<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>

import numpy as np
import queue

class Solution:
    def minimumJumps(self, forbidden: List[int], a: int, b: int, x: int) -> int:
        inf = 999999999

        # Init the forbid array
        forbid = np.zeros((2000 + 1,), dtype=bool)
        for cell in forbidden:
            forbid[cell] = True
        
        # Init the dp array with offset = 4000 to handle negative coordinates
        f = np.ones((20001, 2), dtype=int) * inf
        offset = 10000

        q = queue.Queue()
        q.put((0, 0))
        f[0 + offset, 0] = 0
        while (not q.empty()):
            i_min, d_min = q.get()
            # print(i_min, d_min, f[i_min + offset, d_min])
            # Update the forward direction
            if (i_min + a >= 0):
                if ((i_min + a > 2000) or (i_min + a < 0) or (not forbid[i_min + a])):
                    if ((i_min + a >= -10000) and (i_min + a <= 10000) and (f[i_min + a + offset, 0] == inf)):
                        q.put((i_min + a, 0))
                        f[i_min + a + offset, 0] = f[i_min + offset, d_min] + 1
            
            # Update the backward direction
            if (i_min - b >= 0):
                if ((d_min == 0) and ((i_min - b < 0) or (i_min - b > 2000) or (not forbid[i_min - b]))):
                    if ((i_min - b >= -10000) and (i_min - b <= 10000) and (f[i_min - b + offset, 1] == inf)):
                        q.put((i_min - b, 1))
                        f[i_min - b + offset, 1] = f[i_min + offset, d_min] + 1
                    # print(i_min - b, f[i_min - b, 1])
            
            if ((f[x + offset, 0] != inf) or (f[x + offset, 1] != inf)):
                break

        res = np.min(f[x + offset, :])
        if (res == inf):
            res = -1
        
        return res
388