Binary Decision Diagrams

Transcription

1 Binary Decision Diagrams Logic Circuits Design Seminars WS2010/2011, Lecture 2 Ing. Petr Fišer, Ph.D. Department of Digital Design Faculty of Information Technology Czech Technical University in Prague Evropský sociální fond Praha & EU: Investujeme do vaší budoucnosti PI-SCN-2, ČVUT FIT, Petr Fišer,

2 Binary Decision Diagrams Logic function (!) is represented by a DAG Binary, completely specified, single-output Coming from Binary Decision Trees Binary tree One root Non-terminals are labeled by variable names 2 n leaves (terminals), labeled 0 and 1 Each node represents a decision on a variable Each non-terminal node u has two successors lo(u) and hi(u) PI-SCN-2, ČVUT FIT, Petr Fišer,

3 Binary Decision Tree a b c f = a bc + ab c + abc PI-SCN-2, ČVUT FIT, Petr Fišer,

4 ROBDDs Reduced Ordered Binary Decision Diagrams Rooted directed acyclic graphs (DAGs) Variable ordering a-priori given Obtained from BDTs by applying reduction rules PI-SCN-2, ČVUT FIT, Petr Fišer,

5 From BDTs to ROBDDs Three reduction rules: 1. Eliminate duplicate terminals 2. Eliminate duplicate non-terminals 3. Eliminate redundant nodes PI-SCN-2, ČVUT FIT, Petr Fišer,

6 BDD Reductions 1. Eliminate duplicate terminals Only two terminals as a result PI-SCN-2, ČVUT FIT, Petr Fišer,

7 BDD Reductions 2. Eliminate duplicate non-terminals Merge nodes u, v, where lo(u) = lo(v) & hi(u) = hi(v) PI-SCN-2, ČVUT FIT, Petr Fišer,

8 BDD Reductions 3. Eliminate redundant nodes Eliminate nodes where lo(u) = hi(u) ROBDD PI-SCN-2, ČVUT FIT, Petr Fišer,

9 BDD to SOP Track all paths from the root to 1-terminal f = ac + a bc a b c DSOP (disjoint SOP) is obtained Minimum SOP: f = ac + bc PI-SCN-2, ČVUT FIT, Petr Fišer,

10 Construction of BDDs Obtained by recursively applying Shannon Expansion f ( x, x2,..., xi,..., xn) xi f ( x1,...,1,..., xn) xi f ( x1,...,0,..., x 1 n ) OR f x f x x f x where f x f, f f x1 x x0 (cofactor) PI-SCN-2, ČVUT FIT, Petr Fišer,

11 ROBDDs & BDDs Properties of ROBDDs (recall: Reduced, Ordered) Variable ordering a-priori given (a < b < c) Reduced they are minimal under given variable ordering Canonical under given variable ordering PI-SCN-2, ČVUT FIT, Petr Fišer,

12 Properties of ROBDDs Canonical & Minimal Equivalence of two functions detected in (sub)linear time (NP-hard problem, in general) Two functions are equivalent, iff they have isomorphic ROBDDs Tautology & contradiction is checked in constant time (NP-hard problem, in general) Tautology ROBDD is reduced to 1 terminal Contradiction ROBDD is reduced to 0 terminal Operations upon BDDs performed in linear time PI-SCN-2, ČVUT FIT, Petr Fišer,

13 Where s The Problem Then? Sensitive to variable ordering The BDD size could be exponential with n Its size may explode Finding the best ordering = NP-hard Its size may unpredictably explode Some functions are known to be good, some bad Function Best case Worst case Symmetric Linear Quadratic Adder Linear Exponential Multiplier Exponential Exponential PI-SCN-2, ČVUT FIT, Petr Fišer,

14 BDD Variable Ordering Many algorithms proposed Sifting Symmetric sifting Window permutation reordering Simulated annealing Genetic algorithms PI-SCN-2, ČVUT FIT, Petr Fišer,

15 BDD Operations Reduce Apply Restrict ITE PI-SCN-2, ČVUT FIT, Petr Fišer,

16 Reduce Apply the three reduction rules obtain ROBDD PI-SCN-2, ČVUT FIT, Petr Fišer,

17 APPLY Applies an algebraic (Boolean) operation to two BDDs (AND, OR, XOR) Exploiting Shannon s expansion f op g x f op g x f op g x x x x BDDs are recursively traversed Complexity: O( f. g ) PI-SCN-2, ČVUT FIT, Petr Fišer,

18 RESTRICT Restricts a given variable (x) to a given value Just redirect edges of all nodes labeled by x Complexity: O( f ) ac + bc b = 0 b = 1 PI-SCN-2, ČVUT FIT, Petr Fišer,

19 ITE Operator If-Then-Else operator Each BDD node f = (v, g, h), g = f v h = f v f = if v then g else h Multiplexer. BDD nodes are multiplexers ite( v, g, h) v g v h PI-SCN-2, ČVUT FIT, Petr Fišer,

20 ITE Operator ITE implements any function of 2 variables Application of ITE: Let v be the top variable of f, g, h ite( ( v, ite( f ite( f f, g, h) v, h )) Recursively Complexity: O( f. g. h ), v g v, g v v, h v ), # Function By ITE fg ITE(f, g, 0) 2 fg ITE(f, g, 0) 3 f f 4 f g ITE(f, 0, g) 5 g g 6 f g ITE(f, g, g) 7 f + g ITE(f, 1, g) 8 (f + g) ITE(f, 0, g ) 9 f g ITE(f, g, g ) 10 g ITE(g, 0, 1) 11 f + g ITE(f, 1, g ) 12 f ITE(f, 0, 1) 13 f + g ITE(f, g, 1) 14 (fg) ITE(f, g, 1) PI-SCN-2, ČVUT FIT, Petr Fišer,

21 Efficient BDD Implementation Unique Table Hash table, where nodes are stored Array + collision chain (linked list) Before a node (v, g, h) is added to BDD database, it is looked up in the unique table BDD is constructed bottom up strong canonicity is maintained each ROBDD node represents a unique function PI-SCN-2, ČVUT FIT, Petr Fišer,

22 Efficient BDD Implementation Computed Table Hash table cache Nodes computed by ITE are stored there Maps (f, g, h) to ite(f, g, h) avoids repeated computation of ITE No collision chain PI-SCN-2, ČVUT FIT, Petr Fišer,

23 Efficient BDD Implementation Multi-rooted BDDs Each root represents a distinct function Subgraphs of different functions are shared Nodes in Unique table are shared Canonicity retained PI-SCN-2, ČVUT FIT, Petr Fišer,

24 Efficient BDD Implementation Complemented edges An edge has a complement bit indicating that the connected subtree is complemented (negated) Canonicity? then (1) edge must not be complemented Canonicity retained PI-SCN-2, ČVUT FIT, Petr Fišer,

25 BDDs References S. B. Akers, "Binary decision diagrams", IEEE Transactions on Computers, vol. C-27, No. 6, June 1978, pp R. E. Bryant, Graph based algorithms for Boolean function manipulation, IEEE Transactions on Computers, vol. 35, No. 8, August 1986, pp R. E. Bryant, Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams, Carnegie Mellon University Pittsburgh, PA 15213:, 1992, CMU-CS , p. 35. F. Somenzi, CUDD: Colorado University Decision Diagram Package, PI-SCN-2, ČVUT FIT, Petr Fišer,