Antibandwidth of Hamming graphs

Transcription

1 Antibandwidth of Hamming graphs Štefan Dobrev Rastislav Královič Dana Pardubská Ľubomír Török Imrich Vrťo Workshop on Discrete Mathematics Vienna, November 19-22, 2008

2 Antibandwidth problem Consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. Just another labeling problem (from graph theory point of view)

3 Antibandwidth problem Example (linear)

4 Cyclic antibandwidth Example (cyclic)

5 Motivation Introduced in connection with multiprocessor scheduling problems Dual problem to bandwidth problem Bandwidth = to find vertex ordering of a graph G s.t. the length of the longest edge is minimized. Antibandwidth = to find vertex ordering of a graph G s.t. the length of the shortest edge is maximized.

6 Motivation Radio frequency assignment problem N transmitters, N frequencies. Graph representation: transmitters = vertices, egdes = possible interference Goal: to assign all frequencies to transmitters in a one-to-one manner s.t. interferencing transmitters have as different frequencies as possible.

7 Motivation Enemy facility location problem Graph representation: vertices = facilities, edges = hostility Goal: to arrange facilities in a line (cycle, mesh,...) s.t. the critical (smallest) distance between two enemies is maximized

8 Motivation Coding theory Host graph: n-dimensional hypercube Guest graph: complete graph Then ab K p,q n A is equivalent to an existence of a code with minimal Hamming distance greater or equal to A. K p

9 Previous results NP-complete (Leung, Vornberger) Polynomially solvable for the complements of Interval Arborescent comparability Threshold graphs. (Donnely, Isaak, Hamiltonian powers in graphs)

10 Previous results Trivial upper bound for connected graphs: ab G n 2

11 Previous results Also interesting for disconnected graphs. Exact values for graphs consisting of copies of simple graphs. Bandwidth: Antibandwidth:

12 Previous results Known results for: Paths, cycles, special trees complete and complete bipartite graphs 2D, 3D and n-dimensional meshes tori, hypercubes Cyclic antibandwidth Embedding into cycle

13 Previous results Raspaud, Schröder, Sýkora, Török, Vrťo: Antibandwidth and Cyclic Antibandwidth of Meshes and Hypercubes Calamoneri, Massini, Török, Vrťo: Török, Vrťo: Antibandwidth of complete k-ary trees Antibandwidth of three-dimensional meshes Antibandwidth of d-dimensional meshes

14 Hamming graph Definition Cartesian product of d complete graphs, i.e. K nk, k=1,2,...,d k=1 d K nk Hamming graph K 3 K 4

15 Definition Vertices are d-tuples Hamming graph i 1, i 2,...,i d,i k 0,1,2,..., n k 1 Two vertices i 1, i 2,...,i d and j 1, j 2,..., j d are adjacent iff the two d-tuples differ in precisely one coordinate

16 Work in progress Theorem For d 2, 2 n 1 n 2... n d Results d ab k=1 d ab k=1 K nk =n 1 n 2... n d 1,if n d 1 n d K nk =n 1 n 2... n d 1 1,if n d 1 =n d, n d 2 n d 1 d n 1 n 2... n d 1 min n 1 n 2... n d 2, n q 1... n d 1 ab k=1 K nk n 1 n 2... n d 1 1 where q is minimal index such that q d 2 and n q =n d

17 Results Labelling Case I. n d 1 n d f i 1,i 2,...,i d = N d 1 N d 1 / N 1 i 1 N d 1 N d 1 / N 2 i 2 N d 1 N d 1 / N 3 i 3... N d 1 N d 1 / N d 1 i d 1 N d 1 i d mod N d j N j = k =1 K nk, number of vertices of the product Example K 2 K 3 K 4 f i 1,i 2,i 3 = 9i 1 7i 2 6i 3 mod 24

18 Results Example of labelling of K 2 K 3 K

19 Results Labelling Case II. n d 1 =n d, let q be minimal s.t. q d 1 and n q =n d f i 1,i 2,...,i d = N d 1 N d 1 / N 1 i 1 N d 1 N d 1 / N 2 i 2... N d 1 N d 1 / N q 1 i q 1 N d 1 N d 1 / N q i q... N d 1 N d 1 / N d 1 i d 1 N d 1 i d mod N d

20 Labelling Results Case III. n d 1 =n d, define the alternative labelling f i 1,i 2,...,i d = N d 1 N 0 i 1 N d 1 N 1 i 2... N d 1 N d 2 i d 1 N d 1 i d mod N d

21 Results Antibandwidth and cyclic antibandwidth have the same value Open problem n 2 n ab K n K n K n n 2 1 Conjecture ab K n K n K n =n 2 n

22 The End Thank You