Theoretical Computer Science Cheat Sheet - Advanced Data Structures | CS 6310, Study notes of Data Structures and Algorithms

Download Theoretical Computer Science Cheat Sheet - Advanced Data Structures | CS 6310 and more Study notes Data Structures and Algorithms in PDF only on Docsity! Theoretical Computer Science Cheat Sheet Denitions Series f
n  O
g
n i positive c n such that 
f
n
cg
n n  n nX i
 i  n
n  nX i
 i  n
n 
n  nX i
 i  n
n 
In general nX i
 im   m 
n  m   nX i


i m  im 
m  im nX i
 im   m  mX k
 m   k  Bkn mk
Geometric series nX i
ci  cn   c  c  
X i
ci    c
X i
 ci  c  c c   nX i
ici  ncn 
n cn  c
c  c  
X i
ici  c
  c c  
Harmonic series Hn  nX i
  i nX i
 iHi  n
n  Hn  n
n 
nX i
 Hi 
n  Hn  n nX i
  i m  Hi   n  m   Hn   m   
f
n 
g
n i positive c n such that f
n  cg
n   n  n f
n  
g
n i f
n  O
g
n and f
n 
g
n f
n  o
g
n i limn
f
ng
n   lim n
an  a i   R n such thatjan  aj   n  n sup S least b  R such that b  s s  S inf S greatest b  R such that b
s s  S lim inf n
an limn
inffai j i  n i  Ng lim sup n
an lim n
supfai j i  n i  Ng
n k  Combinations Size k sub sets of a size n set n k  Stirling numbers
st kind Arrangements of an n ele ment set into k cycles
 n k   n
n  kk 
nX k
 n k   n 
 n k    n n k  
 n k   n k  n  k    
 n k    n  k    n  k    
 n m  m k    n k  n k m  k  
X kn  r  k k    r  n  n  
nX k
 k m    n   m  
nX k
 r k  s n  k    r  s n 
 n k  
k  k  n  k 
 n    n n   
 n  n   
 n k  k  n  k   n  k   n k Stirling numbers
nd kind Partitions of an n element set into k nonempty sets n k st order Eulerian numbers Permutations 


n on f


ng with k ascents n k nd order Eulerian numbers Cn Catlan Numbers Binary trees with n  vertices 
n   
n  
n  
n  Hn 
n n    
n k    n k 
n k  
n  n  k   n  k   
 n n   n n     n  
nX k
n k   n 
Cn   n   n n  
 n     n n     
 n k    n n  k  
 n k  
k    n  k  
n k  n  k    
  k   n  if k    otherwise 
 n    n  n  
 n   n 
n  n   n   
xn  nX k
 n k  x k n  
 n m   mX k
 n  k 
m   kn
k 
m  n m  nX k
 n k  k nm  
 n m   nX k
 n k  n k m 
nkmk 
 n     
 n n    for n   
 n k  
k    n   k  
n  k  n  k    
nX k
 n k  
nn n 
 x x n  nX k
 n k   x n  k n  
 n  m    X k  n k  k m  nX k
 k m
m  nk Theoretical Computer Science Cheat Sheet Identities Cont Trees 
n  m     X k n k  k m   nX k
k m  nnk  n nX k
 k k m  
x x n   nX k
 n k  x k n  
 n m  X k  n k  k   m  
nk 
n m   X k n   k    k m 
mk 
 m  n   m  mX k
k  n k k 
m  n   m   mX k
k
n  k n k k  
 n m   X k  n  k   k m 
mk 

nm  n m   X k n  k    k m
mk for n  m 
 n nm  X k  m  n m  k  m n n k  m  k k  
n n m   X k  m  n m  k  m  n n k  m  k k 
 n  m   m    X k  k   n k m  n k  
n  m   m    X k k   n k m  n k 
Every tree with n vertices has n   edges Kraft inequal ity If the depths of the leaves of a binary tree are d


