In a classic story, the famous mathematician Archimedes was asked to determine if a golden crown commissioned by the king was indeed pure gold, and not part silver, as an informant had claimed. Archimedes discovered a way to determine this while stepping into a (Greek) bath. He noted that water spilled out of the bath in proportion to the amount of him that went in. Realizing the implications of this fact, he immediately got out of the bath and ran naked through the city shouting, `Eureka, eureka!,' for he had discovered an analysis tool (displacement), which, when combined with a simple scale, could determine if the king's new crown was good or not. This discovery was unfortunate for the goldsmith, however, for when Archimedes did his analysis, the crown displaced more water than an equal-weight lump of pure gold, indicating that the crown was not, in fact, pure gold.
In this book, we are interested in the design of `good' data structures and algorithms. Simply put, a data structure is a systematic way of organizing and accessing data, and an algorithm is a step-by-step procedure for performing some task in a finite amount of time. These concepts are central to computing, but to be able to classify some data structures and algorithms as `good,' we must have precise ways of analyzing them. The primary analysis tool we will use in this book involves characterizing the running times of algorithms and data structure operations, with space usage also being of interest. Running time is a natural measure of `goodness,' since time is a precious resource. But focusing on running time as a primary measure of goodness raises an interesting question of how running time is defined. We address this question in this chapter.