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Running Time It is convenient to use a function T(n) to represent the number of units of time taken by a program or an algorithm on any input of size n. We shall call T(n) the 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
The Big-O notation: the running time of an algorithm as a function of the size of its input. worst case estimate. asymptotic behavior. O(n2) means that the running time of the algorithm on an input of size n is limited by the quadratic function of n.
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.
Running Time. Why do we need to analyze the running time of a program? • Option 1: Run the program and time it. – Why is this option bad? – What can we do about it? Math Review. • Summation – ∑. • Sum of n consecutive digits = n(n+1)/2. Asymptotic Notation. • 2n + 2. • n=5 -> 12. • n=100 -> 202. • n=1,000,000 -> 2,000,002.
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.
algorithms. However, we will often attempt to obtain even faster running times. A holy grail for many combinatorial optimization problems is to obtain linear time algorithms , that is algorithms with running time O(n) where the input size n. Such a running time implies that the time it takes to solve the problem
Designing better algorithms. Analyzing the asymptotic running time of algorithms is a useful way of thinking about algorithms that often leads to nonobvious improvements. Understanding. An analysis can tell us what parts of an algorithm are crucial for what kinds of inputs, and why.
