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<p>Merge sort is one of the classic sorting algorithms. It divides the input array into two halves, recursively sorts each half, then merges the two sorted halves.</p> |
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<p>In this problem merge sort is used to sort an array of integers in ascending order. The exact behavior is given by the following pseudo-code:</p> |
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<pre>function merge_sort(arr): |
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n = arr.length() |
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if n <= 1: |
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return arr |
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// arr is indexed 0 through n-1, inclusive |
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mid = floor(n/2) |
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first_half = merge_sort(arr[0..mid-1]) |
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second_half = merge_sort(arr[mid..n-1]) |
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return merge(first_half, second_half) |
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function merge(arr1, arr2): |
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result = [] |
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while arr1.length() > 0 and arr2.length() > 0: |
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if arr1[0] < arr2[0]: |
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print '1' // for debugging |
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result.append(arr1[0]) |
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arr1.remove_first() |
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else: |
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print '2' // for debugging |
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result.append(arr2[0]) |
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arr2.remove_first() |
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result.append(arr1) |
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result.append(arr2) |
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return result</pre> |
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<p>A very important permutation of the integers 1 through <strong>N</strong> was lost to a hard drive failure. Luckily, that sequence had been sorted by the above algorithm and the debug sequence of 1s and 2s was recorded on a different disk. You will be given the length <strong>N</strong> of the original sequence, and the debug sequence. Recover the original sequence of integers.</p> |
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<h3>Input</h3> |
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<p>The first line of the input file contains an integer <strong>T</strong>. This is followed by <strong>T</strong> test cases, each of which has two lines. The first line of each test case contains the length of the original sequence, <strong>N</strong>. The second line contains a string of 1s and 2s, the debug sequence produced by merge sort while sorting the original sequence. Lines are separated using Unix-style ("\n") line endings.</p> |
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<h3>Output</h3> |
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<p>To avoid having to upload the entire original sequence, output an integer checksum of the original sequence, calculated by the following algorithm:</p> |
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<pre>function checksum(arr): |
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result = 1 |
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for i=0 to arr.length()-1: |
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result = (31 * result + arr[i]) mod 1000003 |
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return result</pre> |
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<h3>Constraints</h3> |
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<p> |
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5 ≤ <strong>T</strong> ≤ 20<br/> |
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2 ≤ N ≤ 10,000 |
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</p> |
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<h3>Examples</h3> |
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<p>In the first example, N is 2 and the debug sequence is <tt>1</tt>. The original sequence was 1 2 or 2 1. The debug sequence tells us that the first number was smaller than the second so we know the sequence was 1 2. The checksum is 994.</p> |
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<p>In the second example, N is 2 and the debug sequence is <tt>2</tt>. This time the original sequence is 2 1.</p> |
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<p>In the third example, N is 4 and the debug sequence is <tt>12212</tt>. The original sequence is 2 4 3 1.</p> |
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