<p>Given an array of&nbsp;integers <code>arr</code>, you are initially positioned at the first index of the array.</p>

<p>In one step you can jump from index <code>i</code> to index:</p>

<ul>
	<li><code>i + 1</code> where:&nbsp;<code>i + 1 &lt; arr.length</code>.</li>
	<li><code>i - 1</code> where:&nbsp;<code>i - 1 &gt;= 0</code>.</li>
	<li><code>j</code> where: <code>arr[i] == arr[j]</code> and <code>i != j</code>.</li>
</ul>

<p>Return <em>the minimum number of steps</em> to reach the <strong>last index</strong> of the array.</p>

<p>Notice that you can not jump outside of the array at any time.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> arr = [100,-23,-23,404,100,23,23,23,3,404]
<strong>Output:</strong> 3
<strong>Explanation:</strong> You need three jumps from index 0 --&gt; 4 --&gt; 3 --&gt; 9. Note that index 9 is the last index of the array.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> arr = [7]
<strong>Output:</strong> 0
<strong>Explanation:</strong> Start index is the last index. You do not need to jump.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> arr = [7,6,9,6,9,6,9,7]
<strong>Output:</strong> 1
<strong>Explanation:</strong> You can jump directly from index 0 to index 7 which is last index of the array.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= arr.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>-10<sup>8</sup> &lt;= arr[i] &lt;= 10<sup>8</sup></code></li>
</ul>

from collections import deque, defaultdict

class Solution:
    def minJumps(self, arr: List[int]) -> int:
        n = len(arr)
        if n == 1:
            return 0
        
        # Create a dictionary to store the indices of each value
        value_indices = defaultdict(list)
        for i, val in enumerate(arr):
            value_indices[val].append(i)
        
        # Initialize the BFS queue with the starting index and steps count
        queue = deque([(0, 0)])  # (index, steps)
        visited = set()
        
        while queue:
            curr_index, steps = queue.popleft()
            
            # Check if we have reached the last index
            if curr_index == n - 1:
                return steps
            
            # Add next indices to the queue
            next_indices = []
            if curr_index + 1 < n:
                next_indices.append(curr_index + 1)
            if curr_index - 1 >= 0:
                next_indices.append(curr_index - 1)
            for next_index in value_indices[arr[curr_index]]:
                if next_index != curr_index:
                    next_indices.append(next_index)
            
            # Mark current value's indices as visited and add them to the queue
            value_indices[arr[curr_index]].clear()  # Clear to avoid redundant visits
            for next_index in next_indices:
                if next_index not in visited:
                    visited.add(next_index)
                    queue.append((next_index, steps + 1))
        
        # If we haven't found a path to the last index, return -1 (though this should never happen with a valid array)
        return -1
573