Let's start our journey into advanced data structures and algorithms by understanding the complexity analysis of algorithms.

When analyzing algorithms, we often want to know how the algorithm's performance changes as the input size increases. This is where Big O notation comes into play.

Big O notation is a mathematical notation used to describe the upper bound or worst-case scenario of an algorithm's time or space complexity. It allows us to compare and analyze algorithms based on their efficiency.

For example, consider the following code snippet:`java int n = 100; int sum = 0; for (int i = 1; i <= n; i++) { sum += i; }

1The above code calculates the sum of the numbers from 1 to 100. In Big O notation, we would represent the time complexity of this algorithm as O(n), indicating that the number of iterations increases linearly with the input size.
2
3By analyzing the time complexity of an algorithm using Big O notation, we can make informed decisions about which algorithms to use in different scenarios. We can identify algorithms that are more efficient for large input sizes and avoid algorithms that are inefficient or have exponential time complexity.
4
5It's important to note that Big O notation focuses on the growth rate of an algorithm's performance as the input size increases, rather than the actual runtime in seconds or the exact number of operations performed.
6
7In conclusion, Big O notation is a powerful tool that allows us to analyze the efficiency of algorithms based on their worst-case scenario. Understanding Big O notation will help us make informed decisions and choose the most appropriate algorithms for our specific use cases.

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public class Main {

    public static void main(String[] args) {

        int n = 100;

        int sum = 0;

        for (int i = 1; i <= n; i++) {

            sum += i;

        }

        System.out.println("The sum of the numbers from 1 to " + n + " is " + sum);

    }

}

Arrays and Linked Lists

Arrays and linked lists are common data structures used to store and manipulate collections of elements. Although they serve similar purposes, there are notable differences between them.

Arrays

An array is a fixed-size data structure that stores elements of the same type in contiguous memory locations. Each element in an array can be accessed using an index. Arrays provide constant-time access to elements, which means accessing any element takes the same amount of time, regardless of its position.

Here's an example of initializing and printing an array in Java:

1int[] array = new int[5];
2array[0] = 1;
3array[1] = 2;
4array[2] = 3;
5array[3] = 4;
6array[4] = 5;
7
8for (int i = 0; i < array.length; i++) {
9    System.out.print(array[i] + " ");
10}

Linked Lists

A linked list is a dynamic data structure that consists of nodes, each containing a value and a reference to the next node. Unlike arrays, linked lists do not require contiguous memory locations. Instead, each node points to the next node in the sequence.

Here's an example of initializing and printing a linked list in Java:

1import java.util.LinkedList;
2
3LinkedList<Integer> linkedList = new LinkedList<>();
4linkedList.add(1);
5linkedList.add(2);
6linkedList.add(3);
7linkedList.add(4);
8linkedList.add(5);
9
10for (Integer value : linkedList) {
11    System.out.print(value + " ");
12}

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}

import java.util.ArrayList;

import java.util.LinkedList;

​

public class ArraysAndLinkedLists {

​

    public static void main(String[] args) {

        // Arrays

        int[] array = new int[5];

        array[0] = 1;

        array[1] = 2;

        array[2] = 3;

        array[3] = 4;

        array[4] = 5;

​

        System.out.println("Array:");

        for (int i = 0; i < array.length; i++) {

            System.out.print(array[i] + " ");

        }

        System.out.println();

​

        // Linked Lists

        LinkedList<Integer> linkedList = new LinkedList<>();

        linkedList.add(1);

        linkedList.add(2);

        linkedList.add(3);

        linkedList.add(4);

        linkedList.add(5);

​

        System.out.println("Linked List:");

Stacks and Queues

In the world of data structures, stacks and queues are two fundamental concepts that are widely used to store and access data. Both stacks and queues follow the principle of First-In-First-Out (FIFO) or Last-In-First-Out (LIFO) ordering.

Stacks

A stack is a data structure that follows the LIFO ordering, meaning that the most recently added element is the first one to be removed. Think of a stack of plates, where you can only add or remove plates from the top. The operations supported by a stack are:

• push: Add an element to the top.
• pop: Remove and return the top element.
• peek: Return the top element without removing it.

