<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>

class Solution:

    # DP + BFS
    def minJumps(self, arr: List[int]) -> int:
        arr_len = len(arr)
        # initialized the min_steps t0
        min_steps = [-1] * arr_len
        v_dict = {}

        # create a map with all the distinct values as key, and list of their indexes as value
        for i in range(arr_len):
            if arr[i] in v_dict:
                v_dict[arr[i]].append(i)
            else:
                v_dict[arr[i]] = [i]


        q = deque()
        q.append(arr_len -1)
        min_steps[ arr_len -1 ] = 0
        steps = 0

        # use BFS to visit immediate jumps layer by layer
        while (len(q) != 0) :
            q_len = len(q)
            print ("q_len=" + str(q_len))
            for i in range( q_len ):
                j = q.popleft()
                # print("    steps=" + str(steps) + ", j=" + str(j) + ", arr[j]=" + str(arr[j]))
                if ( j + 1 < arr_len and min_steps[j+1] == -1):
                    q.append(j+1)
                    min_steps[j + 1] = steps + 1
                if ( j - 1 >= 0 and min_steps[j-1] == -1):
                    q.append(j-1)
                    min_steps[j - 1] = steps + 1
                if arr[j] in v_dict:
                    indexes = v_dict[arr[j]]
                    # print(indexes)
                    for k in indexes:
                        if k!=j and  min_steps[k] == -1:
                            q.append(k)
                            min_steps[k] = steps + 1
                    # every index with this key has been visited
                    del v_dict[arr[j]]
            # print ("min_steps:" + str(min_steps))
            steps += 1
        # print (min_steps)
        return min_steps[0] 
        
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