where it becomes solvable in polynomial time. + + +## NP vs. NP-Complete Differences + +|**Aspect**|**NP**|**NP-Complete**| +|---|---|---| +|**Definition**|Problems whose solutions can be verified quickly.|Hardest problems in NP. All NP problems reduce to them.| +|**Relation to P**|Contains P (P ⊆ NP).|If any NP-complete problem is in P, P = NP.| +|**Ease of Solution**|Some problems may have unknown solution methods.|Believed to be computationally difficult to solve.| +|**Examples**|Subset Sum, 3-SAT|TSP, Clique, Vertex Cover| + +![](Pasted%20image%2020241203234600.png) +[^1]: a theoretical computing machine that uses a predetermined set of rules to determine its actions. +[^2]: a theoretical model that, unlike a deterministic Turing machine, can make "guesses" to find solutions more efficiently. +[^3]: a method of converting one problem to another in polynomial time, preserving the problem's computational complexity. +[^4]: the complements of NP-Complete problems, where verifying a "no" instance is in NP. + diff --git a/Advanced Algorithms/Pasted image 20241203234600.png b/Advanced Algorithms/Pasted image 20241203234600.png new file mode 100644 index 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algorithm**. It is designed to preprocess a pattern to facilitate efficient searching. + +### Definition +For a string $P$ of length $m$, the prefix function $\pi[i]$ for $i = 1, 2, \ldots, m$ is the length of the longest proper prefix of the substring $P[1 \ldots i]$ that is also a suffix of this substring. + +### Key Points +1. A proper prefix of a string is a prefix that is not equal to the entire string. [^1] +2. $\pi[i]$ helps skip unnecessary comparisons in pattern matching by indicating the next position to check after a mismatch. +3. $\pi[1] = 0$ always, since no proper prefix of a single character can also be a suffix. + +### Example +For the pattern $P = "ababcab"$: +- $P[1] = "a"$: $\pi[1] = 0$. +- $P[1 \ldots 2] = "ab"$: No prefix matches the suffix, so $\pi[2] = 0$. +- $P[1 \ldots 3] = "aba"$: Prefix "a" matches suffix "a", so $\pi[3] = 1$. +- $P[1 \ldots 4] = "abab"$: Prefix "ab" matches suffix "ab", so $\pi[4] = 2$. +- Continue similarly to compute $\pi[i]$ for the entire pattern. + +--- + +## Knuth-Morris-Pratt (KMP) Algorithm + +The KMP algorithm is a pattern matching algorithm that uses the prefix function $\pi$ to efficiently search for occurrences of a pattern $P$ in a text $T$. + +### Key Idea +When a mismatch occurs during the comparison of $P$ with $T$, use the prefix function $\pi$ to determine the next position in $P$ to continue matching, rather than restarting from the beginning. + +### Steps +1. Compute the prefix function $\pi$ for the pattern $P$. +2. Search: + - Compare $P$ with substrings of $T$. + - If there’s a mismatch at $P[j]$ and $T[i]$, use $\pi[j]$ to shift $P$ rather than restarting at $P[1]$. +3. The algorithm runs in $O(n + m)$ time [complexity](Complexity.md), where $n$ is the length of $T$ and $m$ is the length of $P$. + +--- + +## Rabin-Karp Algorithm + +The Rabin-Karp algorithm is another pattern matching algorithm, notable for using hashing to identify potential matches. + +### Key Idea +Instead of comparing substrings character by character, the algorithm compares hash values of the pattern and substrings of the text. + +### Steps +1. Compute the hash value of the pattern $P$ and the first substring of the text $T$ of length $m$. +2. Slide the window over $T$ and compute hash values for the next substrings in constant time using a rolling hash. [^2] +3. If the hash value of a substring matches the hash value of $P$, compare the actual strings to confirm the match. + +### Hash Function +The hash function is typically chosen such that it is fast to compute and minimizes collisions: +$$ +h(s) = (s[1] \cdot p^{m-1} + s[2] \cdot p^{m-2} + \ldots + s[m] \cdot p^0) \mod q, +$$ +where: +- $p$ is a base (e.g., a small prime number), +- $q$ is a large prime to avoid overflow. + +### Complexity +- Best Case: $O(n + m)$, where $n$ is the length of the text and $m$ is the length of the pattern. +- Worst Case: $O(nm)$ due to hash collisions. + +--- + +## KMP v.s. Rabin-Karp + +| Feature | Knuth-Morris-Pratt (KMP) | Rabin-Karp | +| ------------- | ------------------------ | --------------------------------------------------- | +| Technique | Prefix function | Hashing | +| Preprocessing | Compute $\pi$ array | Compute hash of $P$ | +| Efficiency | $O(n + m)$ | $O(n + m)$ (best), $O(nm)$ (worst) | +| Use Case | Best for exact matches | Useful for multiple patterns or approximate matches | +| | | | +_This graphic is too AI generated for me_ -> Use KMP when looking for a pattern, use RK when multiple patterns + +--- + +## Footnotes + +[^1]: A proper prefix of a string $s$ is any prefix of $s$ that is not equal to $s$ itself. For example, proper prefixes of "abc" are "", "a", and "ab". +[^2]: A rolling hash computes the hash of a new substring by updating the hash of the previous substring, avoiding the need to recompute from scratch. diff --git a/Advanced Algorithms/Recurrence relations.md b/Advanced Algorithms/Recurrence