Bad: I present the time complexity of an algorithm in this paper.
OK: We present the time complexity of an algorithm in this paper.
Better: The time complexity of an algorithm is presented in this paper. [traditional style: passive voice]
This paper presents the time complexity of an algorithm. [modern style: active voice]
If the subject of a sentence is truly an author or authors,
the first person plural pronoun "we" is OK.
Otherwise, it is recommended to write the subject explicitly
and avoid a pronoun unless what the pronoun refers to is
definitely obvious to the readers, where the subject is an agent
of the action (i.e., verb) mentioned in a sentence.
Bad: We first sort points in the nondecreasing order of their x-coordinate values.
Then, we find a closest pair of points.
Good: An algorithm first sorts points in the nondecreasing order of their x-coordinate values.
Then, the algorithm finds a closest pair of points.
Examples:
"This paper models a problem of finding shortest paths from a single source to all other vertices in a digraph." "This paper presents an algorithm for determining whether a given positive integer is prime."
Examples of Exceptions:
"In 1883, the puzzle called the Towers of Hanoi was invented by the French mathematician Édouard Lucas." "E. W. Dijkstra proposed a greedy algorithm for finding single-source shortest paths in 1959." "Although we have presented an O(n log n) algorithm for MPT, it remains unknown whether the algorithm has an optimum time complexity. That will be one of our future research topics."
→ Step-wise Refinement in Top-Down Design
One Paragraph ←→ One Issue (Modularity)
"Keywords and Phrases" are sometimes given next to the abstract.
This paper is organized as follows. Section 2
presents necessary definitions and notations on
transactions in a distributed database system.
Section 3 discusses properties of a transaction
that are required for fault tolerance. Based on
the properties, Section 4 presents a distributed
algorithm for recovery from failures of
transactions in a distributed database system.
Then, Section 5 analyzes the communication
complexity of the algorithm and compares it with
those of the previously proposed algorithms.
Finally, Section 6 summaries our results and
discusses future directions of our research.
Each reference should be referred to in the main text at least once by using its label (usually, a serial number).
Example:
"It is widely known that NP-complete problems
are difficult to solve [2,5].
.
.
.
References
[1] E. W. Dijkstra, "A note on two problems
in connexion with graphs," Numerische
Mathematik, vol. 1, pp.269-271, 1959.
[2] M. R. Garey and D. S. Johnson, Computers
and Intractability: A Guide to the Theory
of NP-Completeness, W. H. Freeman and Co.,
1979.
.
.
.
The IEEE Style requires to list references in the order of occurrences of their citations in the main text, although it is also common to list references in the alphabetical order of last names of their first authors with tiebreaking by the chronological order of their publication dates.
A nonempty set F of n factories fi (1 ≤ i ≤ n)where F denotes a set and fi denotes a factory.
...
for each edge (u,v) in E {
x ← 0;
for each vertex w in V that is adjacent to v {
x ← x + 1
}
}
y ← x; // this assignment can cause a fatal error (when? and how?)
...
It's common for a designer to find a flaw in an algorithm while trying to verify it. The designer learns how to avoid design flaws through such mistakes. Thus, verification is very helpful to design a correct algorithm.
For example, an array A[1..n] of integers is to be sorted. An algorithm such as the insertion sort has a loop whose loop invariant at the (i+1)-th iteration is "a subarray A[1..i] is already sorted."
If you are going to prove something that is not an invariant, the textbook's proving method is irrelevant.