Dominating Set. Stephen Grady, Jeremy Poff

Transcription

1 Dominating Set Stephen Grady, Jeremy Poff

2 Outline Questions About us Overview Problem Similarities History Bounds Complexity Algorithms Applications Implementation Open Questions Discussion

3 Questions To what other graph algorithm(s) is dominating set intrinsically linked? Of what other graph algorithm is dominating set a specific case? What is the minimum number of disjoint dominating sets a connected graph can have?

4 About Us Jeremy Undergrad in cs I m working on learning Haskell I have an australian shepard From Greenback, Tennessee Stephen Graduate student in the Genome, Science and Technology Program Advisor: Dr. Michael Langston From Gravette, Arkansas

5 Overview Dominating Set (DS) - Given a graph G = (V,E) a DS is a subset S V s.t. for all vertices v V, v is either in S or is adjacent to a vertex in S Minimum Dominating Set - A dominating set of smallest cardinality on a graph Domination Number (G)= Minimum Dominating Set

6 Example

7 Flavors of Dominating Set Connected Dominating Set: Graph induced by DS must be connected. Total Dominating Set: No isolated vertices on graph induced by DS. Independent Dominating Set: All vertices must be isolated by graph induced by DS. Dominating Clique: Graph induced by DS must be clique. Red-Blue Dominating Set: Partition graph into two sets, Red and Blue. Dominate all vertices in Blue using only vertices in Red.

8 Problem Similarities Vertex Cover: Every vertex cover is a dominating set in connected graphs. Independent Set: Every maximal independent set is a dominating set.

9 History of Dominating Set First instance of the problem thought to be the queen s domination problem. Originally called coefficient of external stability in first published formalization by Claude Berge in First called dominating set by Oystein Ore in 1962 Increased interest in 1970s and 1980s Growth of interest in covering and location problems It s relationship to other NP-complete problems Location - Slater (1975), Harary and Melter (1976) Location and Domination - Henning and Oellermann (2004)

10 Bounds on Dominating Set A lower bound is based on the most a vertex can dominated. γ(g) n/(1+δ) The set of all vertices on a graph is by definition a dominating set Therefore γ(g) n If the graph has no isolated vertices we can improve this bound γ(g) n/2 This is due to the fact that every connected graph has least two disjoint dominating sets. Domatic number 2.

11 Complexity Decision version: Given a graph G and an integer k, is (G) k? NP-Complete Optimization: Given a graph G, what is (G)? NP-Hard

12 Basic Greedy DS Each vertex has a state associated with it: Black= in DS Green= dominated White= not dominated Put gif here Initially all vertices are set to white. DS:= While vertices that are white do v=vertex adjacent to the most white vertices DS:=DS v End do

13 Parallel Greedy DS Initialize like sequential greedy DS DS:= While v is adjacent to white vertices do span=number of white vertices to which v is adjacent Send span to all vertices up to 2 hops If v has greatest span then DS:=DS v End if End do

14 Exact Minimum Dominating Set The best known exact minimum dominating set algorithm runs in O( n ) It does this by reducing the given instance of dominating set to the set cover problem. In actuality dominating set is a specific case of the set cover problem.

15 Permanent Dominating Set Given a dynamic graph G=(V,E) where all updates are known a priori. V is represented as instances of vertices and assumed to be static; only E changes. Find a DS that covers all instances.

16 Permanent Dominating Set

17 Applications of Dominating Set Minimum DS of a network - important exchange points The ideas of Connected DS used in routing (unicast, multicast, ospf, etc..) Ad Hoc wireless networks

18 Applications of Dominating Set Locating and Dominating Set (LDS) Security S is a dominating set and a locating set (every vertex is unique in respect to its vector of distances to vertices of S) Determine best place to install IDS systems based on LDS Eternal Dominating Set

19 Applications of Dominating Set Medicine Find the smallest set of drugs used to treat condition Cancer Research Find a small subset of cancer drugs that affect certain cancer cell lines Controllability of Networks Permutation on nodes of DS thought to drive network.

20 Applications of Dominating Set Spatial Resource Allocation Fire Stations, ice cream trucks, advertisements, etc

21 Implementation of greedy minimal DS Black - In the set, Green - Neighbor of a black node, White - not covered Preprocess - add all 0 degree verts to the set since they have 0 white neighbors and will never be picked up while ( not_dominated ) big_span = vert with the most white neighbors state [ big_span ] = Black for ( neighbor of big_span ) if states [neighbor] is white states [ neighbor ] = Green

22 How to speed this up L1 cache reference 0.5ns Branch mispredict 5ns L2 cache reference 7ns Mutux lock/unlock 25ns Main memory reference 100ns We want to avoid these

23 How to speed this up - Time Complexity Lots of unnecessary O(V) operations In the case where G(E, V) is immutable Pipeline We can use cached values - O(1) vs O(V) for lookups Const qualify calls - allows compiler to optimize better Change If ( states [ vert ] == White ] { white_states++; } to White && white_states++; Decreases branch mis-predictions - these are very expensive Use smaller types Use uchars vs ints for states - can pack more into a cache line This bought a 10-20% speed up on the development box

24 How to make this smaller - Space Complexity Type sizes Using a smaller type or a bit mask allows you to fit more in a cache line Adj list vs Adj matrix vs Adj triangle matrix Adj triangle still has O(1) operations but uses half the space Use intel intrinsics 256 bit, 512 bit vector registers Can do logical operation on 8 or 16 verts at a time We didn t use this as the bookkeeping overhead outweighed the benefits Generating large graphs (> 50k vertices) maxed on the ram on hydra For large graphs, use bits for precomputed neighbor list

25 Timings

26 Some Other Times Number of Verts Density Program Time Set Size 20k k k k k k

27 Comparison to Exact Dominating Set Graph Exact Minimal arenas-jazz foodweb 3 3 scc-fb-forum soc-fb-caltech inf-usaair 36 37

28 To Do Use SIMD instructions Use bits for adj matrix Use 1 byte for 8 verts Use bits for states Use 1 byte for 4 states Reference count white neighbors Use bits for adjacency and neighbor lists 12.5kB vs 100kB per row in adj matrix Use an adj list, increases speed on sparse graphs Dynamic Programming Only cache neighbor lists as you need them This was actually slower Why?

29 Open Questions Vizing s Conjecture: (G H) (G) (H) Does every 3-connected cubic graph satisfy (G) V /3

30 References Intel Intrinsics Guide Application of Dominating Sets in Wireless Sensor Networks, Amir Hassani Karbasi and Reza Ebrahimi Atani Applications and Variations of Domination in Graphs, Paul Andrew Dreyer Connected Dominating Set Problem and its Application to Wireless Sensor Networks, Razieh Asgarnezhad and Javad Akbari Torkestani Bibliography on domination in graphs and some basic definitions of domination parameters, S.T Hedetniemi, R.C. Lasker Domination in Graphs, Jennifer Tarr Design by Measure and Conquer, a Faster Exact Algorithm for Dominating Set, Van Rool, J.M.M.; Bodlaender, H.L. Co-controllability of Drug-Disease-Gene Network, Peng Gang Sun Applications and Variations of Dominating Set, Paul Andrew Dryer JR.

31 References

32 Final thoughts The greedy code is available at

33 Questions To what other graph algorithm(s) is dominating set intrinsically linked? Of what other graph algorithm is dominating set a specific case? What is the minimum number of disjoint dominating sets a connected graph can have?