20 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Floors & Ceilings | Obsidian::STEM | algebra algorithm |
|
Overview
The floor of x is the greatest integer less than x. The ceiling of x is the least integer greater than x. These values are denoted \lfloor x \rfloor and \lceil x \rceil respectively.
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Basic
How is the floor of x denoted?
Back: \lfloor x \rfloor
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What is the floor of x?
Back: The greatest integer less than or equal to x.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
How is the ceiling of x denoted?
Back: \lceil x \rceil
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What is the ceiling of x?
Back: The least integer greater than or equal to x.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
When does \lfloor x / 2 \rfloor = \lceil x / 2 \rceil?
Back: When x is even.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
When does \lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil?
Back: When x is odd.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What does \lceil x \rceil - \lfloor x \rfloor equal?
Back: Either 0 or 1.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What can be said about x if \lceil x \rceil - \lfloor x \rfloor = 0?
Back: x is an integer.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What can be said about x if \lceil x \rceil - \lfloor x \rfloor = 1?
Back: x is not an integer.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lfloor x \rfloor = \lceil x \rceil - 1?
Back: Non-integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lceil x \rceil = \lfloor x \rfloor + 1?
Back: Non-integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lfloor x \rfloor = \lceil x \rceil + 1?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What values of x satisfy \lceil x \rceil = \lfloor x \rfloor - 1?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
For what values k is horizontal and vertical shifting of \lfloor x \rfloor the same?
Back: Integral values.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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%%ANKI Basic What C operator corresponds to floor division? Back: N/A. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
What C operator corresponds to integer division?
Back: /
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
When does C operator / behave like floor division?
Back: When the result is a nonnegative value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
When does C operator / behave like ceiling division?
Back: When the result is a nonpositive value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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%%ANKI Basic What C operator corresponds to ceiling division? Back: N/A. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
How does C evaluate 10 / 3?
Back: 3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate floor(10.f / 3)?
Back: 3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate ceil(10.f / 3)?
Back: 4
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate -10 / 3?
Back: -3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate floor(-10.f / 3)?
Back: -4
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
How does C evaluate ceil(-10.f / 3)?
Back: -3
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c17
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Basic
Given r = \lfloor (p + q) / 2 \rfloor, fair partitioning requires A[r] to be included in which of A[p:r-1] or A[r+1:q]?
Back: A[p:r-1]
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given r = \lfloor (p + q) / 2 \rfloor, when is A[p:r] or A[r+1:q] equally sized?
Back: When A[p:q] has even size.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given r = \lceil (p + q) / 2 \rceil, fair partitioning requires A[r] to be included in which of A[p:r-1] or A[r+1:q]?
Back: A[r+1:q]
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If A[p:q] has odd size, what r most fairly allows partitions A[p:r] and A[r+1:q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If A[p:q] has odd size, what r most fairly allows partitions A[p:r-1] and A[r:q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If A[p:q] has odd size, what r ensures A[p:r-1] has same size as A[r+1:q]?
Back: r = (p + q) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If A[p:q] has even size, what r most fairly allows partitions A[p:r] and A[r+1:q]?
Back: r = \lfloor (p + q) / 2 \rfloor
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If A[p:q] has even size, what r most fairly allows partitions A[p:r-1] and A[r:q]?
Back: r = \lceil (p + q) / 2 \rceil
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given A[p:q] and r = \lfloor (p + q) / 2 \rfloor, how does the size of A[p:r] compare to A[r+1:q]?
Back: It either has zero or one more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given A[p:q] and r = \lfloor (p + q) / 2 \rfloor, what is the size of A[p:r] in terms of n = q - p + 1?
Back: \lceil n / 2 \rceil
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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Basic
Given A[p:q] and r = \lfloor (p + q) / 2 \rfloor, what is the size of A[r+1:q] in terms of n = q - p + 1?
