Count Stepping Numbers in Range
<p>Given two positive integers <code>low</code> and <code>high</code> represented as strings, find the count of <strong>stepping numbers</strong> in the inclusive range <code>[low, high]</code>.</p>

<p>A <strong>stepping number</strong> is an integer such that all of its adjacent digits have an absolute difference of <strong>exactly</strong> <code>1</code>.</p>

<p>Return <em>an integer denoting the count of stepping numbers in the inclusive range</em> <code>[low, high]</code><em>. </em></p>

<p>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p><strong>Note:</strong> A stepping number should not have a leading zero.</p>

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

<pre>
<strong>Input:</strong> low = &quot;1&quot;, high = &quot;11&quot;
<strong>Output:</strong> 10
<strong>Explanation: </strong>The stepping numbers in the range [1,11] are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. There are a total of 10 stepping numbers in the range. Hence, the output is 10.</pre>

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

<pre>
<strong>Input:</strong> low = &quot;90&quot;, high = &quot;101&quot;
<strong>Output:</strong> 2
<strong>Explanation: </strong>The stepping numbers in the range [90,101] are 98 and 101. There are a total of 2 stepping numbers in the range. Hence, the output is 2. </pre>

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

<ul>
	<li><code>1 &lt;= int(low) &lt;= int(high) &lt; 10<sup>100</sup></code></li>
	<li><code>1 &lt;= low.length, high.length &lt;= 100</code></li>
	<li><code>low</code> and <code>high</code> consist of only digits.</li>
	<li><code>low</code> and <code>high</code> don&#39;t have any leading zeros.</li>
</ul>
String; Dynamic Programming
class Solution:
    def countSteppingNumbers(self, low: str, high: str) -> int:
        @cache
        def dfs(pos: int, pre: int, lead: bool, limit: bool) -> int:
            if pos >= len(num):
                return int(not lead)
            up = int(num[pos]) if limit else 9
            ans = 0
            for i in range(up + 1):
                if i == 0 and lead:
                    ans += dfs(pos + 1, pre, True, limit and i == up)
                elif pre == -1 or abs(i - pre) == 1:
                    ans += dfs(pos + 1, i, False, limit and i == up)
            return ans % mod

        mod = 10**9 + 7
        num = high
        a = dfs(0, -1, True, True)
        dfs.cache_clear()
        num = str(int(low) - 1)
        b = dfs(0, -1, True, True)
        return (a - b) % mod
