An algorithm is a procedure consisting of computational steps that transforms an input value into an output value, where
For example, "sorting" is a problem where the input is a sequence of items of a certain type, with a total order defined between any pair of them, and the output should be a sequence of the same items, arranged according to the order. A sorting algorithm should specify, step by step, how to turn any valid input into the corresponding output in finite time, then stop (halt) there.
An algorithm is correct if for every valid input instance, it halts with the output as specified in the problem. An algorithm is incorrect if there is a valid input instance for which the algorithm produces an incorrect answer or no answer at all (i.e. does not halt).
For a given problem, if there are multiple candidate algorithms, which one should be used? There are several factors to be considered:
When the (input, output, or intermediate) data of a problem contain multiple components, they are usually organized in a data structure, which represents both the data items and the relevant relation among them.
A data structure can be specified either abstractly, in terms of the operations (as computations) that can be carried out on it, or concretely, in terms of the storage organization and the algorithms implementing the operations. An abstract data structure often corresponds to multiple concrete ones.
In the design and selection of data structures, the analysis of the algorithms involved is a central topic.
Programming means to implement an algorithm in a computer language. A program is language-specific, while an algorithm is language-independent.
For a given program, the actual time it takes to solve a given problem instance depends on
As a starting point, the following instructions are usually assumed to be directly executable and each takes a constant amount of time.
To measure the time complexity of algorithms, the common practice is to define a size (usually using n) for each instance of the problem (which intuitively measures the relative difficulty of processing the instance), then to represent the running time as a function of this instance size (as T(n) in the following). Finally, the increasing rate of the function, with respect to the increasing of the instance size, is used as the indicator of the efficiency or complexity of the algorithm.
With such a procedure, algorithm analysis becomes a mathematical problem, which is well-defined, and the result has universal validity.
Though it is a powerful technique, we need to keep in mind that many relevant factors have been ignored in this process, and therefore, if for certain reason some of the factors have to be taken into consideration, the traditional algorithm analyzing approach may become improper to use.
Insertion sort works by repeatedly insert an element into the sorted part of the array.
The correctness of the algorithm is proven by checking the loop invariant, which is a proposition about a relation among certain variables, such as
At the start of each iteration of the for loop of line 1-8, the subarray A[1 : i−1] consists of the elements originally in A[1 : i−1] but in sorted order.We must show three things about a loop invariant:
For sorting, it is natural to use the number of keys to be sorted as input size, and it is assumed that a constant amount of time is required to execute each line of the pseudocode (except comments).
Now let us mark the cost and the number of execution times of each line:
In line 5-7, ti is the number of executions of Line 5 for the i value.
The running time of the algorithm is
For a given n, T(n) depends on the values of tj, which changes from instance to instance.
For a given size n, the best case of the algorithm happens when the array is already sorted, so that ti = 1 for i = 2, 3, ..., n, and the function becomes
which is a linear function of n.
The worst case of the algorithm happens when the array is reverse sorted, so that ti = i for i = 2, 3, ..., n, and the function becomes
which is a quadratic function of n.
Usually, analysis of algorithm concentrates on the worst case.
Merge sort is an algorithm that sorts an array by cutting it into two halves, sorting them recursively, then merging the two sorted subarrays.
The merge procedures Merge(A, p, q, r) first moves the two sorted subarrays A[p : q] and A[q+1 : r] into two separate arrays L and R, put two special sentinel values at the end of each of them, then merge the two back into the original array A[p : r].
Its time expense is a linear function of n, because every data item is moved 2 times, and compared less than 1 time in average.
The following is the merge-sort algorithm.
The correctness and efficiency of this algorithm can be analyzed similarly.