<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
from typing import List

class Solution:
    def minJumps(self, arr: List[int]) -> int:
        MAX_INT = 999999999999999
        final_index = len(arr) - 1
        portal_dict = {}

        # Populate the portal_dict with sets of indices where each value occurs
        for i in range(final_index + 1):
            if arr[i] not in portal_dict:
                portal_dict[arr[i]] = {i}
            else:
                portal_dict[arr[i]].add(i)
        
        # Use deque for efficient popping from the front
        queue = deque([(0, 0)])  # (index, steps)
        visited = {0}

        while queue:
            index, steps = queue.popleft()

            # If we've reached the final index, return the number of steps
            if index == final_index:
                return steps

            # Next step
            steps += 1

            # Collect all possible next indices to visit
            next_indices = set()

            # Add next index
            if index + 1 <= final_index and index + 1 not in visited:
                next_indices.add(index + 1)

            # Add previous index
            if index > 0 and index - 1 not in visited:
                next_indices.add(index - 1)

            # Add indices connected through the value portals
            if arr[index] in portal_dict:
                next_indices.update(portal_dict[arr[index]])
                # Remove the list of indices associated with arr[index] after use to prevent reprocessing
                del portal_dict[arr[index]]

            # Enqueue all valid next indices and mark them as visited
            for next_index in next_indices:
                if next_index not in visited:
                    visited.add(next_index)
                    queue.append((next_index, steps))

        return -1
37873