P4_Search3x3Puzzle

P4_Search3x3Puzzle

A 3x3 puzzle, often referred to as a 3x3 sliding puzzle or 8-puzzle, is a classic puzzle game played on a 3x3 grid with 8 numbered tiles and one empty space. The objective of the puzzle is to arrange the tiles in ascending order (from left to right, top to bottom), starting with the number 1 in the top-left corner and leaving the empty space in the bottom-right corner. You can slide tiles into the empty space to rearrange them.

Here's an example of a scrambled 3x3 puzzle:

2  3  1
5     6
4  8  7

The goal is to reach the following configuration:

1  2  3
4  5  6
7  8

To solve the 3x3 puzzle using the A* algorithm, you can follow these steps:

1. State Representation: Define a representation of the puzzle's state. This typically involves using a 2D array or matrix to represent the arrangement of tiles, where each tile is assigned a number, and an empty space is represented as a blank or zero.

2. Heuristic Function (h): Create a heuristic function that estimates the cost from the current state to the goal state. The most common heuristic for the 8-puzzle is the Manhattan distance, which calculates the sum of the distances each tile is away from its goal position (measured horizontally and vertically).

3. Start and Goal States: Define the initial state (the scrambled puzzle) and the goal state (the solved puzzle).

4. Open and Closed Lists: Create two lists, the open list and the closed list. The open list stores states that need to be explored, and the closed list stores states that have already been explored.

5. A Algorithm*:

   • Initialize the open list with the initial state.
   • While the open list is not empty:
     • Select the state with the lowest cost (f = g + h, where g is the cost to reach the current state from the initial state).
     • Expand the selected state by generating its possible successor states (by moving a tile into the empty space).
     • For each successor state:
       • If it is the goal state, the puzzle is solved.
       • If it is not in the closed list, calculate its f-value and add it to the open list.
     • Add the current state to the closed list.

6. Backtracking: Once the goal state is reached, you can backtrack from the goal state to the initial state to find the sequence of moves that solved the puzzle.

Here's a high-level pseudocode of the A* algorithm for solving the 3x3 puzzle:

1. Initialize open list with the initial state.
2. Initialize closed list as empty.
3. While open list is not empty:
     a. Select the state with the lowest f-value from the open list.
     b. Generate successor states by moving a tile into the empty space.
     c. For each successor state:
          i. If it is the goal state, the puzzle is solved.
          ii. If it is not in the closed list, calculate its f-value and add it to the open list.
     d. Add the current state to the closed list.
4. Backtrack from the goal state to the initial state to find the solution path.

The A* algorithm guarantees the shortest path to the goal state while considering the heuristic function, making it an efficient way to solve the 3x3 puzzle.

heapq is a Python module that provides a collection of heap queue algorithms. A heap is a specialized tree-based data structure that satisfies the heap property. In a min-heap, for example, the parent nodes always have values less than or equal to their child nodes, making it efficient to extract the smallest element from the heap.

The heapq module allows you to use a list as a min-heap or max-heap and provides functions for performing various operations on the heap, such as inserting elements, removing elements, and finding the smallest (or largest) element. It's particularly useful for solving problems involving priority queues and sorting elements efficiently.

Here are some of the main functions provided by the heapq module:

1. heapify(iterable): This function takes an iterable (e.g., a list) and transforms it into a valid heap in-place. It rearranges the elements within the iterable so that they satisfy the heap property.

2. heappush(heap, item): This function adds an element item to the heap heap while maintaining the heap property.

3. heappop(heap): This function removes and returns the smallest element from the heap heap. After removing the element, it rearranges the remaining elements to restore the heap property.

4. heappushpop(heap, item): This function pushes a new element item onto the heap heap and then immediately pops and returns the smallest element. This can be more efficient than calling heappush() followed by heappop() separately.

5. heapreplace(heap, item): This function is similar to heappushpop(), but it returns the popped element first and then pushes the new element onto the heap. This can also be more efficient in certain scenarios.

6. nlargest(n, iterable, key=None): Returns the n largest elements from an iterable (e.g., a list). You can specify a custom key function to determine the ordering.

7. nsmallest(n, iterable, key=None): Returns the n smallest elements from an iterable, similar to nlargest(). The key function can be used to customize the ordering.

8. merge(*iterables, key=None, reverse=False): Merges multiple sorted iterables into a single sorted iterable. It's useful for merging sorted lists or other data structures efficiently.

heapq provides an efficient way to maintain a priority queue, which is a common data structure used in algorithms like Dijkstra's shortest path algorithm, Prim's minimum spanning tree algorithm, and various other graph and tree traversal algorithms.

Here's an example of using heapq to find the three smallest elements in a list:

import heapq

data = [5, 3, 8, 1, 2, 9, 4, 7, 6]

# Convert the list into a min-heap
heapq.heapify(data)

# Find the three smallest elements
smallest_three = heapq.nsmallest(3, data)
print(smallest_three)  # Output: [1, 2, 3]

In this example, we first convert the list data into a min-heap using heapq.heapify(), and then we use heapq.nsmallest() to find the three smallest elements efficiently.

Files

• main.py