Here's an example of creating a stack, pushing elements into it, and popping and peeking elements from it in Java:

1import java.util.Stack;
2
3class Main {
4  public static void main(String[] args) {
5    // Create a stack
6    Stack<String> stack = new Stack<String>();
7
8    // Push elements into the stack
9    stack.push("element1");
10    stack.push("element2");
11    stack.push("element3");
12
13    // Pop elements from the stack
14    String poppedElement = stack.pop();
15
16    // Print the popped element
17    System.out.println("Popped Element: " + poppedElement);
18
19    // Get the top element of the stack
20    String topElement = stack.peek();
21
22    // Print the top element
23    System.out.println("Top Element: " + topElement);
24  }
25}

Queues

A queue is a data structure that follows the FIFO ordering, meaning that the element that has been in the queue the longest is the first one to be removed. Think of a queue of people waiting in a line, where the person who has been waiting the longest gets served first. The operations supported by a queue are:

• enqueue: Add an element to the end.
• dequeue: Remove and return the first element.
• peek: Return the first element without removing it.

Here's an example of creating a queue, enqueueing elements into it, and dequeuing and peeking elements from it in Java:

1import java.util.LinkedList;
2import java.util.Queue;
3
4class Main {
5  public static void main(String[] args) {
6    // Create a queue
7    Queue<String> queue = new LinkedList<String>();
8
9    // Enqueue elements into the queue
10    queue.add("element1");
11    queue.add("element2");
12    queue.add("element3");
13
14    // Dequeue elements from the queue
15    String dequeuedElement = queue.remove();
16
17    // Print the dequeued element
18    System.out.println("Dequeued Element: " + dequeuedElement);
19
20    // Get the first element of the queue
21    String firstElement = queue.peek();
22
23    // Print the first element
24    System.out.println("First Element: " + firstElement);
25  }
26}

Understanding the concepts of stacks and queues and their respective applications is crucial in developing efficient algorithms and solving various problems.

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class Main {

  public static void main(String[] args) {

    // Create a stack

    Stack<String> stack = new Stack<String>();

​

    // Push elements into the stack

    stack.push("element1");

    stack.push("element2");

    stack.push("element3");

​

    // Pop elements from the stack

    String poppedElement = stack.pop();

​

    // Print the popped element

    System.out.println("Popped Element: " + poppedElement);

​

    // Get the top element of the stack

    String topElement = stack.peek();

​

    // Print the top element

    System.out.println("Top Element: " + topElement);

  }

}

Trees

In computer science, a tree is a widely-used abstract data type that resembles a hierarchical structure. It consists of nodes connected by edges. The topmost node in the tree is called the root, while the nodes at the bottom are called leaves.

Trees provide an organized way of storing and accessing data. They are especially useful when we need to represent hierarchical relationships or perform operations such as searching, insertion, and deletion efficiently.

There are various types of trees, including binary trees, AVL trees, B-trees, and heaps. Each type has its own characteristics and operations associated with it.

Binary Trees

A binary tree is a tree in which each node can have at most two children, referred to as the left child and the right child. The left child is smaller than the parent node, while the right child is greater. Binary trees are commonly used and implemented in various algorithms and data structures.

Here's an example of creating a binary tree and accessing its nodes in Java:

1class Main {
2  public static void main(String[] args) {
3
4    // Creating a binary tree
5    TreeNode rootNode = new TreeNode(1);
6    TreeNode leftChild = new TreeNode(2);
7    TreeNode rightChild = new TreeNode(3);
8    rootNode.left = leftChild;
9    rootNode.right = rightChild;
10
11    // Accessing tree nodes
12    System.out.println("Root Node: " + rootNode.val);
13    System.out.println("Left Child: " + rootNode.left.val);
14    System.out.println("Right Child: " + rootNode.right.val);
15  }
16}
17
18class TreeNode {
19  int val;
20  TreeNode left;
21  TreeNode right;
22
23  public TreeNode(int val) {
24    this.val = val;
25  }
26}

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class Main {

  public static void main(String[] args) {

​

    // Creating a binary tree

    TreeNode rootNode = new TreeNode(1);

    TreeNode leftChild = new TreeNode(2);

    TreeNode rightChild = new TreeNode(3);

    rootNode.left = leftChild;

    rootNode.right = rightChild;

​

    // Accessing tree nodes

    System.out.println("Root Node: " + rootNode.val);

    System.out.println("Left Child: " + rootNode.left.val);

    System.out.println("Right Child: " + rootNode.right.val);

  }

}

​

class TreeNode {

  int val;

  TreeNode left;

  TreeNode right;

​

  public TreeNode(int val) {

    this.val = val;

  }

}

Binary Search Trees

Binary search trees (BSTs) are a type of binary tree where the left child of a node is smaller than the node, and the right child is greater. This organization allows for efficient search, insert, and delete operations.

