Explanation
- A **Binary Decision Diagram (BDD)** is a data structure used to represent boolean functions. It uses a directed acyclic graph where each node represents a decision based on a variable, and the edges represent the outcomes of that decision.
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Steps
- Construct the decision diagram: Represent the boolean function in terms of binary decisions.
- The diagram is built such that each node represents a binary decision based on a variable.
- The edges represent the outcome of that decision (true or false).
- Minimize the diagram: After constructing the initial BDD, apply techniques (like reduction rules) to minimize the number of nodes and edges. This step makes the BDD more efficient for function evaluation.
- Evaluate the boolean function: Once the BDD is constructed, evaluate the function by traversing the tree-like structure and applying the decisions.
Time Complexity
- O(n) for evaluating the boolean function, but complexity can vary depending on the size and structure of the diagram.
- Construction: O(n) where
nis the number of variables, assuming the boolean function is represented as a decision tree. - Evaluation: O(n) for evaluating the function, where
nis the number of variables in the BDD (traversing each node once).
class BDDNode:
def __init__(self, variable=None, high=None, low=None, value=None):
self.variable = variable # Variable name
self.high = high # Pointer to high branch (True)
self.low = low # Pointer to low branch (False)
self.value = value # Value for terminal nodes (True/False)
class BDD:
def __init__(self):
self.nodes = {} # Dictionary to store nodes for caching
self.var_count = 0 # Variable counter to name the variables
def add_node(self, variable, high, low):
"""Create a new node for the BDD."""
if (variable, high, low) in self.nodes:
return self.nodes[(variable, high, low)] # Return cached node
node = BDDNode(variable, high, low)
self.nodes[(variable, high, low)] = node
return node
def create_bdd(self, expression):
"""Create a BDD for a given boolean expression."""
if not expression:
return None
if expression == "True":
return BDDNode(value=True)
if expression == "False":
return BDDNode(value=False)
# Handle variable (e.g., "x1" or "x2")
var = expression[0]
high = self.create_bdd(expression[1:])
low = self.create_bdd(expression[1:])
return self.add_node(var, high, low)
def evaluate(self, node, assignments):
"""Evaluate the BDD with the given variable assignments."""
if node is None:
return False
if node.value is not None: # Terminal node (True or False)
return node.value
# Traverse according to the variable's value in assignments
var_value = assignments.get(node.variable)
if var_value: # If True, move to the 'high' branch
return self.evaluate(node.high, assignments)
else: # If False, move to the 'low' branch
return self.evaluate(node.low, assignments)
# Example usage
bdd = BDD()
# Constructing a simple BDD for the expression "x1 AND x2"
# This represents the boolean function x1 AND x2
bdd_root = bdd.create_bdd("x1x2")
# Define variable assignments for testing
assignments = {"x1": True, "x2": True}
# Evaluate the BDD with the assignments
result = bdd.evaluate(bdd_root, assignments)
print(f"Result of BDD evaluation: {result}") # Expected output: True