Combinatorial Designs and Their Applications ( 組合設計及其應用 ) 應用數學系 傅恆霖

Combinatorial Design? Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in Sudoku grids. combinatorialmathematicssystems of finite setsblock designsSudoku grids

System of Sets A design defined on X is a collection of subsets of X denoted by Ɓ. If all the subsets are of the same cardinality, then it is called a block design.

The Fano plane Seven points Three points on each line Every two points define a line Seven lines Three lines through each point Every two lines meet at a point 4

The Fano plane as a set system {0,1,4}, {0,2,5}, {0,3,6}, {1,2,6}, {4,2,3}, {4,5,6}, {1,3,5}

Round robin tournament 6 Directed edge between every pair of vertices X  Y means X beats Y {(1,2),(1,4),(2,4),(3,1),(3,2),(4,3)}

Doubles tournament Each game: a, b v c, d Tournament has many games Tournament usually has structure (e.g. everyone plays in the same number of games) 7

Whist tournament every pair of players partner once and oppose twice. Tournament is played in rounds. Example: Whist with 8 players 8 Table 1Table 2 Round 1∞0v4513v26 Round 2∞1v5624v30 Round 3∞2v6035v41 Round 4∞3v0146v52 Round 5∞4v1250v63 Round 6∞5v2361v04 Round 7∞6v3402v15

Research Strategies Use theoretical techniques to prove that a given design exists (or doesn’t exist) for certain sizes. Use experimental techniques to prove that a given design exists (or doesn’t exist) for certain sizes. 9

Quiz Problem: I have at most two favor numbers in my pocket and the number are in {1,2,3,…,63}. Can you find the numbers by asking (me) as few queries ( 問題 ) as possible?

Group Testing Applications!

A Latin square of order n is an n×n array based on an n-set S such that each element of S occurs exactly once in each row and each column. We can take. A latin square of order 3

Array Presentations We can use L = [ l i,j ] nxn to represent a latin square of order n. For convenience, l i,j is read as the (i,j)-entry of the latin square L. Two latin squares of the same order are distinct if there is an ordered pair (i,j) such that their corresponding entries are not the same.

How many? Let L(n) denote the number of distinct latin squares of order n. L(1) = 1 L(2) = 2 L(3) = 12 L(4) = 576 …

L(5) = L(6) = L(7) = L(8) = … L(11) = What ’ s next?

Sudoku Sudoku, or Su Doku, is a Japanese word (or phrase) meaning something like Number Place. There are about 5.525x10 27 latin squares of order 9 and 6.671x10 21 valid Sudoku grids. Note here that a Sudoku grid is a latin square with special properties.

Sudoku puzzle

A latin square of order n defines a quasigroup on 3 elements. is a quasigroup if is a groupoid and have unique solution.

How much do you know? A quasigroup is not a group due to the “ associative law ”. It is not difficult to prove that an associative quasigroup is a group. Equivalently, if a quasigroup is also a semigroup, then it is in fact a group! Group?

Orthogonal Latin squares 2123:32:21 36 officers ( Euler 1779 ) Latin square ---- Euler 的困 惑

Two latin squares and of order n are orthogonal if 。 Two orthogonal latin squares of order 4

are mutually orthogonal latin squares of order n (MOLS(n)), if for 。 MOLS(4) Theorem Let n be a prime power and n≠2. Then there exists n-1 latin squares (best possible!) of order n which is a collection of mutually orthogonal latin squares.

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An Updated Result 2523:32:21 NEXT!

利用 L ⊥ M 我們可定義函數 f ， f 為 的一個 Permutation ， 或者說 f 為由 對映至 的 1-1, onto 函數。 例： f((1,1))=(0,3) f((2,3))=(1,0)

( ＊ ) 令 α 為的一個排列， β 也是排列， 則 ， 其中 。 由上述的結果，我們可以發現當 L ⊥ M 時，與 L 垂直的 n 階拉丁方陣至少有 n ！個。所以，要判斷 key 是由哪兩個方陣所形成並不容易！

Cryptosystem from MOLS(n) 1. 使用 n 階拉丁方陣。 2. 有 個 Messages (Plaintexts) 。 3. 以 n 階拉丁方陣建構一個圖 (OLSG) ， G ，則有 |E(G)| 個 keys 。 ( 最早使用 k 個 MOLS(n) ，則有 個 keys.)

討論： 1. 當 n 相當大時， MOLS(n) 的個數也會很大。 2. 如果只考慮 MOLS(n) ， key space 較小。 3.Orthogonal mate 。不容易找！ ( 全部找出來！ )

分散模式解 (Sharing Scheme) 在近代有很多重大的決定，為了確保決策過程 沒有暇疵，通常會採用由多個人都同意的情況下才 執行；例如開金庫，發射核彈 … 。所以，建立一個 系統使得較小的人數無法開啟是有它的必要性。

A critical set C in a latin square is a set (partial latin square) with the following two properties ： 1. L is the only latin square of order n which has symbol k in (i,j)-cell for each ； and 2. no proper subset of c has property (1). A critical set

例 上面圈出的三個位置，任兩個都會形成一個臨界 集 (Critical Set) 。少了，或多了都不是臨界；然而多 了 (3 個 ) 也可以繼續填成唯一的拉丁方陣。

Critical Set for Sudoku If we expect the solution of a Sudoku puzzle is unique, then the partial latin square shown must have contain a “ critical ” set in the sense of satisfying the requirements of a Sudoku game. Sometimes, we did find more than one solution for some game.

Fact 1 A critical set C of a latin square L provides minimal infos from which L can be reconstructed. Fact 2 Deciding whether a partial latin square is a critical set is NP-complete. (From completion point of view.) Fact 3 Denote the minimum size of a critical set of order n by M(n). [D. Curran & G.H.J.van Rees, Cong. Numer. 1979]

Critical sets, n=5

分散模式解 1.Key → L ( 拉丁方陣 ) n public 2. 選一個集合它是 L 中多個臨界集的聯集： S 。 3. 把其中 t’ ≦ |S| 。 ( 可以容許高階者持有多一些 (i,j ; k), 甚至一個 Critical Set ！ ) 4. 足夠多的人即可得到 “Key“ 。

A (s,t)-secret sharing scheme is a system where k pieces of information called shares or shadows of a secret key K are distributed so that each participant has a share such that: 1. the key K can be reconstructed from knowledge of any t or more shares; and 2. the key K can not be reconstructed from knowledge of fewer than t shares.

臨界集的選擇有很多！ ↑ 對於臨集的了解不多。 ↓ 增加破解難度

More Applications Coding Theory Group Testing Experimental Designs More to be introduced!