3. (a) Suppose we could multiply two 33 matrices using 25 scalar multiplications and a constant...

Transcribed Image Text:

3. (a) Suppose we could multiply two 33 matrices using 25 scalar multiplications and a constant number of scalar additions and subtractions. Set up and solve the recurrence relations to analyze the resulting divide-and-conquer algorithm for matrix multiplication. (b) Suppose we could multiply two 33 matrices using r scalar multiplications and a constant number of scalar additions and subtractions. How small would r have to be to make the resulting divide-and-conquer algorithm for matrix multiplication asymptotically faster than Strassen's matrix multiplication algorithm? Justify your answer. A6. a. Draw the digital circuit for following Boolean expression AB+BC (B+C). b. Simplify the Boolean expression using laws of Boolean algebra. c. Draw the truth table for the simplified Boolean expression. d. Define a relation R on (1, 2, 3, 4) by R-((a, b)/a+b is even) and Write R as a list of pairs. 1. 11. 111. Construct the matrix representation of R. Draw the graph representation of the relation R. Solution: We define the Longest MatPath as the longest path of integers in a matrix of unique integers, such that each integer forming the path is adjacent to one another and increasing by a value of one. For example: suppose we have the following matrix: 4 5 7 8 1 2 6 9 11 3 13 12 [10 14 15 16 The longest path in the above matrix is {6, 7, 8, 9}. (a) Write a recursive algorithm that, given an n n matrix and a starting index (where 6 in the above example has index (2, 3)), returns the longest path starting at this integer. (b) Write another algorithm that runs the above algorithm as a subroutine and finds the longest path in the matrix overall. (c) Modify your origianl algorithm, so that it stores longest paths it has already calculated and uses this storage to avoid making unnecessary recursive calls. 3. (a) Suppose we could multiply two 33 matrices using 25 scalar multiplications and a constant number of scalar additions and subtractions. Set up and solve the recurrence relations to analyze the resulting divide-and-conquer algorithm for matrix multiplication. (b) Suppose we could multiply two 33 matrices using r scalar multiplications and a constant number of scalar additions and subtractions. How small would r have to be to make the resulting divide-and-conquer algorithm for matrix multiplication asymptotically faster than Strassen's matrix multiplication algorithm? Justify your answer. A6. a. Draw the digital circuit for following Boolean expression AB+BC (B+C). b. Simplify the Boolean expression using laws of Boolean algebra. c. Draw the truth table for the simplified Boolean expression. d. Define a relation R on (1, 2, 3, 4) by R-((a, b)/a+b is even) and Write R as a list of pairs. 1. 11. 111. Construct the matrix representation of R. Draw the graph representation of the relation R. Solution: We define the Longest MatPath as the longest path of integers in a matrix of unique integers, such that each integer forming the path is adjacent to one another and increasing by a value of one. For example: suppose we have the following matrix: 4 5 7 8 1 2 6 9 11 3 13 12 [10 14 15 16 The longest path in the above matrix is {6, 7, 8, 9}. (a) Write a recursive algorithm that, given an n n matrix and a starting index (where 6 in the above example has index (2, 3)), returns the longest path starting at this integer. (b) Write another algorithm that runs the above algorithm as a subroutine and finds the longest path in the matrix overall. (c) Modify your origianl algorithm, so that it stores longest paths it has already calculated and uses this storage to avoid making unnecessary recursive calls.