Basic algorithms for arithmetic, when first originated thousands of years ago, have probably had a far more profound effect on the world than any recent computer-based algorithm. Particularly in accounting and engineering, they would have contributed to the ability to have economies that could support large civilizations. The earliest records found of ancient languages are sometimes accounting records.
Interpreting "algorithms in computing" more widely than the questioner meant of course. :-)
Algebra itself started out as a bunch of recipes for doing computations back in Babylonian times, "fathered" by Diophantus, and Al-Khwarizmi who named and codified Algebra.
Name of author: Abu Ja'far Muhammad ibn Musa al-Khwarizmi. Name of work: al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala ("A Handbook of Calculation by Completion and Reduction").
John Derbyshire examines al-Khwarizmi's work in chapter 3 of Unknown Quantity: A Real and Imaginary History of Algebra [ISBN 978-0-452-28853-9], which I just finished reading a few days ago.
The first part of al-Khwarizmi's book concerns finding the roots of first- and second-order polynomials of one unknown. He classified the polynomials into 6 fundamental types, with all positive coefficients. Keep in mind, that negative numbers did not exist at the time, though subtraction did. He showed how to manipulate polynomials into a suitable one of the 6 types by adding a term (al-jabr, completing) or subtracting a term (al-muqabala, reducing).
Says Derbyshire, "al-Khwarizmi has no literal symbolism--no way to lay out equations in letters and numbers, no sign for the unknown quantity and its powers." The problems, the fundamental types, the procedures are all presented in words.
Interestingly, Diophantus 600 years earlier did addition and subtraction of polynomial terms using "a rich literal symbolism to aid the manipulations."
Interpreting "algorithms in computing" more widely than the questioner meant of course. :-)