![APPENDIX A
Useful Formulas for the
Analysis of Algorithms
This appendix contains a list of useful formulas and rules that are helpful in the
mathematical analysis of algorithms. More advanced material can be found in
[Gra94], [Gre07], [Pur04], and [Sed96].
Properties of Logarithms
All logarithm bases are assumed to be greater than 1 in the formulas below; lg x
denotes the logarithm base 2, ln x denotes the logarithm base e = 2.71828 . . . ;
x, y are arbitrary positive numbers.
1. loga 1 = 0
2. loga a = 1
3. loga xy = y loga x
4. loga xy = loga x + loga y
5. loga
x
y
= loga x − loga y
6. alogb x = xlogb a
7. loga x =
logb x
logb a
= loga b logb x
Combinatorics
1. Number of permutations of an n-element set: P(n) = n!
2. Number of k-combinations of an n-element set: C(n, k) = n!
k!(n − k)!
3. Number of subsets of an n-element set: 2n
475](https://image.slidesharecdn.com/formulasdaa-170718202501/85/Formulas-para-algoritmos-1-320.jpg)
Formulas para algoritmos
- 1. APPENDIX A Useful Formulas for the Analysis of Algorithms This appendix contains a list of useful formulas and rules that are helpful in the mathematical analysis of algorithms. More advanced material can be found in [Gra94], [Gre07], [Pur04], and [Sed96]. Properties of Logarithms All logarithm bases are assumed to be greater than 1 in the formulas below; lg x denotes the logarithm base 2, ln x denotes the logarithm base e = 2.71828 . . . ; x, y are arbitrary positive numbers. 1. loga 1 = 0 2. loga a = 1 3. loga xy = y loga x 4. loga xy = loga x + loga y 5. loga x y = loga x − loga y 6. alogb x = xlogb a 7. loga x = logb x logb a = loga b logb x Combinatorics 1. Number of permutations of an n-element set: P(n) = n! 2. Number of k-combinations of an n-element set: C(n, k) = n! k!(n − k)! 3. Number of subsets of an n-element set: 2n 475
- 2. 476 Useful Formulas for the Analysis of Algorithms Important Summation Formulas 1. u i=l 1 = 1 + 1 + . . . + 1 u−l+1 times = u − l + 1 (l, u are integer limits, l ≤ u); n i=1 1 = n 2. n i=1 i = 1 + 2 + . . . + n = n(n + 1) 2 ≈ 1 2 n2 3. n i=1 i2 = 12 + 22 + . . . + n2 = n(n + 1)(2n + 1) 6 ≈ 1 3 n3 4. n i=1 ik = 1k + 2k + . . . + nk ≈ 1 k + 1 nk+1 5. n i=0 ai = 1 + a + . . . + an = an+1 − 1 a − 1 (a = 1); n i=0 2i = 2n+1 − 1 6. n i=1 i2i = 1 . 2 + 2 . 22 + . . . + n2n = (n − 1)2n+1 + 2 7. n i=1 1 i = 1 + 1 2 + . . . + 1 n ≈ ln n + γ , where γ ≈ 0.5772 . . . (Euler’s constant) 8. n i=1 lg i ≈ n lg n Sum Manipulation Rules 1. u i=l cai = c u i=l ai 2. u i=l (ai ± bi) = u i=l ai ± u i=l bi 3. u i=l ai = m i=l ai + u i=m+1 ai, where l ≤ m < u 4. u i=l (ai − ai−1) = au − al−1
- 3. Useful Formulas for the Analysis of Algorithms 477 Approximation of a Sum by a Definite Integral u l−1 f (x)dx ≤ u i=l f (i) ≤ u+1 l f (x)dx for a nondecreasing f (x) u+1 l f (x)dx ≤ u i=l f (i) ≤ u l−1 f (x)dx for a nonincreasing f (x) Floor and Ceiling Formulas The floor of a real number x, denoted x , is defined as the greatest integer not larger than x (e.g., 3.8 = 3, −3.8 = −4, 3 = 3). The ceiling of a real number x, denoted x , is defined as the smallest integer not smaller than x (e.g., 3.8 = 4, −3.8 = −3, 3 = 3). 1. x − 1 < x ≤ x ≤ x < x + 1 2. x + n = x + n and x + n = x + n for real x and integer n 3. n/2 + n/2 = n 4. lg(n + 1) = lg n + 1 Miscellaneous 1. n!≈ √ 2πn n e n as n → ∞ (Stirling’s formula) 2. Modular arithmetic (n, m are integers, p is a positive integer) (n + m) mod p = (n mod p + m mod p) mod p (nm) mod p = ((n mod p)(m mod p)) mod p