A binary search tree is commonly represented by a root node, which is the topmost node in the tree. Each node in the tree has a value and two child nodes, referred to as the left child and the right child.

Here's an example of creating and searching for an element in a binary search tree in Java:

1class Main {
2  public static void main(String[] args) {
3
4    // Creating a binary search tree
5    TreeNode rootNode = new TreeNode(8);
6    rootNode.left = new TreeNode(3);
7    rootNode.right = new TreeNode(10);
8    rootNode.left.left = new TreeNode(1);
9    rootNode.left.right = new TreeNode(6);
10    rootNode.left.right.left = new TreeNode(4);
11    rootNode.left.right.right = new TreeNode(7);
12    rootNode.right.right = new TreeNode(14);
13    rootNode.right.right.left = new TreeNode(13);
14
15    // Searching in the binary search tree
16    TreeNode result = search(rootNode, 6);
17
18    // Printing the result
19    if (result != null) {
20      System.out.println("Element found: " + result.val);
21    } else {
22      System.out.println("Element not found.");
23    }
24  }
25
26  public static TreeNode search(TreeNode rootNode, int value) {
27    if (rootNode == null || rootNode.val == value) {
28      return rootNode;
29    }
30
31    if (value < rootNode.val) {
32      return search(rootNode.left, value);
33    }
34
35    return search(rootNode.right, value);
36  }
37}
38
39class TreeNode {
40  int val;
41  TreeNode left;
42  TreeNode right;
43
44  public TreeNode(int val) {
45    this.val = val;
46  }
47}

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}

class Main {

  public static void main(String[] args) {

​

    // Creating a binary search tree

    TreeNode rootNode = new TreeNode(8);

    rootNode.left = new TreeNode(3);

    rootNode.right = new TreeNode(10);

    rootNode.left.left = new TreeNode(1);

    rootNode.left.right = new TreeNode(6);

    rootNode.left.right.left = new TreeNode(4);

    rootNode.left.right.right = new TreeNode(7);

    rootNode.right.right = new TreeNode(14);

    rootNode.right.right.left = new TreeNode(13);

​

    // Searching in the binary search tree

    TreeNode result = search(rootNode, 6);

​

    // Printing the result

    if (result != null) {

      System.out.println("Element found: " + result.val);

    } else {

      System.out.println("Element not found.");

    }

  }

​

  public static TreeNode search(TreeNode rootNode, int value) {

    if (rootNode == null || rootNode.val == value) {

      return rootNode;

    }

In graph theory, a graph is a collection of nodes connected by edges. It is a powerful data structure that represents relationships between objects.

The nodes in a graph can be any type of object, and the edges represent connections or relationships between the nodes. Graphs can be used to model various real-world scenarios, such as social networks, transportation networks, and computer networks.

Here's an example of creating a graph in Java:

1class Node {
2  public int value;
3  public List<Node> neighbors;
4
5  public Node(int value) {
6    this.value = value;
7    this.neighbors = new ArrayList<>();
8  }
9}
10
11public class Graph {
12  public List<Node> nodes;
13
14  public Graph() {
15    this.nodes = new ArrayList<>();
16  }
17
18  public void addNode(Node newNode) {
19    this.nodes.add(newNode);
20  }
21}
22
23public class Main {
24  public static void main(String[] args) {
25    // Creating a graph
26    Graph graph = new Graph();
27
28    Node nodeA = new Node(1);
29    Node nodeB = new Node(2);
30    Node nodeC = new Node(3);
31    Node nodeD = new Node(4);
32
33    nodeA.neighbors.add(nodeB);
34    nodeA.neighbors.add(nodeC);
35    nodeB.neighbors.add(nodeD);
36    nodeC.neighbors.add(nodeD);
37
38    graph.addNode(nodeA);
39    graph.addNode(nodeB);
40    graph.addNode(nodeC);
41    graph.addNode(nodeD);
42  }
43}

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}

class Node {

  public int value;

  public List<Node> neighbors;

​

  public Node(int value) {

    this.value = value;

    this.neighbors = new ArrayList<>();

  }

}

​

public class Graph {

  public List<Node> nodes;

​

  public Graph() {

    this.nodes = new ArrayList<>();

  }

​

  public void addNode(Node newNode) {

    this.nodes.add(newNode);

  }

}

​

public class Main {

  public static void main(String[] args) {

    // Creating a graph

    Graph graph = new Graph();

​

    Node nodeA = new Node(1);

    Node nodeB = new Node(2);

Hash tables, also known as hash maps, are a fundamental data structure in computer science. They allow efficient insertion, deletion, and retrieval of data by using a technique called hashing.

