Stanford Cheatsheet 2012

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Sparse max-flow (C++)1.

Min-cost max-flow (C++)2.

Push-relabel max-flow (C++)3.

Min-cost matching (C++)4.

Max bipartite matching (C++)5.

Global min cut (C++)6.

Convex hull (C++)7.

Miscellaneous geometry (C++)8.

Java geometry (Java)9.

3D geometry (Java)10.

Slow Delaunay triangulation (C++)11.

Number theoretic algorithms (modular, Chinese remainder, linear Diophantine) (C++)12.

Systems of linear equations, matrix inverse, determinant (C++)13.Reduced row echelon form, matrix rank (C++)14.

Fast Fourier transform (C++)15.

Simplex algorithm (C++)16.

Fast Dijkstra's algorithm (C++)17.

Strongly connected components (C)18.

Suffix arrays (C++)19.

Binary Indexed Tree20.

Union-Find Set (C/C++)21.

KD-tree (C++)22.

Longest increasing subsequence (C++)23.

Dates (C++)24.

Regular expressions (Java)25.

Prime numbers (C++)26.

Knuth-Morris-Pratt (C++)27.

// Adjacency list implementation of Dinic's blocking flow algorithm.// This is very fast in practice, and only loses to push-relabel flow.

//// Running time:// O(|V|^2 |E|)//// INPUT:// - graph, constructed using AddEdge()// - source// - sink//// OUTPUT:// - maximum flow value// - To obtain the actual flow values, look at all edges with// capacity > 0 (zero capacity edges are residual edges).

#include

#include#include#include

using namespace std;

const intINF = 2000000000;

structEdge {intfrom, to, cap, flow, index;

Edge(intfrom, intto, intcap, intflow, intindex) : from(from), to(to), cap(cap), flow(flow), index(index) {}};

structDinic {intN;

vector G; vector dad; vector Q;

Dinic(intN) : N(N), G(N), dad(N), Q(N) {}

voidAddEdge(intfrom, intto, intcap) { G[from].push_back(Edge(from, to, cap, 0, G[to].size()));

if(from == to) G[from].back().index++; G[to].push_back(Edge(to, from, 0, 0, G[from].size() - 1)); }

long longBlockingFlow(ints, intt) { fill(dad.begin(), dad.end(), (Edge *) NULL); dad[s] = &G[0][0] - 1;

inthead = 0, tail = 0; Q[tail++] = s;

while(head < tail) {intx = Q[head++];for(inti = 0; i < G[x].size(); i++) {Edge &e = G[x][i];if(!dad[e.to] && e.cap - e.flow > 0) { dad[e.to] = &G[x][i]; Q[tail++] = e.to;}

} }

if(!dad[t]) return0;

long longtotflow = 0;

for(inti = 0; i < G[t].size(); i++) { Edge *start = &G[G[t][i].to][G[t][i].index];

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// expected: -0.233333 0.166667 0.133333 0.0666667// 0.166667 0.166667 0.333333 -0.333333// 0.233333 0.833333 -0.133333 -0.0666667// 0.05 -0.75 -0.1 0.2

cout


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Theoretical Computer Science Cheat Sheet

Series Eschers KnotExpansions:

1

(1 x)n+1 ln 1

1 x =i=0

(Hn+i Hn)

n + i

i

xi,

1

x

n=

i=0

i

n

xi,

xn =i=0

n

i

xi, (ex 1)n =

i=0

i

n

n!xi

i! ,

ln

1

1 xn

=

i=0

i

n

n!xi

i! , x cot x =

i=0

(4)iB2ix2i(2i)!

,

tan x =i=1

(1)i1 22i(22i 1)B2ix2i1

(2i)! , (x) =

i=1

1

ix,

1

(x) =

i=1(i)

ix ,

(x 1)(x)

=

i=1(i)

ix ,

Stieltjes Integration(x) =p

11 px ,

2(x) =i=1

d(i)

xi whered(n) =

d|n1,

(x)(x 1) =i=1

S(i)

xi whereS(n) =

d|n d,

(2n) =2 2n1|B2n|

(2n)! 2n, n N,

x

sin x =

i=0

(1)i1 (4i 2)B2ix2i

(2i)! ,

1

1 4x2x

n

=

i=0

n(2i + n 1)!i!(n + i)!

xi,

ex sin x =i=1

2i/2 sin i4i!

xi,

1 1 x

x =

i=0

(4i)!

16i

2(2i)!(2i + 1)!xi,

arcsin x

x

2=

i=0

4ii!2

(i + 1)(2i + 1)!x2i.

IfG is continuous in the interval [a, b] andF is nondecreasing then ba

G(x) dF(x)

exists. Ifa b c then ca

G(x) dF(x) =

ba

G(x) dF(x) +

cb

G(x) dF(x).

If the integrals involved exist ba

G(x) + H(x)

dF(x) =

ba

G(x) dF(x) +

ba

H(x) dF(x),

b

a

G(x) d

F(x) + H(x)

=

b

a

G(x) dF(x) +

b

a

G(x) dH(x),

b

a

c G(x) dF(x) = b

a

G(x) d

c F(x) =c ba

G(x) dF(x),

ba

G(x) dF(x) = G(b)F(b) G(a)F(a) ba

F(x) dG(x).

If the integrals involved exist, andFpossesses a derivativeFat everypoint in [a, b] then b

a

G(x) dF(x) =

ba

G(x)F(x) dx.

Cramers Rule0 0 4 7 1 8 7 6 2 9 9 3 8 5 3 4 6 1 5 2

8 6 1 1 5 7 2 8 7 0 3 9 9 4 4 5 0 2 6 3

9 5 8 0 2 2 6 7 3 8 7 1 4 9 5 6 1 3 0 4

5 9 9 6 8 1 3 3 0 7 4 8 7 2 6 0 2 4 1 5

7 3 6 9 9 0 8 2 4 4 1 7 5 8 0 1 3 5 2 6

6 8 7 4 0 9 9 1 8 3 5 5 2 7 1 2 4 6 3 0

3 7 0 8 7 5 1 9 9 2 8 4 6 6 2 3 5 0 4 1

1 4 2 5 3 6 4 0 5 1 6 2 0 3 7 7 8 8 9 9

2 1 3 2 4 3 5 4 6 5 0 6 1 0 8 9 9 7 7 8

4 2 5 3 6 4 0 5 1 6 2 0 3 1 9 8 7 9 8 7

Fibonacci Numbers

If we have equations:a1,1x1+ a1,2x2+ + a1,nxn = b1a2,1x1+ a2,2x2+

+ a2,nxn = b2

......

...

an,1x1+ an,2x2+ + an,nxn= bnLetA= (ai,j) andB be the column matrix (bi). Thenthere is a unique solution iff det A= 0. Let Ai b eAwith columni replaced by B . Then

xi=det Ai

det A.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

Definitions:

Fi = Fi1+Fi2, F0 = F1= 1,Fi= (1)i1Fi,

Fi = 1

5

i i

,

Cassinis identity: for i >0:

Fi+1Fi1 F2i = (1)i.Additive rule:

Fn+k= FkFn+1+ Fk1Fn,

F2n= FnFn+1+ Fn1Fn.

Calculation by matrices:Fn2 Fn1Fn1 Fn

=

0 11 1

n.

The Fibonacci number system:Every integer n has a uniquerepresentation

n= Fk1+ Fk2+ + Fkm,where ki ki+1 + 2 for all i ,1 i < m and km 2.

Improvement makes strait roads, but the crookedroads without Improvement, are roads of Genius. William Blake (The Marriage of Heaven and Hell)

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Stanford Cheatsheet 2012

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