Adventures in Tech

Another great example of randomized algorithms in the introductory chapters of the book " Probability and Computing " is verifying matrix multiplication. Suppose we have 3 matrices of compatible dimensions $A, B$, and $C$, and we want to verify that \[A\cdot B = C\] For simplicity, let's assume that all the matrices are square, and of dimension $n \times n$. The straightforward way to do the verification is to explicitly multiple the matrices $A$ and $B$ together, an operation that is of the order of magnitude of $O(n^3)$. As an aside, we can do better than $O(n^3)$ for matrix multiplication for larger matrices. Wikipedia has an excellent writeup on  Strassen's algorithm  which accomplishes matrix multiplication in $O(n^{2.8074})$ through divide and conquer. First the matrices $A$, $B$, and $C$ are padded with zero columns and rows so that their dimensions are of the form $2^m \times 2^m$. The matrices are then divided into equally sized  block matrices of the f...

Think of a search query, and go to your favorite search engine and conduct the search. Of the millions of matching documents returned, it is most likely that the first page or two contain what you are looking for. Have you ever wondered how the search engines know how to present the most relevant results for us in the first couple of pages, when all the results returned match the query we were looking for? There are a lot of signals search engines use to determine the importance of the matched results: some are intrinsic to the page such as where in the page the match occurs, whether it is highlighted or not, and the importance of the web page; and some are extrinsic such as how many users clicked on the result for a similar query. The signals, their weights, and the formulas to rank the results are usually guarded secrets by search engines as they are the secret sauce for giving users the best results to their queries. One such signal is the importance of a web page. How would yo...