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ROBDD - Reduced Ordered Binary Decision Diagram

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Team-VLSI-ITMU
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Reduced ordered binary decision diagram

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REDUCED ORDERED BINARY DECISION DIAGRAM (ROBDD) Devi Sivaraman 12ECP026 M.Tech-VLSI
Binary Decision Diagrams • Boolean function of m variables defines a Boolean space of 2m points • The data structure to represent Boolean function should be compact and easy to manipulate • A binary decision diagram (BDD) is a data structure that is used to represent a Boolean function, directly derivable from Shannon’s expansion
Shannon’s Expansion • Shannon's expansion is a method by which a Boolean function can be represented by the sum of two sub-functions of the original • For Example:
BINARY TREE REPRESENTATION 11 1 0 0 0 x yy ( ) ( ) x x xy x y xy x y f xf x f x yf y f x yf y f       xyfx y fxy fx y f
ROBDD Flow
Example 1: XOR Truth Table Binary Decision Tree A B F 0 0 0 0 1 1 1 0 1 1 1 0 A BB 0 1 1 0 0 0 1 1 0 1
Binary Decision Diagram
OBDD Ordered BDD (OBDD) : Input variables are ordered – each path from root to sink visits nodes with labels (variables) in ascending order. a c c b 0 1 ordered order = a,c,b a b c c 0 1 not ordered b
Ordered Binary Decision Tree (OBDT) • This graph representation is called OBDT or OBDD (Ordered binary Decision Diagram) which has a directed tree structure • Each vertex has two children. Two edges originated from a vertex are called high (positive co-factor) & low (negative co- factor)
OBDT to ROBDD ROBDD can be obtained from an OBDT (Ordered Binary Decision Tree) by repeatedly applying following reduction rules (until none of the them can be applied anymore): • Remove duplicate terminal (leaf) nodes • Remove duplicate non-terminal (internal) nodes • Remove nodes with redundant tests
Reduction rules of OBDD Elimination rule Reduction Rule
OBDT to ROBDD Rule 1: Collapse Leaf Nodes to remove redundant tests It’s no longer a tree
OBDT to ROBDD Rule 2: Isomorphic sub-graphs
OBDT to ROBDD Final Representation
ROBDDs • Directed acyclic graph (DAG) • One root node per function, two terminals 0, 1 • Each node, two children, and a variable • Shannon co-factoring tree, except reduced and ordered (ROBDD) Reduced: – any node with two identical children is removed – two nodes with isomorphic BDD’s are merged Ordered: Co-factoring variables (splitting variables) always follow the same order from a root to a terminal xi1 < xi2 < xi3 < … < xin BDDs 15
Example 2 Binary Decision Tree 16 a c b f a b b cccc 0 0 1 0 0 1 1 1 0 1 0 0 0 01 11 1 1 1 0 0Graph representation of a Boolean function. Leaf nodes
Ordered Binary Decision Diagram (OBDD) 17 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 a b b cccc 0 0 1 0 0 1 1 1 0 1 0 0 0 01 11 1 1 1 0 0 Tree OBDD
OBDD With Different Input Ordering 18 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 c b b a 0 1 0 1 1 0 1 0 0 1 1 0 a 0 1 0 1
Reduction: OBDD to ROBDD 19 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 a 10 cc b b 0 0 0 0 1 1 1 1 1 0
Properties of ROBDD • The 0- and 1-succesor of any node should not be identical • ROBDDs provide canonical representation of switching functions • ROBDDs can be manipulated efficiently • ROBDDs representations are small for many important switching functions • Isomorphic graph should not be there i.e. sub-graphs that yields same information about function several times
Thank You

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ROBDD - Reduced Ordered Binary Decision Diagram

  • 1. REDUCED ORDERED BINARY DECISION DIAGRAM (ROBDD) Devi Sivaraman 12ECP026 M.Tech-VLSI
  • 2. Binary Decision Diagrams • Boolean function of m variables defines a Boolean space of 2m points • The data structure to represent Boolean function should be compact and easy to manipulate • A binary decision diagram (BDD) is a data structure that is used to represent a Boolean function, directly derivable from Shannon’s expansion
  • 3. Shannon’s Expansion • Shannon's expansion is a method by which a Boolean function can be represented by the sum of two sub-functions of the original • For Example:
  • 4. BINARY TREE REPRESENTATION 11 1 0 0 0 x yy ( ) ( ) x x xy x y xy x y f xf x f x yf y f x yf y f       xyfx y fxy fx y f
  • 6. Example 1: XOR Truth Table Binary Decision Tree A B F 0 0 0 0 1 1 1 0 1 1 1 0 A BB 0 1 1 0 0 0 1 1 0 1
  • 8. OBDD Ordered BDD (OBDD) : Input variables are ordered – each path from root to sink visits nodes with labels (variables) in ascending order. a c c b 0 1 ordered order = a,c,b a b c c 0 1 not ordered b
  • 9. Ordered Binary Decision Tree (OBDT) • This graph representation is called OBDT or OBDD (Ordered binary Decision Diagram) which has a directed tree structure • Each vertex has two children. Two edges originated from a vertex are called high (positive co-factor) & low (negative co- factor)
  • 10. OBDT to ROBDD ROBDD can be obtained from an OBDT (Ordered Binary Decision Tree) by repeatedly applying following reduction rules (until none of the them can be applied anymore): • Remove duplicate terminal (leaf) nodes • Remove duplicate non-terminal (internal) nodes • Remove nodes with redundant tests
  • 11. Reduction rules of OBDD Elimination rule Reduction Rule
  • 12. OBDT to ROBDD Rule 1: Collapse Leaf Nodes to remove redundant tests It’s no longer a tree
  • 13. OBDT to ROBDD Rule 2: Isomorphic sub-graphs
  • 14. OBDT to ROBDD Final Representation
  • 15. ROBDDs • Directed acyclic graph (DAG) • One root node per function, two terminals 0, 1 • Each node, two children, and a variable • Shannon co-factoring tree, except reduced and ordered (ROBDD) Reduced: – any node with two identical children is removed – two nodes with isomorphic BDD’s are merged Ordered: Co-factoring variables (splitting variables) always follow the same order from a root to a terminal xi1 < xi2 < xi3 < … < xin BDDs 15
  • 16. Example 2 Binary Decision Tree 16 a c b f a b b cccc 0 0 1 0 0 1 1 1 0 1 0 0 0 01 11 1 1 1 0 0Graph representation of a Boolean function. Leaf nodes
  • 17. Ordered Binary Decision Diagram (OBDD) 17 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 a b b cccc 0 0 1 0 0 1 1 1 0 1 0 0 0 01 11 1 1 1 0 0 Tree OBDD
  • 18. OBDD With Different Input Ordering 18 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 c b b a 0 1 0 1 1 0 1 0 0 1 1 0 a 0 1 0 1
  • 19. Reduction: OBDD to ROBDD 19 a c b f a b b cc 0 1 0 0 1 1 0 1 0 0 01 1 1 1 0 a 10 cc b b 0 0 0 0 1 1 1 1 1 0
  • 20. Properties of ROBDD • The 0- and 1-succesor of any node should not be identical • ROBDDs provide canonical representation of switching functions • ROBDDs can be manipulated efficiently • ROBDDs representations are small for many important switching functions • Isomorphic graph should not be there i.e. sub-graphs that yields same information about function several times