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    <p>Sure it can be done in polynomial time. It's an excellent exercise in <a href="http://en.wikipedia.org/wiki/Dynamic_programming" rel="noreferrer">dynamic programming</a> or <a href="http://en.wikipedia.org/wiki/Memoization" rel="noreferrer">memoization</a>.</p> <p>Lets assume N (the number of digits) equals 10 for the example.</p> <p>Think of it recursively like this: <em>How many numbers can I construct using 10 digits starting from <code>1</code>?</em></p> <p>Answer is</p> <pre><code>[number of 9-digit numbers starting from 8] + [number of 9-digit numbers starting from 6]. </code></pre> <p>So <em>how many "9-digit numbers starting from 8" are there?</em> Well,</p> <pre><code>[number of 8-digit numbers starting from 1] + [number of 8-digit numbers starting from 3] </code></pre> <p>and so on. Base case is reached when you get the question <em>"How many 1-digit numbers are there starting from <code>X</code>"</em> (and the answer is obviously 1).</p> <p>When it comes to complexity, the key observation is that you reuse previously computed solutions. That is for instance, the answer to <em>"how many 5-digit numbers starting from <code>3</code>"</em> there are, can be used both when answering <em>"how many 6-digit numbers are there starting from <code>8</code>"</em> AND <em>"how many 6-digit numbers are there starting from <code>4</code>"</em>. This reuse make the complexity collapse from exponential to polynomial.</p> <p>Let's take a closer look at the complexity of a dynamic programming solution:</p> <p>Such implementation would fill in a matrix in the following way:</p> <pre><code>num[1][i] = 1, for all 0&lt;=i&lt;=9 -- there are one 1-digit number starting from X. for digits = 2...N for from = 0...9 num[digits][from] = num[digits-1][successor 1 of from] + num[digits-1][successor 2 of from] + ... num[digits-1][successor K of from] return num[N][1] -- number of N-digit numbers starting from 1. </code></pre> <p>The algorithm simply fills the matrix one cell at a time, and the matrix is of dimension 10*N, and thus runs in <strong>linear time</strong>.</p> <hr> <p>Wrote it down from the top of my head, please correct me if there are any typos.</p>
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