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For example, a procedure that adds up all elements of a list requires time proportional to the length of the list, if the adding time is constant, or, at least, bounded by a constant. Linear time is the best possible time complexity in situations where the algorithm has to sequentially read its entire input.
For algorithms and data structures not necessarily mentioned here, see list of algorithms and list of data structures. This list of terms was originally derived from the index of that document, and is in the public domain, as it was compiled by a Federal Government employee as part of a Federal Government work. Some of the terms defined are:
Say that the actions carried out in step 1 are considered to consume time at most T 1, step 2 uses time at most T 2, and so forth. In the algorithm above, steps 1, 2 and 7 will only be run once. For a worst-case evaluation, it should be assumed that step 3 will be run as well. Thus the total amount of time to run steps 1–3 and step 7 is:
Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a n × n integer matrix is O ( n 3 ) {\displaystyle O(n^{3})} for the usual algorithms ( Gaussian elimination ).
function lookupByPositionIndex(i) node ← head i ← i + 1 # don't count the head as a step for level from top to bottom do while i ≥ node.width[level] do # if next step is not too far i ← i - node.width[level] # subtract the current width node ← node.next[level] # traverse forward at the current level repeat repeat return node.value end ...
Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n.
Since the time taken on different inputs of the same size can be different, the worst-case time complexity () is defined to be the maximum time taken over all inputs of size . If T ( n ) {\displaystyle T(n)} is a polynomial in n {\displaystyle n} , then the algorithm is said to be a polynomial time algorithm.
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.