Minimum Jumps to Reach Home
<p>A certain bug&#39;s home is on the x-axis at position <code>x</code>. Help them get there from position <code>0</code>.</p>

<p>The bug jumps according to the following rules:</p>

<ul>
	<li>It can jump exactly <code>a</code> positions <strong>forward</strong> (to the right).</li>
	<li>It can jump exactly <code>b</code> positions <strong>backward</strong> (to the left).</li>
	<li>It cannot jump backward twice in a row.</li>
	<li>It cannot jump to any <code>forbidden</code> positions.</li>
</ul>

<p>The bug may jump forward <strong>beyond</strong> its home, but it <strong>cannot jump</strong> to positions numbered with <strong>negative</strong> integers.</p>

<p>Given an array of integers <code>forbidden</code>, where <code>forbidden[i]</code> means that the bug cannot jump to the position <code>forbidden[i]</code>, and integers <code>a</code>, <code>b</code>, and <code>x</code>, return <em>the minimum number of jumps needed for the bug to reach its home</em>. If there is no possible sequence of jumps that lands the bug on position <code>x</code>, return <code>-1.</code></p>

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

<pre>
<strong>Input:</strong> forbidden = [14,4,18,1,15], a = 3, b = 15, x = 9
<strong>Output:</strong> 3
<strong>Explanation:</strong> 3 jumps forward (0 -&gt; 3 -&gt; 6 -&gt; 9) will get the bug home.
</pre>

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

<pre>
<strong>Input:</strong> forbidden = [8,3,16,6,12,20], a = 15, b = 13, x = 11
<strong>Output:</strong> -1
</pre>

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

<pre>
<strong>Input:</strong> forbidden = [1,6,2,14,5,17,4], a = 16, b = 9, x = 7
<strong>Output:</strong> 2
<strong>Explanation:</strong> One jump forward (0 -&gt; 16) then one jump backward (16 -&gt; 7) will get the bug home.
</pre>

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

<ul>
	<li><code>1 &lt;= forbidden.length &lt;= 1000</code></li>
	<li><code>1 &lt;= a, b, forbidden[i] &lt;= 2000</code></li>
	<li><code>0 &lt;= x &lt;= 2000</code></li>
	<li>All the elements in <code>forbidden</code> are distinct.</li>
	<li>Position <code>x</code> is not forbidden.</li>
</ul>
Breadth-First Search; Array; Dynamic Programming
class Solution:
    def minimumJumps(self, forbidden: List[int], a: int, b: int, x: int) -> int:
        s = set(forbidden)
        q = deque([(0, 1)])
        vis = {(0, 1)}
        ans = 0
        while q:
            for _ in range(len(q)):
                i, k = q.popleft()
                if i == x:
                    return ans
                nxt = [(i + a, 1)]
                if k & 1:
                    nxt.append((i - b, 0))
                for j, k in nxt:
                    if 0 <= j < 6000 and j not in s and (j, k) not in vis:
                        q.append((j, k))
                        vis.add((j, k))
            ans += 1
        return -1
