The Division Breakthroughs Eric Allender 1 Summary: The Division Breakthroughs Eric Allender 1 1 Introduction All of us learn to do arithmetic in grade school. The algorithms for addition and subtraction take some time to master, and the multiplication algorithm is even more complicated. Eventually students learn the division algorithm; most students find it to be complicated, time-consuming, and tedious. Is there a better way to divide? For most practical purposes, the correct way to answer this question is to con- sider the time-complexity of division; what is the fastest division algorithm? That is not the subject of this article. I am not aware of any recent breakthrough on this question; any good textbook on design and analysis of algorithms will tell you about the current state of the art on that front. Complexity theory gives us an equally-valid way to ask about the complexity of division: In what complexity class does division lie? One of the most important subclasses of P (and one of the first to be defined and studied) is the class L (deterministic logarithmic space). It is easy to see how to add and subtract in L. It is a simple exercise to show that multiplication can be computed in logspace, too. However, it had been an open question since the 1960's if logspace machines can divide. Collections: Computer Technologies and Information Sciences