relations.md new file mode 100644 index 0000000..bf623c9 --- /dev/null +++ b/Advanced Algorithms/Recurrence relations.md @@ -0,0 +1,126 @@ +--- +type: mixed +--- + +In algorithmics, a **recurrence relation** is often used to describe the time complexity of recursive algorithms. It expresses how the running time of an algorithm depends on the size of the input and the cost of recursive calls. + +- **Example**: Consider the merge sort algorithm, which divides the input of size $n$ into two halves, recursively sorts each half, and then merges the two sorted halves. Its recurrence relation is: + $$ + T(n) = 2T\left(\frac{n}{2}\right) + O(n), + $$ + where: + - $2T\left(\frac{n}{2}\right)$ represents the two recursive calls on halves of the input. + - $O(n)$ represents the time to merge the two halves. + +--- + +## Solving Recurrence Relations for Time Complexity + +To determine the time complexity of an algorithm, we solve the recurrence relation to find an explicit formula for $T(n)$. + +### 1. **Backtracking** +This involves repeatedly substituting the recurrence relation into itself until a pattern emerges. + +- **Example**: For the recurrence $T(n) = 2T\left(\frac{n}{2}\right) + O(n)$: + - First substitution: + $$ + T(n) = 2\left[2T\left(\frac{n}{4}\right) + O\left(\frac{n}{2}\right)\right] + O(n) + = 4T\left(\frac{n}{4}\right) + 2O\left(\frac{n}{2}\right) + O(n). + $$ + - Second substitution: + $$ + T(n) = 8T\left(\frac{n}{8}\right) + 4O\left(\frac{n}{4}\right) + 2O\left(\frac{n}{2}\right) + O(n). + $$ + - General pattern: + $$ + T(n) = 2^k T\left(\frac{n}{2^k}\right) + O\left(n \log n\right). + $$ + - When $k = \log n$, the base case $T(1)$ is reached, so: + $$ + T(n) = O(n \log n). + $$ + +--- + +### 2. **Master Theorem** +The **Master Theorem** provides a direct way to analyze recurrence relations of the form: +$$ +T(n) = aT\left(\frac{n}{b}\right) + O(n^d), +$$ +where: +- $a$ is the number of recursive calls, +- $b$ is the factor by which the input size is divided, +- $d$ is the exponent of the non-recursive work. [^1] + +The solution depends on the comparison of $a$ with $b^d$: +1. **Case 1 ($a < b^d$)**: The work outside recursion dominates. + $$ + T(n) = O(n^d). + $$ + +2. **Case 2 ($a = b^d$)**: The work is evenly distributed. + $$ + T(n) = O(n^d \log n). + $$ + +3. **Case 3 ($a > b^d$)**: The recursion dominates. + $$ + T(n) = O(n^{\log_b a}). + $$ + +- **Example**: For $T(n) = 2T\left(\frac{n}{2}\right) + O(n)$: + - $a = 2$, $b = 2$, $d = 1$. + - Compare $a$ with $b^d$: $2 = 2^1$, so it falls under **Case 2**. + - Solution: + $$ + T(n) = O(n \log n). + $$ + +--- + +### 3. **Substitution Method** +This involves guessing a solution and proving it correct using **mathematical induction**. + +- **Example**: For $T(n) = 2T\left(\frac{n}{2}\right) + O(n)$, guess $T(n) = O(n \log n)$: + 1. Assume $T(k) = O(k \log k)$ for $k < n$ (inductive hypothesis). + 2. Substitute into the recurrence: + $$ + T(n) = 2T\left(\frac{n}{2}\right) + O(n). + $$ + Using the hypothesis: + $$ + T\left(\frac{n}{2}\right) = O\left(\frac{n}{2} \log \frac{n}{2}\right) = O\left(\frac{n}{2} (\log n - 1)\right). + $$ + Simplify: + $$ + T(n) = 2 \cdot O\left(\frac{n}{2} \log n\right) + O(n) = O(n \log n). + $$ + 3. Conclusion: The guess is correct. + +--- + +## Applications in Algorithm Design + +1. **Divide and Conquer**: Many divide-and-conquer algorithms, like merge sort, quicksort, and binary search, have their time complexity described by recurrence relations. + +2. **Dynamic Programming**: Recurrences are also used to describe subproblem dependencies in dynamic programming algorithms. + +3. **Graph Algorithms**: Recurrences appear in graph traversal techniques and optimization algorithms (e.g., shortest paths). + +--- + +## Tips for Mastering Recurrence Relations + +1. Understand the **nature of recursion** in the algorithm (e.g., how input is divided, base cases, etc.). +2. Identify the **dominant term** in the recurrence to estimate growth. +3. Use tools like **backtracking**, the **Master Theorem**, or **substitution** to solve recurrences efficiently. +4. Practice interpreting recurrences in terms of algorithm behavior. + +--- + +### Backlinks + +- [Recurrence relations](Discrete%20Structures/Recurrence%20relations.md) +- [Complexity](Complexity.md) + +[^1]: The Master Theorem simplifies solving divide-and-conquer recurrences by comparing the relative growth of recursion and non-recursive work. diff --git a/Advanced Algorithms/Sorting algorithms.md b/Advanced Algorithms/Sorting algorithms.md new file mode 100644 index 0000000..c941494 --- /dev/null +++ b/Advanced Algorithms/Sorting algorithms.md @@ -0,0 +1,39 @@ +--- +date: 02.09.2024 +type: mixed +--- + + +## Sorting algorithms and their [complexity](Complexity.md) + +| **Algorithm** | **Time Complexity** | +| ------------- | ------------------------------------------------------------- | +| Selection | $\mathcal{O}(n^2)$ | +| Insertion | $\mathcal{O}(n^2)$ | +| Bubble | $\mathcal{O}(n^2)$ | +| Quick | $\mathcal{O}(n^2)$ worst case, $\mathcal{O}(n\log n)$ average | +| Merge | 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