Course Info. Instructor: Office hour: 804, Tuesday, 2pm-4pm course homepage:

Transcription

1 Combinatorics

2 Course Info Instructor: Office hour: 804, Tuesday, 2pm-4pm course homepage:

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

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

5 Reference Books Aigner and Ziegler. Proofs from THE BOOK. Alon and Spencer. The Probabilistic Method. Cook, Cunningham, Pulleyblank, and Schrijver. Combinatorial Optimization.

6 Combinatorics solution: combinatorial object constraint: combinatorial structure Enumeration (counting): combinatorial discrete finite How many solutions to these constraints? Existence: Extremal: Ramsey: Does a solution exist? How large/small a solution can be to preserve/avoid certain structure? When a solution is sufficiently large, some structure must emerge. Optimization: Find the optimal solution. Construction (design): Construct a solution.

7 Tools (and prerequisites) Combinatorial (elementary) techniques; Algebra (linear & abstract); Probability theory; Analysis (calculus).

8 Enumeration (counting) How many ways are there: 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?

9 The Twelvefold Way Stanley, Enumerative Combinatorics, Volume 1 Gian-Carlo Rota ( )

10 The twelvefold way f : N M N = n, M = m elements of N distinct elements of M distinct any f 1-1 on-to identical distinct distinct identical identical identical

11 Knuth s version (in TAOCP vol.4a) n balls are put into m bins 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

12 Counting (labeled) trees How many different trees can be formed from n distinct vertices?

13 Cayley s formula: There are n n 2 trees on n distinct vertices. Arthur Cayley ( )

14 Algorithmic Enumeration enumeration algorithm: for i =1,2,3,... n n-2 output the i-th tree; counting algorithm: input: undirected graph G(V, E) t(g) : The number of different spanning trees of G(V,E).

15 Graph Laplacian Graph G(V,E) 1 2 adjacency matrix A A(i, j) = D(i, j) = ( 1 {i, j} 2 E diagonal matrix D 0 {i, j} 62 E ( deg(i) i = j 0 i 6= j 4 D = 2 d 1 d dn graph Laplacian L L = D A L =

16 L i,i : submatrix of L by removing ith row and ith collumn i i Gustav Kirchhoff ( ) t(g) : number of spanning trees in G Kirchhoff s Matrix-Tree Theorem: 8i, t(g) =det(l i,i )

17 Bipartite Perfect Matching bipartite graph perfect matchings G([n],[n],E) permutation of [n] (i, (i)) E s.t. n n matrix A : # of P.M. in G A i,j = 1 (i, j) E 0 (i, j) E = S n i [n] A i, (i)

18 Permanent n n matrix A : perm(a) = A i, (i) S n i [n] #P-hard to compute determinant: det(a) = S n ( 1) r( ) i [n] A i, (i) poly-time by Gaussian elimination

19 Ryser s formula S n i [n] A i, (i) = I [n]( 1) n I i [n] j I A i,j O(n!) time O(n2 n ) time PIE (Principle of Inclusion-Exclusion): I S ( 1) S I = 1 S = 0 otherwise

20 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

21 Inversion V: 2 n -dimensional vector space of all mappings f :2 [n] linear transformation N : V V S [n], f(s) f(t ) T T S [n] then its inverse: S [n], 1 f(s) = ( 1) T \S f(t ) T T S [n]

22 Fibonacci number F n 1 + F n 2 if n 2, F n = 1 if n =1 0 if n =0. F n = 1 5 n ˆn = by generating functions... ˆ = 1 2 5

23 Quicksort input: an array A of n numbers 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); Complexity: number of comparisons worst-case: average-case: Θ(n 2 )?

24 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); T n : average # of comparisons used by Qsort pivot: the k-th smallest number in A L = k-1 R = n-k Recursion: n T n = 1 n (nn 1+T1 k 1 + T n k ) =2n ln n + O(n) k=1 T 0 = T 1 =0 generating functions

25 Counting with Symmetry Rotation : Rotation & Reflection:

26 Symmetries

27 Pólya s Theory of Counting George Pólya ( )

28 pattern inventory : (multi-variate) generating function F G (y 1,y 2,...,y m )= X ~v=(n 1,...,nm) n 1 + +nm=n a ~v y n 1 1 yn 2 2 yn m m a ~v : # of config. (up to symmetry) with ni many color i Pólya s enumeration formula (1937): F G (y 1,y 2,...,y m )=P G m X y i, mx y 2 i,..., mx y n i! i=1 i=1 i=1 = `1 z } { ( ) cycle index: `2 z } { ( ) `k z } { ( ) {z } k cycles M (x 1,x 2,...,x n )= P G (x 1,x 2,...,x n )= 1 G X 2G ky i=1 x`i M (x 1,x 2,...,x n )

29 F D20 (r, q, l)

30 Existing Does there exist: 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?

31 Circuit Complexity Boolean function f : {0, 1} n {0, 1} Boolean circuit x 1 x 2 x 3

32 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 Claude Shannon ( )

33 Combinatorics solution: combinatorial object constraint: combinatorial structure Enumeration (counting): combinatorial discrete finite How many solutions to these constraints? Existence: Extremal: Ramsey: Does a solution exist? How large/small a solution can be to preserve/avoid certain structure? When a solution is sufficiently large, some structure must emerge. Optimization: Find the optimal solution. Construction (design): Construct a solution.