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"Snakes and Ladders: The Quickest Way Up" problem, we can use the Breadth-First Search (BFS) algorithm. Below is the PHP solution for this problem:

<?php

function quickestWayUp($ladders, $snakes) {
    $board = array_fill(1, 100, 0);

    // Set up ladders
    foreach ($ladders as $ladder) {
        $board[$ladder[0]] = $ladder[1] - $ladder[0];
    }

    // Set up snakes
    foreach ($snakes as $snake) {
        $board[$snake[0]] = $snake[1] - $snake[0];
    }

    // BFS to find the shortest path
    $queue = [[1, 0]];
    $visited = array_fill(1, 100, false);
    $visited[1] = true;

    while (!empty($queue)) {
        list($current, $moves) = array_shift($queue);

        for ($dice = 1; $dice <= 6; $dice++) {
            $next = $current + $dice;
            if ($next <= 100) {
                $next += $board[$next];

                if ($next == 100) {
                    return $moves + 1;
                }

                if (!$visited[$next]) {
                    $visited[$next] = true;
                    array_push($queue, [$next, $moves + 1]);
                }
            }
        }
    }

    return -1;
}

?>

Explanation:

1. Initialization:

   • The board array stores the effect of ladders and snakes on each cell.
   • queue is used for BFS, initialized with the starting position (1, 0) representing the start of the board and 0 moves.
   • visited array keeps track of visited cells to avoid revisiting them.

2. Setup Ladders and Snakes:

   • For ladders, the board at the start cell of the ladder has the value of the end cell minus the start cell.
   • Similarly, for snakes, the board at the start cell of the snake has the value of the end cell minus the start cell.

3. BFS to Find Shortest Path:

   • Dequeue the current position and number of moves.
   • For each dice roll (1 to 6), calculate the next position.
   • Apply any ladder or snake effect.
   • Check if the next position is 100 (end of the board).
   • If not visited, mark it as visited and enqueue it with incremented move count.

4. Handling Input:

   • The readInput function reads the input from standard input and processes it.
   • For each test case, read the number of ladders and snakes, and their respective positions.
   • Call the quickestWayUp function for each test case and print the result.

Pthon

BFS

def bfs(node, squares):
    visited = set()
    q = []
    q.append(node)
    sentinel = None
    q.append(sentinel)
    movement = 0
    while(len(q)>0):
        n = q.pop(0)
       
        if(n is sentinel):
            if(q):
                movement += 1
                q.append(sentinel)
                continue
            else:
                return -1
        if (n.finish):
            return movement
        if(n in visited):
            continue
        visited.add(n)
        for i in n.edges:
            q.append(squares[i])
            
def quickestWayUp(ladders, snakes):
    squares = {}
    dladders = {}
    dsnakes = {}
    for i in ladders:
        s1 = i[0]
        s2 = i[1]
        dladders[s1] = s2
    for i in snakes:
        s1 = i[0]
        s2 = i[1]
        dsnakes[s1] = s2
    for i in range(100, 0, -1):
        n = Node(i)
        squares[i] = n
        for j in range(i+1, i+7):
            if (j<=100):
                if(j in dladders):
                    n.edges.append(dladders[j])
                elif (j in dsnakes):
                    n.edges.append(dsnakes[j])
                else:
                    n.edges.append(j)  
    # for i in range(1,101):
    #     print("n", squares[i].val)
    #     print(squares[i].edges)
    squares[100].finish=True
    return bfs(squares[1], squares)