dn nX i
 di
 and equality holds only if every in ternal node has sons Recurrences Master method T
n  aT
nb  f
n a   b   If    such that f
n  O
nlogb a then T
n  
nlogb a
If f
n  
nlogb a then T
n  
nlogb a log n
If    such that f
n 
nlogb a and c   such that af
nb
cf
n for large n then T
n  
f
n
Substitution
example Consider the following recurrence Ti  i  T i T 
Note that Ti is always a power of two Let ti  log Ti Then we have ti  i  ti t  
Let ui  ti i Dividing both sides of the previous equation by i we get ti i  i i  ti i
Substituting we nd ui     ui u   which is simply ui  i  So we nd that Ti has the closed form Ti  ii  Summing factors
example Consider the following recurrence Ti  Tn
  n T  n
Rewrite so that all terms involving T are on the left side Ti  Tn
  n
Now expand the recurrence and choose a factor which makes the left side tele scope 
T
n T
n   n 
T
n  T
n   n     log
n
T
 T
   log
n
T
    Summing the left side we get T
n Sum ming the right side we get log
nX i
n i i
Let c   and m  log n Then we have n mX i
ci  n  cm   c     n
c  clog
n    n
c  ck logc n    nk  n  n  n where k 
log    Full history recur rences can often be changed to limited his tory ones
example Consider the follow ing recurrence Ti    iX j
Tj T  
Note that Ti    iX j
Tj
Subtracting we nd Ti  Ti    iX j
Tj   iX j
Tj  Ti
And so Ti  Ti  i Generating functions  Multiply both sides of the equa tion by xi  Sum both sides over all i for which the equation is valid  Choose a generating function G
x Usually G
x  P
i
x i  Rewrite the equation in terms of the generating function G
x  Solve for G
x  The coecient of xi in G
x is gi Example gi  gi   g  
Multiply and sum X i gix i  X i gix i  X i xi
We choose G
x  P i x i Rewrite in terms of G
x G
x g x  G
x  X i xi
Simplify G
x x  G
x    x
Solve for G
x G
x  x
 x
 x
Expand this using partial fractions G
x  x   x    x   x   X i ixi  X i xi  A  X i
i  xi
So gi  i   Theoretical Computer Science Cheat Sheet Number Theory Graph Theory The Chinese remainder theorem There ex ists a number C such that C  r modm    C  rn modmn if mi and mj are relatively prime for i  j Eulers function
x is the number of positive integers less than x relatively prime to x If Qn i
 p ei i is the prime fac torization of x then
x  nY i
 peii
pi  
Eulers theorem If a and b are relatively prime then   a b mod b
Fermats theorem   ap mod p
The Euclidean algorithm if a  b are in tegers then gcd
a b  gcd
a mod b b
If Qn i
 p ei i is the prime factorization of x then S
x  X djx d  nY i
 peii   pi  
Perfect Numbers x is an even perfect num ber i x  n
n and n is prime Wilsons theorem n is a prime i
n    mod n
Mobius inversion
i     if i    if i is not squarefree
r if i is the product of r distinct primes If G
a  X dja F
d then F
a  X dja
dG a d 
Prime numbers pn  n lnn n ln lnn  n n ln lnn lnn  O  n lnn  
n  n lnn  n
lnn  n
lnn  O  n
lnn 
Denitions Loop An edge connecting a ver tex to itself Directed Each edge has a direction Simple Graph with no loops or multiedges Walk A sequence vev


e v  Trail A walk with distinct edges Path A trail with distinct vertices Connected A graph where there exists a path between any two vertices Component A maximal connected subgraph Tree A connected acyclic graph Free tree A tree with no root DAG Directed acyclic graph Eulerian Graph with a trail visiting each edge exactly once Hamiltonian Graph with a path visiting each vertex exactly once Cut A set of edges whose re moval increases the num ber of components Cutset A minimal cut Cut edge A size  cut kConnected A graph connected with the removal of any k   vertices kTough S  V S   we have k  c
G S
jSj kRegular A graph where all vertices have degree k kFactor A kregular spanning subgraph Matching A set of edges no two of which are adjacent Clique A set of vertices all of which are adjacent Ind
set A set of vertices none of which are adjacent Vertex cover A set of vertices which cover all edges Planar graph A graph which can be em beded in the plane Plane graph An embedding of a planar graphX vV deg
v  m
If G is planar then nm  f   so f
n m
n
Any planar graph has a vertex with de gree
 Notation E
G Edge set V
G Vertex set c
G Number of components GS Induced subgraph deg
v Degree of v
G Maximum degree 
G Minimum degree 
G Chromatic number E
G Edge chromatic number Gc Complement graph Kn Complete graph Knn
Complete bipartite graph r
k  Ramsey number Geometry Projective coordinates triples
x y z not all x y and z zero
x y z 
cx cy cz c  
Cartesian Projective
x y
x y  y  mx b
m b x  c
 c Distance formula Lp and L
metric p
x  x 
x  xjx  xjp  jx  xjp
p lim p
jx  xjp  jx  xjp
p
Area of triangle
x y
x y and
x y   abs x  x y  yx  x y  y 
Angle formed by three points 
x y
x y  
  cos  
x y 
x y 
Line through two points
x y and
x y  x y  x y  x y    
Area of circle volume of sphere A  r V  r 
If I have seen farther than others it is because I have stood on the shoulders of giants  Issac Newton Theoretical Computer Science Cheat Sheet  Calculus Wallis identity                          Brounckers continued fraction expansion        
 

 




Gregrorys series                Newtons series                       Sharps series   p                       Eulers series 
  
  
  
  
  
    
   
  
  
 
 
    
   
 
 
 
 
