Proposed Definition of Efficiency

Polynomial Time (多项式时间):
There are absolute constants c > 0 and d > 0 so that on every input instance of size N, its running time is bounded by cN^d primitive computational steps. (In other words, its running time is at most proportional to N^d.)

Proposed Definition of Efficiency: An algorithm is efficient if it has a polynomial running time.

Where our previous definition seemed overly vague, this one seems much too prescriptive. Wouldn’t an algorithm with running time proportional to n¹⁰⁰—and hence polynomial—be hopelessly inefficient? Wouldn’t we be relatively pleased with a nonpolynomial running time of n^{1+.02(log n)}?
The answers are, of course, “yes” and “yes.” And indeed, however much one may try to abstractly motivate the definition of efficiency in terms of polynomial time, a primary justification for it is this: It really works. Problems for which polynomial-time algorithms exist almost invariably turn out to have algorithms with running times proportional to very moderately growing polynomials like n, nlogn, n², or n³. Conversely, problems for which no polynomial-time algorithm is known tend to be very difficult in practice. There are certainly exceptions to this principle in both directions: there are cases, for example, in which an algorithm with exponential worst-case behavior generally runs well on the kinds of instances that arise in practice; and there are also cases where the best polynomial-time algorithm for a problem is completely impractical due to large constants or a high exponent on the polynomial bound. All this serves to reinforce the point that our emphasis on worst-case, polynomial-time bounds is only an abstraction of practical situations. But overwhelmingly, the concrete mathematical definition of polynomial time has turned out to correspond surprisingly well in practice to what we observe about the efficiency of algorithms, and the tractability of problems, in real life.