Back: \lfloor n / 2 \rfloor
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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Basic
Given A[p:q] and r = \lceil (p + q) / 2 \rceil, how does the size of A[p:r-1] compare to A[r:q]?
Back: It either has zero or one fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given A[p:q] and r = \lceil (p + q) / 2 \rceil, what is the size of A[r:q] in terms of n = q - p + 1?
Back: \lceil n / 2 \rceil.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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Basic
Given A[p:q] and r = \lceil (p + q) / 2 \rceil, what is the size of A[p:r-1] in terms of n = q - p + 1?
Back: \lfloor n / 2 \rfloor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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Basic
Given A[p:q] and r = \lfloor (p + q) / 2 \rfloor, how does the size of A[p:r-1] compare to A[r:q]?
Back: It either has one or two fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given A[p:q] and r = \lceil (p + q) / 2 \rceil, how does the size of A[p:r] compare to A[r+1:q]?
Back: It either has one or two more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given A[p:q] and r = \lfloor (p + q) / 2 \rfloor, why is the size of A[p:r] potentially larger than A[r+1:q]?
Back: If A[p:q] has odd size, A[p:r] contains the midpoint.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Identities
For integers x and y > 0, \begin{align*} \left\lfloor \frac{x}{y} \right\rfloor & = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil \ \left\lceil \frac{x}{y} \right\rceil & = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor \end{align*}
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Basic
If n is even, what integer value does \lfloor n / 2 \rfloor evaluate to?
Back: n / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If n is odd, what integer value does \lfloor n / 2 \rfloor evaluate to?
Back: (n - 1) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If n is even, what integer value does \lceil n / 2 \rceil evaluate to?
Back: n / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
If n is odd, what integer value does \lceil n / 2 \rceil evaluate to?
Back: (n + 1) / 2
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Basic
Given integers x and y > 0, what value of Bias satisfies the following identity? \left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + Bias \right\rfloor$$
Back: (y - 1) / y
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
Given integers x and y > 0, what value of Bias satisfies the following identity? \left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + Bias}{y} \right\rfloor$$
Back: (y - 1)
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
Given integers x and y > 0, what operator satisfies the following identity? \left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} ;\square; \frac{y - 1}{y} \right\rfloor$$
Back: +
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
What intuition explains why the following identity holds for integers x and y > 0? \left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor$$
Back: (y - 1) / y only affects the RHS if and only if x / y is not an integer.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
Given integers x and y > 0, what value of Bias satisfies the following identity? \left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - Bias \right\rceil$$
Back: (y - 1) / y
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
Given integers x and y > 0, what value of Bias satisfies the following identity? \left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x - Bias}{y} \right\rceil$$
Back: (y - 1)
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
Given integers x and y > 0, what operator satisfies the following identity? \left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} ;\square; \frac{y - 1}{y} \right\rceil$$
Back: -
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
What intuition explains why the following identity holds for integers x and y > 0? \left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil$$
Back: (y - 1) / y only affects the RHS if and only if x / y is not an integer.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Cloze
For any integer n, floor expression {\lfloor n / 2 \rfloor} is equal to ceiling expression {\lceil (n - 1) / 2 \rceil}.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Cloze
For any integer n, ceiling expression {\lceil n / 2 \rceil} is equal to floor expression {\lfloor (n + 1) / 2 \rfloor}.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
What identity generalizes the following? \left\lfloor \frac{n}{2} \right\rfloor = \left\lceil \frac{n - 1}{2} \right\rceil$$
Back: \left\lfloor \frac{n}{d} \right\rfloor = \left\lceil \frac{n - (d - 1)}{d} \right\rceil$$
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Basic
What identity generalizes the following? \left\lceil \frac{n}{2} \right\rceil = \left\lfloor \frac{n + 1}{2} \right\rfloor$$
Back: \left\lceil \frac{n}{d} \right\rceil = \left\lfloor \frac{n + (d - 1)}{d} \right\rfloor$$
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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Bibliography
- Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
- Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
- Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).