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
import java.math.BigInteger;

class Solution {
    private final int mod = (int) 1e9 + 7;
    private String num;
    private Integer[][] f;

    public int countSteppingNumbers(String low, String high) {
        f = new Integer[high.length() + 1][10];
        num = high;
        int a = dfs(0, -1, true, true);
        f = new Integer[high.length() + 1][10];
        num = new BigInteger(low).subtract(BigInteger.ONE).toString();
        int b = dfs(0, -1, true, true);
        return (a - b + mod) % mod;
    }

    private int dfs(int pos, int pre, boolean lead, boolean limit) {
        if (pos >= num.length()) {
            return lead ? 0 : 1;
        }
        if (!lead && !limit && f[pos][pre] != null) {
            return f[pos][pre];
        }
        int ans = 0;
        int up = limit ? num.charAt(pos) - '0' : 9;
        for (int i = 0; i <= up; ++i) {
            if (i == 0 && lead) {
                ans += dfs(pos + 1, pre, true, limit && i == up);
            } else if (pre == -1 || Math.abs(pre - i) == 1) {
                ans += dfs(pos + 1, i, false, limit && i == up);
            }
            ans %= mod;
        }
        if (!lead && !limit) {
            f[pos][pre] = ans;
        }
        return ans;
    }
}