Art, Problem Solving, and Algorithms Are Related After All
This story was first published in Coronadiet’s old site on May 31, 2020.
I’ve never seen myself as mathematically-minded by any means, but thinking about the world and recent events from a more objective, mathematical standpoint has been an effective way to refresh my mind and view things from a different perspective. I find myself exposed to an overload of personal statements and beliefs during many points throughout the day. Countless of these are undeniably factual, but they are nonetheless subjective positions that people decided to autonomously adopt and spread. It’s definitely important to take these into consideration, but at the end of the day, it’s more essential to constantly learn and inform yourself in ways that allow you to develop your own opinions. Hopefully, this presents a new frame of thinking that can act as a reset or help you in this important process. It’s one that’s more important now than ever.
Give me a piece of writing, visual art, or anything else that piques my interest, and you’ll get my mind running. But who knew a Wiki page on a mathematical algorithm would do the trick? After reading about problem-solving heuristics in mathematics known as greedy algorithms, I began thinking about how some of the consequences of short-term thinking and actions seem to be modeled by them. Greedy algorithms select the most optimal short-term options available in front of them, achieving local rather than global maximums. In other words, following these choices leads to small wins that may feel big in the present but turn out to not be so in the long run.
An algorithm of this kind can be put to practice in an exercise that aims to find the path leading to the largest sum. At each step, the algorithm selects the largest possible number, not realizing that the optimal approach is to actually select the smaller number since it leads to a significantly larger number at the next step. The algorithm fails to recognize that making decisions solely based on information at any single step disregards the overall big picture.
Two conditions need to be met in order for a greedy algorithm to successfully solve a problem. Crucially, one of them is the following:
the optimal solution to the entire problem must contain the optimal solutions to the sub-problems.
This presents an issue if we’re modeling approaches to real world problems after greedy algorithms. The algorithm (or in real life, the person solving the problem) may very well be unaware of a larger number down the line that has yet to be seen. This means that their solution to the sub-problem for finding the largest sum doesn’t appear in the solution to the entire problem. More concisely put, acting upon the belief of doing what seems to be the best solution for your current problem won’t solve your larger, more deeply-rooted issue. While greedy algorithms provide efficient solutions to select mathematical problems, they seem too risky to be applied to real life issues with very real consequences. Instead, informed, effective approaches and solutions are what we need right now.
