Combinatorics Course Info Instructor: yinyt@nju.edu.cn, - - PowerPoint PPT Presentation

Combinatorics

Course Info

• Instructor:
• yinyt@nju.edu.cn, yitong.yin@gmail.com
• Office hour: 804 , Tuesday, 2pm-4pm
• course homepage:
• http://tcs.nju.edu.cn/wiki/

Textbook

van Lint and Wilson, A course in Combinatorics, 2nd Edition. Jukna, Extremal Combinatorics: with applications in computer science, 2nd Edition.

Reference Books

Graham, Knuth, and Patashnik, Concrete Mathematics: A Foundation for Computer Science Stanley, Enumerative Combinatorics, Volume 1

Alon and Spencer. The Probabilistic Method.

Reference Books

Cook, Cunningham, Pulleyblank, and Schrijver.

Combinatorial Optimization. Aigner and Ziegler. Proofs from THE BOOK.

Combinatorics

• Enumeration (counting):
• Existence:
• Extremal:
• Ramsey:
• Optimization:
• Construction (design):

How many solutions to these constraints? Does a solution exist? When a solution is sufficiently large, some structure must emerge. How large/small a solution can be to preserve/avoid certain structure? Find the optimal solution. Construct a solution.

solution: combinatorial object constraint: combinatorial structure combinatorial≈discrete finite

Tools (and prerequisites)

• Combinatorial (elementary) techniques;
• Algebra (linear & abstract);
• Probability theory;
• Analysis (calculus).

Enumeration

(counting)

• to rank n people?
• to assign m zodiac signs to n people?
• to choose m people out of n people?
• to partition n people into m groups?
• to distribute m yuan to n people?
• to partition m yuan to n parts?
• ... ...

How many ways are there:

Gian-Carlo Rota (1932-1999)

The Twelvefold Way

Stanley, Enumerative Combinatorics, Volume 1

The twelvefold way

f : N → M |N| = n, |M| = m

elements

• f N

elements

• f M

any f 1-1

• n-to

distinct distinct identical distinct distinct identical identical identical

balls per bin: unrestricted

≤ 1 ≥ 1

n distinct balls, m distinct bins n identical balls, m distinct bins n distinct balls, m identical bins n identical balls, m identical bins

Knuth’s version (in TAOCP vol.4A)

n balls are put into m bins

Counting (labeled) trees

“How many different trees can be formed from n distinct vertices?”

2 2 1 2 1 2

Arthur Cayley (1821-1895)

There are nn−2 trees on n distinct vertices.

Cayley’s formula:

Algorithmic Enumeration

“The number of different spanning trees of G(V,E).” input: undirected graph G(V, E) t(G) : for i =1,2,3,... nn-2

• utput the i-th tree;

enumeration algorithm: counting algorithm:

Graph Laplacian

1 2 3 4

Graph G(V,E) adjacency matrix A

A(i, j) = ( 1 {i, j} 2 E {i, j} 62 E D(i, j) = ( deg(i) i = j i 6= j

diagonal matrix D

D =      d1 d2 ... dn     

graph Laplacian L L = D − A

L =     3 −1 −1 −1 −1 2 −1 −1 −1 3 −1 −1 −1 2    

Li,i : submatrix of L by removing ith row and ith collumn i i t(G) : number of spanning trees in G Kirchhoff ’s Matrix-Tree Theorem: ∀i, t(G) = det(Li,i) Gustav Kirchhoff (1824-1887)

Bipartite Perfect Matching

bipartite graph G([n],[n],E) perfect matchings permutation π of [n] (i, π(i)) ∈ E s.t.

