There is certainly a correlation (although I hesitate to say that challenging algorithms-focused interviews are inherently "better").

This is simply a factual statement. There is a correlation, regardless of whether or not you'd like there to be.

• Companies like Microsoft, Google, Amazon, Facebook, and Palantir all do problem solving / algorithms for software engineers.
  See http://www.careercup.com/page?pid=google-interview-questions
• Companies like Goldman Sachs and Motorola ask a mix of problem solving / algorithms statements with some "trivia" (knowledge-based) questions.
  See http://www.careercup.com/page?pid=goldman-sachs-interview-questions
  and http://www.careercup.com/page?pid=motorola-interview-questions
  
• Companies like American Airlines (and other companies where tech is not a core part of the strategy) tend to focus more on knowledge:
  http://www.careercup.com/page?pid=american-airlines-interview-questions
  

You can argue plenty about whether or not there should be a correlation, but there certainly is. Correlation, of course, is not necessarily causation.

This is, to a limited extent, causal.

The challenging, algorithms-focused interview questions are designed to test intelligence / problem solving skills. By asking these questions, a company is demonstrating that their prioritize problem solving skills -- that is, aptitude -- over knowledge of the details of a language.

A software company that assumes that a good coder can pick up any language easily will prioritize problem solving skills / intelligence / aptitude over knowledge. They also have harder, more interesting work in many cases. So, yes, asking these challenging questions does reflect something about the values of the company.

Moreover, asking the challenging, problem solving questions reflects something about the caliber of the interviewers, and therefore the employees. A company filled with programmers who aren't very smart and also lack a CS degree is just generally not going to be able to ask the standard algorithm questions. First, the interviewers won't have the knowledge or the capability to ask these questions. Second, if the employees don't have that skill set, then the candidates probably aren't great. They would be asking questions beyond the range of many of their candidates. That wouldn't be a very wise thing to do.

So the correlation between the interview questions and the types of questions is not just a coincidence. It exists for good reason.

But, the correlation is limited.

A company can "fake it." If they see algorithms-based questions as the way that the elite software companies do it, they can mimic this process (to some extent -- they still can't ask questions way beyond the range of their own employees).

Additionally, within the algorithms-focused questions, how difficult the questions are may not have much correlation with how good the employees are. An easier question does not make it any easier to get hired. If company A asks harder questions than company B, company A may just put a lower bar on performance (that is, they may allow for more mistakes and more struggling while still thinking the candidate is good). And there are companies which intentionally ask harder questions to make it sound like they're really "elite", when in fact they're not much different than the other top tech companies.

So, yes, there is a correlation, and it is partially causal and partially not. But there are better ways of figuring out how interesting the company is.