As we have learned in lecture, the Huffman code satisfies the Prefix Property, which states that the bit string representing each symbol is not a prefix of the bit string representing any other symbol. One nice property of such codes is that, given a bit string, there is at most one way to decode it back to a sequence of symbols. However, this is not true anymore once we are working with codes that do not satisfy the Prefix Property. For example, consider the code that maps A to 1, B to 01 and C to 101. A bit string 101 can be interpreted in two ways: as C or as AB.

Your task is to, given a bit string s, determine how many ways one can interpret s. The mapping from symbols to bit strings of the code will be given to you as a dictionary d (e.g., in the example, d = {A : 1, B : 01, C : 101}); you may assume that you can access each symbol in the dictionary in constant time. Your algorithm should run in time at most O(nm`) where n is the length of the input bit string s, m is the number of symbols, and ` is an upper bound on the length of the bit strings representing symbols. Clearly describe your algorithm, prove its correctness and runtime.