Advanced Topics in Algorithms and Data Structures 1 An example.

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Presentation transcript:

Advanced Topics in Algorithms and Data Structures 1 An example

Advanced Topics in Algorithms and Data Structures 2 An example

Advanced Topics in Algorithms and Data Structures 3 Evaluation of arithmetic expressions If we evaluate an expression tree bottom-up, it will take O( n ) time for a long and skinny tree. Hence we apply tree contraction.

Advanced Topics in Algorithms and Data Structures 4 Evaluation of arithmetic expressions We do not completely evaluate each internal node. We evaluate the internal nodes partially. For each internal node v, we associate a label ( a v, b v ). a v and b v are constants. The value of the expression at node is: ( a v X + b v ), where X is an unknown value for the expression of the subtree rooted at v.

Advanced Topics in Algorithms and Data Structures 5 Keeping an invariant Invariant: Let u be an internal node which holds the operation  , . Let v and w are the children of u with labels ( a v, b v ) and ( a w, b w ). Then the value at u is: val ( u ) = ( a v val ( v ) + b v )  ( a w val ( w ) + b w )

Advanced Topics in Algorithms and Data Structures 6 Keeping an invariant

Advanced Topics in Algorithms and Data Structures 7 Applying the rake operation The value at node u is: val ( u ) = ( a v c v + b v )  ( a w X + b w ) X is the unknown value at node w.

Advanced Topics in Algorithms and Data Structures 8 Applying the rake operation The contribution of val ( u ) to the value of node p ( u ) is: a u  val ( u ) + b u = a u [( a v c v + b v )  ( a w X + b w )] + b u We can adjust the labels of node w to ( a ’ w, b ’ w ) a ’ w = a u ( a v c v + b v ) a w b ’ w = a u ( a v c v + b v ) b w + b u

Advanced Topics in Algorithms and Data Structures 9 Complexity of expression evaluation The correctness of the expression evaluation depends on correctly maintaining the invariants. We start with a label (1, 0) for each leaf and correctly maintain the invariant at each rake operation. We have already proved the correctness of the rake operation. Hence, evaluation of an expression given as a binary tree takes O ( n ) work and O (log n ) time.

Advanced Topics in Algorithms and Data Structures 10 An example

Advanced Topics in Algorithms and Data Structures 11 An example

Advanced Topics in Algorithms and Data Structures 12 An example