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running time of the program. For example, a program may have a running time T(n) = cn, where c is some constant. Put another way, the running time of this program is linearly proportional to the size of the input on which it is run. Such a Linear-time program or algorithm is said to be linear time, or just linear.
Analysis of Algorithm 4. Limitations of Experiments. It is necessary to implement the algorithm, which may be difficult. Results may not be indicative of the running time on other inputs not included in the experiment. In order to compare two algorithms, the same hardware and software environments must be used.
We can understand an algorithm’s cost by nding its complexity class: { If T(N) = k, where k is some constant, then we can say T(N) is a constant time algorithm. This is a O(1) algorithm. { If T(N) = kN, where k is some constant, then we can say T(N) is a linear time algorithm. This is a O(N) algorithm.
There is active research and important open problems in getting all sorts of running times, i.e. occasionally solving problems in sublinear time (which is asymptotically less than the input size) or quasipolynomial time
Goal: Analyze an algorithm written in pseudocode and describe its running time. Without having to write code. In a way that is independent of the computer used. To achieve that, we need to. Make simplifying assumptions about the running time of each basic (primitive) operations.
Running time of programs. For any program Pand any input x, let t. P(x) denote the number of \steps" Ptakes on input x. We need to specify what we mean by a \step." A \step" typically corresponds to machine instructions being executed, or some indication of time or resources expended.
We measure the running time of a program as a function of the size of its input. Thus, if a program runs in linear time, its running time grows as a constant times the size of the input.
