Algorithms and AI: Data Structures and Searching

• The text discusses algorithms and AI, covering data structures, searching, sorting, efficiency, and the P vs. NP problem.

Data Structures

• A stack is a LIFO (Last In, First Out) data structure, where the last element added is the first one removed
  • The push operation adds new elements to the stack
  • The pop operation removes the top element
• A queue is a FIFO (First In, First Out) data structure, where the first element added is the first one removed.
  • The enqueue operation adds new elements to the end of the queue
  • The dequeue operation removes elements from the beginning of the queue

Searching

• Linear search involves checking each element in a list from start to finish
  • The efficiency is O(n), as it may require checking all elements
  • It is used for unsorted lists or when dealing with only a few elements
• Binary search works by repeatedly dividing the search interval in half
  • It functions only on sorted lists
  • The efficiency is O(log n), as each step halves the search space
• A binary search tree (BST) is a data structure where each node has at most two children
  • The left subtree contains values less than the node's value
  • The right subtree contains values greater than the node's value
  • Search begins at the root, moves to the left for smaller values, and to the right for larger values, repeating until the element is found or an empty node is reached
  • The best-case efficiency is O(log n), but it can be O(n) if the tree is unbalanced

Sorting

• Bubble sort compares and swaps adjacent elements if they are in the wrong order, repeating until the list is sorted
  • The best-case complexity is O(n), when the list is already sorted
  • The average and worst-case complexity is O(n²)
  • It is inefficient for large datasets but easy to implement
• Selection sort finds the smallest element and swaps it with the first element, then the second smallest with the second element, and so on
  • Its complexity is O(n²) in the best, average, and worst cases
  • It is slow for large datasets
  • It involves fewer swaps than bubble sort
• Both bubble sort and selection sort have a time complexity of O(n²) but selection sort performs fewer swaps and both are inefficient for large lists, but easy to implement

Efficiency

• Efficiency is how quickly an algorithm can perform a task, where inefficient algorithms can become unusable with large datasets
• Algorithms should be optimized to save space and computing time
• Big O notation describes the runtime of algorithms relative to input size n
  • O(1) is constant runtime
  • O(log n) is logarithmic runtime
  • O(n) is linear runtime
  • O(n log n) are efficient sorting algorithms
  • O(n²) is quadratic runtime
  • O(2ⁿ) is exponential runtime
• To determine number of commands depending on n
  • Analyze the number of loops and recursive calls
  • A single loop over n elements is O(n)
  • A nested loop is O(n²)
  • Recursive halving is O(log n)

P vs. NP Problem

• P (Polynomial Time) is a set of problems that can be solved in polynomial time
• NP (Non-Deterministic Polynomial Time) is a set of problems where a solution can be quickly verified, but finding a solution may take exponentially long
• It is unknown whether P = NP or P ≠ NP
  • If P = NP, many complex problems could be solved efficiently
  • If P ≠ NP, finding quick solutions for many problems remains difficult

Conclusion

• Data structures like stacks and queues help manage data efficiently
• Search algorithms like linear and binary search have different uses
• Sorting algorithms like bubble sort and selection sort are inefficient for large datasets
• The efficiency of algorithms is crucial, with O-notation helping in analysis
• The P vs. NP problem remains unsolved in computer since

Explore fundamental algorithms and AI concepts. Covers data structures like stacks and queues, including push, pop, enqueue, and dequeue operations. Discusses searching algorithms such as linear and binary search, including their efficiency and applications.