Hashing is a process that converts data into a fixed-size value called a hash code or hash value. The hash code is then used as an index to store the data in an array-like structure called the hash table.

The key idea behind hash tables is that they provide constant-time average-case performance for basic operations such as insertion, deletion, and retrieval. This makes them incredibly useful in scenarios where fast data access is required.

Let's take a look at an example of using a hash table in Java:

1import java.util.HashMap;
2
3public class Main {
4  public static void main(String[] args) {
5    // Creating a hash table
6    HashMap<String, Integer> hashTable = new HashMap<>();
7
8    // Inserting data
9    hashTable.put("apple", 1);
10    hashTable.put("banana", 2);
11    hashTable.put("cherry", 3);
12
13    // Retrieving data
14    int value = hashTable.get("banana");
15    System.out.println(value); // Output: 2
16
17    // Deleting data
18    hashTable.remove("apple");
19    boolean contains = hashTable.containsKey("apple");
20    System.out.println(contains); // Output: false
21  }
22}

In this example, we create a HashMap object from the Java util library. We then insert key-value pairs into the hash table using the put method. We can retrieve values using the get method and remove entries using the remove method.

Hash tables have various applications in computer science, such as indexing and searching data, implementing caches, and counting occurrences of elements. They are also used as a building block for other data structures like hash sets and hash maps.

Understanding hash tables and their use cases is essential for developing efficient algorithms and data structures. It is also important to consider factors such as hash function quality, load factor, and collision resolution strategies when working with hash tables.

Now that you have a basic understanding of hash tables, let's move on to exploring other advanced data structures and algorithms. Stay tuned for more exciting topics!

Heaps are a fundamental data structure in computer science that represent a complete binary tree. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible.

Heaps are commonly used to implement the priority queue data structure. A priority queue is an abstract data type that provides efficient access to the element with the highest (or lowest) priority.

In Java, the PriorityQueue class is often used to represent a heap. Here's an example of creating a min heap using PriorityQueue:

1import java.util.PriorityQueue;
2
3public class Main {
4  public static void main(String[] args) {
5    // Creating a min heap
6    PriorityQueue<Integer> minHeap = new PriorityQueue<>();
7
8    // Inserting elements
9    minHeap.offer(5);
10    minHeap.offer(1);
11    minHeap.offer(3);
12
13    // Peeking the smallest element
14    int minElement = minHeap.peek();
15    System.out.println(minElement); // Output: 1
16
17    // Removing the smallest element
18    minHeap.poll();
19
20    // Peeking the new smallest element
21    minElement = minHeap.peek();
22    System.out.println(minElement); // Output: 3
23  }
24}

In this example, we create a PriorityQueue object to represent a min heap. We can insert elements into the heap using the offer method. The peek method returns the smallest element without removing it, and the poll method removes the smallest element from the heap.

Heaps have various applications in computer science, such as sorting algorithms (e.g., heapsort), graph algorithms (e.g., Dijkstra's algorithm), and data compression algorithms (e.g., Huffman coding). They are also used in priority scheduling, event-driven simulations, and system resource allocation.

Understanding the heap data structure and its applications is essential for developing efficient algorithms and solving complex problems.

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import java.util.PriorityQueue;

​

public class Main {

  public static void main(String[] args) {

    // Creating a min heap

    PriorityQueue<Integer> minHeap = new PriorityQueue<>();

​

    // Inserting elements

    minHeap.offer(5);

    minHeap.offer(1);

    minHeap.offer(3);

​

    // Peeking the smallest element

    int minElement = minHeap.peek();

    System.out.println(minElement); // Output: 1

​

    // Removing the smallest element

    minHeap.poll();

​

    // Peeking the new smallest element

    minElement = minHeap.peek();

    System.out.println(minElement); // Output: 3

  }

}

When it comes to sorting algorithms, there are various options to choose from. One of the advanced sorting algorithms that you might come across is the Quicksort algorithm.

The Quicksort algorithm is an efficient divide-and-conquer algorithm that sorts an array by selecting a pivot element and partitioning the other elements into sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively.