    Derivatives
d
cu dx  c du dx 
d
u v dx  du dx  dv dx 
d
uv dx  u dv dx  v du dx 
d
un dx  nun du dx 
d
uv dx  v
du dx   u
dv dx  v 
d
ecu dx  cecu du dx 
d
cu dx 
ln ccu du dx 
d
lnu dx   u du dx
d
sinu dx  cosu du dx
d
cos u dx   sinudu dx
d
tanu dx  sec u du dx 
d
cotu dx  csc u du dx 
d
sec u dx  tanu sec u du dx 
d
csc u dx   cotu csc udu dx 
d
arcsinu dx  p  u du dx 
d
arccos u dx  p  u du dx 
d
arctanu dx    u du dx 
d
arccot u dx    u du dx
d
arcsec u dx   u p  u du dx 
d
arccsc u dx   u p  u du dx 
d
sinhu dx  coshu du dx 
d
coshu dx  sinhu du dx 
d
tanhu dx  sech u du dx 
d
cothu dx   csch udu dx 
d
sech u dx   sech u tanhudu dx 
d
csch u dx   csch u cothudu dx 
d
arcsinhu dx  p   u du dx 
d
arccosh u dx  p u   du dx 
d
arctanhu dx    u du dx 
d
arccoth u dx   u   du dx 
d
arcsech u dx   u p  u du dx 
d
arccsch u dx   jujp  u du dx
Integrals
Z cu dx  c Z u dx 
Z
u v dx  Z u dx Z v dx 
Z xn dx   n  xn n   
Z  x dx  lnx 
Z ex dx  ex 
Z dx   x  arctanx 
Z u dv dx dx  uv  Z v du dx dx 
Z sinx dx   cosx
Z cosx dx  sinx
Z tanx dx   ln j cosxj
Z cotx dx  ln j cosxj 
Z sec x dx  ln j sec x tanxj 
Z csc x dx  ln j cscx cotxj 
Z arcsin xadx  arcsin x a  p a  x a   Partial Fractions Let N
x and D
x be polynomial func tions of x We can break down N
xD
x using partial fraction expan sion First if the degree of N is greater than or equal to the degree of D divide N by D obtaining N
x D
x  Q
x  N 
x D
x where the degree of N  is less than that of D Second factor D
x Use the follow ing rules For a nonrepeated factor N
x
x aD
x  A x a  N 
x D
x where A  N
x D
x  x
a
For a repeated factor N
x
x amD
x  mX k
Ak
x amk N 
x D
x where Ak   k dk dxk  N
x D
x  x
a
The reasonable man adapts himself to the world! the unreasonable persists in trying to adapt the world to himself Therefore all progress depends on the unreasonable  George Bernard Shaw Theoretical Computer Science Cheat Sheet Calculus Cont 
Z arccos xadx  arccos x a  p a  x a   
Z arctan xadx  x arctan x a  a ln
a  x a   
Z sin
axdx  a
ax sin
ax cos
ax 
Z cos
axdx  a
ax sin
ax cos
ax
Z sec x dx  tanx 
Z csc x dx   cotx 
Z sinn x dx   sin n x cosx n  n  n Z sinn x dx 
Z cosn x dx  cosn x sinx n  n  n Z cosn x dx 
Z tann x dx  tann x n   Z tann x dx n   
Z cotn x dx  cot n x n   Z cotn x dx n   
Z secn x dx  tanx secn x n    n n  Z secn x dx n   
Z cscn x dx  cot x csc n x n    n n  Z cscn x dx n   
Z sinhx dx  coshx 
Z coshx dx  sinhx 
Z tanhx dx  ln j coshxj 
Z cothx dx  ln j sinhxj 
Z sech x dx  arctan sinhx 
Z csch x dx  ln tanh x   
Z sinh x dx   sinh
x x 
Z cosh x dx   sinh
x   x 
Z sech x dx  tanhx 
Z arcsinh x a dx  x arcsinh x a  p x  a a   
Z arctanh x a dx  x arctanh x a  a ln ja  xj 
Z arccosh xadx    x arccosh x a  p x  a if arccosh xa   and a   x arccosh x a  p x  a if arccosh xa   and a   
Z dxp a  x  ln  x p a  x  a   
Z dx a  x  a arctan x a a   
Z p a  x dx  x p a  x  a
 arcsin xa a   
Z
a  x
dx  x
a  x p a  x  a arcsin xa a   
Z dxp a  x  arcsin x a a   
Z dx a  x   a ln a xa x  
Z dx
a  x
  x a p a  x 
Z p a x dx  x p a x a
 ln xpa x 
Z dxp x  a  ln xpx  a a   
Z dx ax  bx   a ln  xa bx  
Z x p a bxdx 
bx a
a bx
 b 
Z p a bx x dx  p a bx a Z  x p a bx dx 
Z xp a bx dx  p ln  p a bxpap a bx p a  a   
Z p a  x x dx  p a  x  a ln a p a  x x  
Z x p a  x dx  
a  x
 
Z x p a  x dx  x
x  a p a  x  a arcsin xa a   
Z dxp a  x    a ln a  p a  x x  
Z x dxp a  x   p a  x 
Z x dxp a  x   x  p a  x  a
 arcsin xa a   
Z p a  x x dx  p a  x  a ln a p a  x x  
Z p x  a x dx  p x  a  a arccos ajxj a   
Z x p x a dx  
x a
 
Z dx x p x  a  a ln  xa pa  x