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Theoretical Computer Science Cheat Sheet
Number Theory Graph Theory
The Chinese remainder theorem: There ex-ists a number C such
that:
C r1 mod m1...
..
.
..
.C rn mod mn
ifmi and mj are relatively prime for i = j.Eulers function: (x)
is the number ofpositive integers less than x relativelyprime to x.
If
ni=1p
eii is the prime fac-
torization of x then
(x) =n
i=1
pei1i (pi 1).
Eulers theorem: If a and b are relativelyprime then
1 a(b)
mod b.Fermats theorem:
1 ap1 mod p.The Euclidean algorithm: if a > b are in-tegers
then
gcd(a, b) = gcd(a mod b, b).
Ifn
i=1peii is the prime factorization of x
then
S(x) =d|x
d =n
i=1
pei+1i 1pi 1 .
Perfect Numbers: x is an even perfect num-ber iffx = 2n1(2n1)
and 2n1 is prime.Wilsons theorem: n is a prime iff
(n 1)! 1 mod n.Mobius inversion:
(i) =
1 if i = 1.0 if i is not square-free.(1)r if i is the product
of
r distinct primes.
IfG(a) =
d|a
F(d),
thenF(a) =
d|a
(d)Ga
d
.
Prime numbers:
pn = n ln n + n lnln n n + n lnln nln n
+ O
n
ln n
,
(n) =n
ln n+
n
(ln n)2+
2!n
(ln n)3
+ On
(ln n)4.
Definitions:
Loop An edge connecting a ver-tex to itself.
Directed Each edge has a direction.Simple Graph with no loops
or
multi-edges.Walk A sequence v0e1v1 . . . ev.Trail A walk with
distinct edges.Path A trail with distinct
vertices.
Connected A graph where there existsa path between any
twovertices.
Component A maximal connectedsubgraph.
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 cycle visitingeach vertex exactly
once.
Cut A set of edges whose re-moval increases the num-ber of
components.
Cut-set A minimal cut.Cut edge A size 1 cut.k-Connected A graph
connected with
the removal of any k 1vertices.
k-Tough S V, S = we havek c(G S) |S|.
k-Regular A graph where all verticeshave degree k.
k-Factor A k-regular spanningsubgraph.
Matching A set of edges, no two ofwhich are adjacent.
Clique A set of vertices, all ofwhich are adjacent.
Ind. set A set of vertices, none ofwhich are adjacent.
Vertex cover A set of vertices whichcover all edges.
Planar graph A graph which can be em-beded in the plane.
Plane graph An embedding of a planargraph.
vVdeg(v) = 2m.
If G is planar then n m + f = 2, sof 2n 4, m 3n 6.
Any planar graph has a vertex with de-
gree 5.
Notation:
E(G) Edge setV(G) Vertex setc(G) Number of componentsG[S]
Induced subgraph
deg(v) Degree ofv(G) Maximum degree(G) Minimum degree(G)
Chromatic numberE(G) Edge chromatic numberGc Complement graphKn
Complete graphKn1,n2 Complete bipartite grapr(k, ) Ramsey
number
Geometry
Projective coordinates: triple(x,y,z), not all x, y and z
zero.
(x,y,z) = (cx,cy,cz) c = 0.Cartesian Projective
(x, y) (x,y, 1)y = mx + b (m,1, b)x = c (1, 0,c)Distance
formula, Lp and Lmetric:
(x1 x0)2 + (y1 y0)2,|x1 x0|p + |y1 y0|p1/p,lim
p |x1 x0|p + |y1 y0|p
1/p
Area of triangle (x0, y0), (x1, y1and (x2, y2):
12 abs
x1 x0 y1 y0x2 x0 y2 y0 .
Angle formed by three points:
(0, 0) (x1, y1)
(x2, y2)
2
1
cos =(x1, y1) (x2, y2)
12.
Line through two points (x0, y0and (x1, y1):
x y 1x0 y0 1x1 y1 1
= 0.Area of circle, volume of sphere:
A = r2, V = 43r3.
If I have seen farther than othersit is because I have stood on
theshoulders of giants.
Issac Newton
..
aph.pa l v s t ngl
ctly once.o
a cycle visictex exactlyl
o e geso eoval ncreasova c
ber of comc
A minie s z
nnectet
- oug- o
-
deedegree
atic numbge chromatic t
omplemenCompletl
1 2ompll
(( , ) Ra
.quare- ree.uae pro uct oe o
inct primesc me
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