Programming Praxies – Finding Digit Strings In Powers Of Two

Today’s boredom lead me to solving another programming praxies problem:

Search every power of two below 2¹⁰⁰⁰⁰ and return the index of the first power of two in which a target string appears. For instance, if the target is 42, the correct answer is 19 because 2¹⁹ = 524288, in which the target 42 appears as the third and fourth digits, and no smaller power of two contains the string 42.

The naive solution is simple: keep doubling a number (starting at 1) until you find a sequence that matches the target. Of course once you get 2^64 storing the number as a primitive type will not suffice, where you would need to come up with a solution to store larger numbers.

I  improved on the naive approach by caching the number sequences for each exponent. For the cache I used a type of suffix tree:

private class SuffixTreeNode {
	public short MinExponent;
	public SuffixTreeNode[] Children;
}

Unlike your traditional suffix tree, this one does not compress string sequences. It’s really just a trie, where the number in each sequence is stored implicitly as the index in Children (i.e. these arrays are of length 10). However the data structure is populated and access like a suffix tree: where each suffix of a number sequence is inserted into the tree. Each node in the tree is annotated with the smallest exponent that the sequence can be found in.

Here is the full code:

public static class DigitStringPowerTwoSearch
{
	private static SuffixTreeNode suffixRoot;
	private static short currentExponent;
	private static List<int> currentDigits;

	static DigitStringPowerTwoSearch () {
		FlushCache ();
	}

	internal static void FlushCache() {
		// Exposed as internal really for unit tests only
		currentExponent = 0;
		currentDigits = new List<int> { 1 };
		suffixRoot = new SuffixTreeNode ();
		AddDigitsToTree (currentDigits, 0, 0);
	}

	public static int FindMinBaseTwoExponent(ulong target) {
		var targetDigits = ToDigitArray(target);
		while (currentExponent <= 10000) {
			short exponent = FindMinExponentInTree(targetDigits);
			if (exponent >= 0)
				return exponent;
			AddNextExponentToTree();
		}
		throw new ArgumentOutOfRangeException ("target", 
		                                       "target's digits do not exist in a base two number with exponent " +
		                                       "less or equal to 10,000");
	}

	private static void AddNextExponentToTree() {
		DoubleDigits (currentDigits);
		currentExponent++;

		for (int i = 0; i < currentDigits.Count; i++) {
			AddDigitsToTree (currentDigits, i, currentExponent);
		}
	}

	private static void AddDigitsToTree(IList<int> digits, int startIndex, short exponent) {
		SuffixTreeNode current = suffixRoot;
		for (int i = startIndex; i < digits.Count; i++) {
			int digit = digits [i];
			if (current.Children == null) {
				current.Children = new SuffixTreeNode[10];
			}
			if (current.Children [digit] == null) {
				var newNode = new SuffixTreeNode { MinExponent = exponent };
				current.Children [digit] = newNode;
				current = newNode;
			} else {
				current = current.Children [digit];
				// Here we assume exponent is always the largest exponent,
				// so no need to check/update current.MinExponent
			}
		}
	}

	private static short FindMinExponentInTree(int[] targetDigits) {
		SuffixTreeNode current = suffixRoot;
		foreach(var digit in targetDigits) {
			if (current == null || current.Children == null)
				return -1;
			current = current.Children[digit];
		}
		if (current == null)
			return -1;
		return current.MinExponent;
	}

	private static int[] ToDigitArray(ulong n) {
		if (n == 0) 
			return new int[] { 0 };

		var digits = new List<int>();

		for (; n != 0; n /= 10)
			digits.Add((int)(n % 10));

		var arr = digits.ToArray();
		Array.Reverse(arr);
		return arr;
	}

	private static void DoubleDigits(List<int> digits) {
		bool carry = false;
		for (int i = digits.Count - 1; i >= 0; i--) {
			int d = digits [i] * 2;
			if (carry)
				d++;
			if (d >= 10) {
				d -= 10;
				carry = true;
			} else {
				carry = false;
			}
			digits [i] = d;
		}
		if (carry)
			digits.Insert (0, 1);
	}

	private class SuffixTreeNode {
		public short MinExponent;
		public SuffixTreeNode[] Children;
	}
}

When the function is invoked, if a sequence of numbers have been seen before, then we get O(N) performance (where N = amount of digits in target). Effectively the cache becomes a hash-trie.

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