Here's an example implementation of the Quicksort algorithm in Java:

1// Quicksort algorithm
2
3public class Quicksort {
4  public static void quicksort(int[] arr, int low, int high) {
5    if (low < high) {
6      int pivotIndex = partition(arr, low, high);
7      quicksort(arr, low, pivotIndex - 1);
8      quicksort(arr, pivotIndex + 1, high);
9    }
10  }
11
12  public static int partition(int[] arr, int low, int high) {
13    int pivot = arr[high];
14    int i = low - 1;
15
16    for (int j = low; j < high; j++) {
17      if (arr[j] < pivot) {
18        i++;
19        swap(arr, i, j);
20      }
21    }
22
23    swap(arr, i + 1, high);
24    return i + 1;
25  }
26
27  public static void swap(int[] arr, int i, int j) {
28    int temp = arr[i];
29    arr[i] = arr[j];
30    arr[j] = temp;
31  }
32
33  public static void main(String[] args) {
34    int[] arr = {12, 4, 5, 6, 7, 3, 1, 15};
35
36    System.out.println("Original array:");
37    for (int num : arr) {
38      System.out.print(num + " ");
39    }
40    System.out.println();
41
42    quicksort(arr, 0, arr.length - 1);
43
44    System.out.println("Sorted array:");
45    for (int num : arr) {
46      System.out.print(num + " ");
47    }
48    System.out.println();
49  }
50}

In this example, we have an array of integers that we want to sort using the Quicksort algorithm. The quicksort function takes in the array, the starting index, and the ending index. It partitions the array based on a pivot element and recursively sorts the sub-arrays. The partition function helps in selecting the pivot element and partitioning the array.

Understanding advanced sorting algorithms like Quicksort can be beneficial in cases where you need to efficiently sort large datasets or optimize the performance of your code. It is widely used in practice due to its average-case time complexity of O(n log n) and its ability to handle large amounts of data.

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}

// Quicksort algorithm

​

public class Quicksort {

  public static void quicksort(int[] arr, int low, int high) {

    if (low < high) {

      int pivotIndex = partition(arr, low, high);

      quicksort(arr, low, pivotIndex - 1);

      quicksort(arr, pivotIndex + 1, high);

    }

  }

​

  public static int partition(int[] arr, int low, int high) {

    int pivot = arr[high];

    int i = low - 1;

​

    for (int j = low; j < high; j++) {

      if (arr[j] < pivot) {

        i++;

        swap(arr, i, j);

      }

    }

​

    swap(arr, i + 1, high);

    return i + 1;

  }

​

  public static void swap(int[] arr, int i, int j) {

    int temp = arr[i];

    arr[i] = arr[j];

Dynamic programming is a powerful technique used to solve complex problems by breaking them down into smaller, overlapping subproblems. It allows us to efficiently solve problems by storing and reusing computed results, rather than recomputing them.

In dynamic programming, we divide a problem into smaller subproblems and solve each subproblem only once. We then combine the results of the subproblems to solve the original problem.

Dynamic programming is particularly useful when a problem has overlapping subproblems, meaning that multiple subproblems share subproblems in common.

For example, let's say we want to calculate the nth Fibonacci number. We can use dynamic programming to store the results of calculating the Fibonacci numbers up to n, so that we can reuse them when calculating subsequent Fibonacci numbers.

Here's an example of calculating the nth Fibonacci number using dynamic programming in Java:

1public class Fibonacci {
2  public static int fib(int n) {
3    int[] memo = new int[n + 1];
4    return fibHelper(n, memo);
5  }
6
7  public static int fibHelper(int n, int[] memo) {
8    if (n <= 1) {
9      return n;
10    }
11
12    if (memo[n] != 0) {
13      return memo[n];
14    }
15
16    memo[n] = fibHelper(n - 1, memo) + fibHelper(n - 2, memo);
17    return memo[n];
18  }
19
20  public static void main(String[] args) {
21    int n = 10;
22    int result = fib(n);
23    System.out.println("The " + n + "th Fibonacci number is: " + result);
24  }
25}

In this example, we store the calculated Fibonacci numbers in the memo array to avoid redundant calculations. The fibHelper function checks if the result for the current fib(n) has already been computed and stored in the memo array. If it has, it returns the stored result. Otherwise, it calculates and stores the result before returning it.

Dynamic programming is a powerful technique that can significantly improve the efficiency of solving complex problems. By breaking down problems into smaller, overlapping subproblems and reusing computed results, we can achieve faster and more scalable solutions.

