A logarithmic time complexity algorithm for pattern searching using product‐sum property

Product–sum property states that an ordered pair $(s_n,p_n)$ is unique for any ordered set $a_1,a_2,\dots ,a_n$ where $a_i,n\in N$ , and $s_n$ and $p_n$ are the sum and product of the elements of the set, respectively. This fact has been exploited to develop an $O(log\left(M\right))$ time complexity algorithm for pattern searching in a large dataset, where $M$ is the number of records in the dataset. Two potential applications (from databases and computational biology) of this property have been demonstrated to show the effectiveness and working of the proposed algorithm. The space complexity of the algorithm rises to the quadratic order.