Maze generation using the hole-digging method starts with an initial grid space that is completely filled with walls, and constructs a maze through a recursive space expansion process that continues digging from random coordinates as long as two adjacent spaces ahead are walls.

This method does not necessarily create a closed space, but rather forms a single continuous network of paths.

On the other hand, the Botoshi method is a method of separating passages by knocking down walls in either the up, down, left, or right directions from wall pillars placed at regular intervals.It requires very little calculation and can be generated at high speed, but it requires specific rules for processing the top and left edges.

Then, the fractal algorithm using depth-first search uses a stacked data structure to randomly proceed from the current location to unvisited neighboring cells, and when it reaches a dead end, it repeats the process of backtracking to the past branching point.

This allows us to mathematically derive a highly difficult grid structure with long straight lines and complex branches. All generation calculations are completed within a single thread on the browser, and a huge maze array is instantly constructed in memory without relying on the computing power of an external server.

The solver function for conquering the constructed maze grid is centered on the A-star algorithm, which is breadth-first search and agent-based heuristic search.

Breadth-first search sequentially stores cells that are close to the starting coordinates in a queue data structure and expands the search area isotropically, so it derives an exact solution that guarantees the shortest number of steps in a maze where the path weights are uniform.

However, as the board becomes larger, the search space increases exponentially, so the A-star algorithm is incorporated as a more advanced route search.

In the A-star algorithm, the straight-line distance or Manhattan distance from the current location to the goal is calculated as the estimated cost, and the node with the minimum sum with the actual cost already traveled is taken out from the prioritized queue and the search proceeds.

This dramatically reduces searches in unnecessary directions, making it possible to identify the optimal route coordinate array in milliseconds while avoiding complicated dead ends.

The calculated shortest path is overlaid on the maze array data as an independent layer and visualized as a continuous set of coordinates from the start point to the end point.

All operations, from the maze generation process to the execution of the search algorithm, are handled locally on the client side, completely eliminating the overhead of network delays and server-side state synchronization.

When the player challenges the generated maze, movement input using the direction keys is captured in real time through an event listener and immediately converted into an update process for the current position coordinates on a two-dimensional array.

Collision with a wall is determined by checking whether the array value of the movement destination coordinates is a passage or a wall at a constant time, and only valid movement is reflected in the screen drawing.

At the same time, a time measurement model that launches in conjunction with the game start trigger utilizes the Performance API's microsecond-accurate timer to closely track player completion times.

The timer stops the moment the goal coordinate arrival event fires, and the elapsed time is converted from milliseconds to human-readable format and recorded as performance data.

This series of input processing and state transitions are all performed in synchronization with the browser's redraw cycle, resulting in extremely smooth interaction.

The shortest path derived by the analysis algorithm and the history of manually traversed paths are dynamically overlaid on the screen by drawing directly onto DOM elements or via the Canvas API.

In addition, the constructed maze structure data and solution path data are connected to the ability to export in multiple formats for external reuse and printing purposes.

In scalable vector graphics output, which is a vector format, the straight lines that make up the maze walls are mathematically described as path elements, and the answer path is defined as a path with separate stroke attributes, thereby generating a high-quality image file that does not degrade in resolution no matter how enlarged it is.

Portable network graphics output builds pixel data on the drawing context of an HTML5 canvas and converts it into a byte stream as a compressed image including an alpha channel.

In addition, for print output, a complete output pipeline has been built that calculates the physical dimensions of page size and margins, automatically scales the maze, and generates a document object that can be used as an offline writing puzzle.

The data structure that forms the basis of the maze engine consists of securing the states of passages and walls in memory as a one-dimensional flattened array or a nested two-dimensional array.

Each cell uses bit flags to densely pack and maintain attributes such as the presence or absence of walls on the top, bottom, left and right, the visited state in the search algorithm, and whether it is included in the shortest path in a single integer value.

This bitwise state management minimizes the amount of space computation even in large grids and prevents performance degradation caused by browser garbage collection.

Additionally, to avoid the problem of depth-first search where the recursive call depth exceeds the call stack limit, the algorithm has been rewritten to iteratively loop using an explicit stack array.

As a result, we have established a robust processing platform that can safely and stably generate and analyze even extremely large maze spaces containing tens of thousands of cells, while completely eliminating the risk of stack overflow.

Furthermore, through this optimization, the design is designed to minimize the load on the browser's rendering engine even when drawing large-scale mazes using CSS grid layout.

This maze generation and pathfinding system does more than just provide puzzles; it also serves as an advanced learning platform for visually understanding the behavior of graph theory and search algorithms in computer science.

Every time the user switches the generation algorithm from the digging method to the stick-throw method or depth-first search, qualitative changes in the topology and difficulty of the generated maze are immediately reflected on the screen, making it possible to intuitively grasp the mathematical characteristics of each algorithm.

In addition, by visualizing step-by-step animations of the solver execution process using breadth-first search and A-star algorithms, the search node selection criteria based on the queue expansion status and heuristic evaluation values ​​are clearly shown.

This converts abstract program code execution logic into concrete visual information of expanding the search area on a two-dimensional grid.

By continuously intersecting spatial comprehension ability training through interactive play and theoretical observation of route optimization mechanisms on the same interface, it maximizes its value as an experiential programming algorithm learning guide.