Greedy algorithms are a class of algorithms that make locally optimal choices at each stage with the hope of finding a global optimum. Unlike dynamic programming, greedy algorithms do not always guarantee the optimal solution, but they are often faster and simpler to implement.

One example of a problem that can be solved using a greedy algorithm is the Knapsack problem. In the knapsack problem, we are given a set of items, each with a weight and a value, and a knapsack with a certain weight capacity. The goal is to find the most valuable combination of items that can fit into the knapsack without exceeding its capacity.

Here's an example of solving the knapsack problem using a greedy algorithm in Java:

1class Main {
2  public static void main(String[] args) {
3    int[] weights = {10, 20, 30};
4    int[] values = {60, 100, 120};
5    int capacity = 50;
6    int maxValue = knapSack(weights, values, capacity);
7    System.out.println("The maximum value that can be obtained is: " + maxValue);
8  }
9
10  public static int knapSack(int[] weights, int[] values, int capacity) {
11    int n = weights.length;
12    int[][] dp = new int[n + 1][capacity + 1];
13    
14    for (int i = 1; i <= n; i++) {
15      for (int j = 1; j <= capacity; j++) {
16        if (weights[i - 1] <= j) {
17          dp[i][j] = Math.max(values[i - 1] + dp[i - 1][j - weights[i - 1]], dp[i - 1][j]);
18        } else {
19          dp[i][j] = dp[i - 1][j];
20        }
21      }
22    }
23    
24    return dp[n][capacity];
25  }
26}

In this example, we use a two-dimensional array dp to store the maximum value that can be obtained for each combination of items and capacity. We iterate through every item and every capacity and make a choice to either include the current item or exclude it based on whether its weight exceeds the remaining capacity. We update the dp array accordingly to store the maximum value.

Greedy algorithms offer a simple and elegant solution to many optimization problems. However, it is important to note that they do not always produce the optimal solution. It is crucial to analyze the problem and make reasonable assumptions to ensure that a greedy approach will lead to a reasonably good solution.

Make sure to familiarize yourself with greedy algorithms and when they are suitable for solving problems! You will encounter many scenarios in which a greedy algorithm can be utilized to find an efficient solution.

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class Main {

  public static void main(String[] args) {

    int[] weights = {10, 20, 30};

    int[] values = {60, 100, 120};

    int capacity = 50;

    int maxValue = knapSack(weights, values, capacity);

    System.out.println("The maximum value that can be obtained is: " + maxValue);

  }

​

  public static int knapSack(int[] weights, int[] values, int capacity) {

    int n = weights.length;

    int[][] dp = new int[n + 1][capacity + 1];

    for (int i = 1; i <= n; i++) {

      for (int j = 1; j <= capacity; j++) {

        if (weights[i - 1] <= j) {

          dp[i][j] = Math.max(values[i - 1] + dp[i - 1][j - weights[i - 1]], dp[i - 1][j]);

        } else {

          dp[i][j] = dp[i - 1][j];

        }

      }

    }

    return dp[n][capacity];

  }

}

Congratulations! You have completed the tutorial on Introduction to Advanced Data Structures and Algorithms. In this tutorial, we covered various important topics including Big O notation, arrays and linked lists, stacks and queues, trees, binary search trees, graphs, hash tables, heaps, advanced sorting algorithms, dynamic programming, and greedy algorithms.

Throughout the course, we discussed the complexity analysis of algorithms using Big O notation and compared and contrasted different data structures such as arrays and linked lists. We explored the concepts of stacks and queues, along with their applications. We learned about tree data structures and binary search trees, and how they can be used to solve various problems. We also introduced graphs as a data structure and discussed graph traversal algorithms.

We explored the concept of hash tables and their use cases, as well as the heap data structure and its applications. We covered advanced sorting algorithms such as quicksort and mergesort. We also introduced dynamic programming as a technique for solving problems efficiently. Lastly, we discussed greedy algorithms and their applications.

By completing this tutorial, you have gained a solid understanding of advanced data structures and algorithms. You now have the knowledge and skills to analyze the complexity of algorithms, choose appropriate data structures for different scenarios, and solve problems efficiently using various algorithms.

Keep practicing and applying these concepts in your coding projects to reinforce your understanding and become a more proficient software engineer. Advanced data structures and algorithms play a crucial role in designing efficient and scalable software systems. Good luck on your learning journey!