Ai,j =

• 1

(i, j) E (i, j) E

n × n matrix A : =

• π∈Sn
• i∈[n]

Ai,π(i) # of P .M. in G

Permanent

n × n matrix A : =

• π∈Sn
• i∈[n]

Ai,π(i) perm(A) det(A) =

• π∈Sn

(−1)r(π)

i∈[n]

Ai,π(i) determinant: poly-time by Gaussian elimination #P-hard to compute

Ryser’s formula

• π∈Sn
• i∈[n]

Ai,π(i) =

• I⊆[n]

(−1)n−|I|

i∈[n]

• j∈I

Ai,j O(n!) time O(n2n) time

• I⊆S

(−1)|S|−|I| =

• 1

S = ∅

• therwise

PIE (Principle of Inclusion-Exclusion):

PIE

(Principle of Inclusion-Exclusion) |A ∪ B| = |A| + |B| −|A ∩ B| |A ∪ B ∪ C| = |A| + |B| + |C| −|A ∩ B| − |A ∩ C| − |B ∩ C| +|A ∩ B ∩ C|

Inversion

f : 2[n] → N

V: 2n-dimensional vector space of all mappings

φ : V → V

linear transformation

∀S ⊆ [n], ∀S ⊆ [n],

then its inverse:

φf(S)

• T ⊇S

T ⊆[n]

f(T) φ−1f(S) =

• T ⊇S

T ⊆[n]

(−1)|T \S|f(T)

Fibonacci number

Fn =      Fn−1 + Fn−2 if n ≥ 2, 1 if n = 1 if n = 0. φ = 1 + √ 5 2 ˆ φ = 1 − √ 5 2 Fn = 1 √ 5

• φn − ˆ

φn⇥ by generating functions ...

Quicksort

Qsort(A): choose a pivot x = A[1]; partition A into L with all L[i ] < x , R with all R[i ] > x ; Qsort(L) and Qsort(R);

input: an array A of n numbers Complexity: number of comparisons worst-case: average-case:

?

Θ(n2)

Qsort(A): choose a pivot x = A[1]; partition A into L with all L[i ] < x , R with all R[i ] > x ; Qsort(L) and Qsort(R);

Tn :

average # of comparisons used by Qsort

= 1 n

n

• k=1

(n − 1 + Tk−1 + Tn−k)

Tn pivot: the k-th smallest number in A |L| = k-1 |R| = n-k

n − 1

Recursion:

T0 = T1 = 0

= 2n ln n + O(n)

generating functions

Counting with Symmetry

Rotation : Rotation & Reflection:

Symmetries

Pólya’s Theory of Counting

George Pólya (1887-1985)

a~

v : # of config. (up to symmetry) with ni many color i

pattern inventory :

FG(y1, y2, . . . , ym) = X

~ v=(n1,...,nm) n1+···+nm=n

a~

v yn1 1 yn2 2 · · · ynm m

(multi-variate) generating function

FG(y1, y2, . . . , ym) = PG m X

i=1

yi,

m

X

i=1

y2

i , . . . , m

X

i=1

yn

i

!

Pólya’s enumeration formula (1937):

π =

`1

z }| { (· · · )

`2

z }| { (· · · ) · · ·

`k

z }| { (· · · ) | {z }

k cycles

M⇡(x1, x2, . . . , xn) =

k

Y

i=1

x`i

PG(x1, x2, . . . , xn) = 1 |G| X

π∈G

Mπ(x1, x2, . . . , xn)

cycle index:

FD20(r, q, l)

Existing

• a configuration satisfying this condition?
• a counterexample for this method?
• an efficient algorithm for this problem?
• a problem which is hard to solve in this

computation model?

• ...

Does there exist:

Circuit Complexity

∧

¬

x1 x2 x3

∨ ∧ ∨

f : {0, 1}n → {0, 1}

Boolean function Boolean circuit

Claude Shannon (1916 - 2001)

Theorem (Shannon 1949) There is a boolean function f : {0, 1}n → {0, 1} which cannot be computed by any circuit with 2n

3n gates.

no constructive proof is known

Combinatorics

• Enumeration (counting):
• Existence:
• Extremal:
• Ramsey:
• Optimization:
• Construction (design):

How many solutions to these constraints? Does a solution exist? When a solution is sufficiently large, some structure must emerge. How large/small a solution can be to preserve/avoid certain structure? Find the optimal solution. Construct a solution.

solution: combinatorial object constraint: combinatorial structure combinatorial≈discrete finite