Chapter 1 addtion - exp Mod N Fall 2017 PDF.pdf

Chapter 1 addtion - exp Mod N Fall 2017 PDF.pdf - Chapter 1...

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Chapter 1 Numerical Algorithms
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Addition on a computer is done by circuits Modern computers 64 bits at a time. (Aside, C int is still 32 bits for compatibility) One bit adder demonstration is by Marble machine (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons Because we are working with asymptotic notation. A 64-bit adder is a constant 64 times as e cient as a 1-bit adder In the bit model we assume we are working with a 1-bit adder because we can simply extract out the factor of 64 and treat everything as 1-bit adder. Note we only need the bit model when working with numbers that have no fixed upper bound (as that upper bound can be treated as another constant value). 1 1 0 1 0 + 1 1 1 0 1 16’s 8’s 4’s 2’s 1’s 26 +29
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Binary Addition Using the Bit Model In the bit-model, we consider adding three binary digits to be one step (i.e. takes a constant amount of time) Adding three single digit numbers results in a number at most two digits long! (This is true for addition in any base 2) As a function of the input size, how long does it take to add two n-bit binary numbers? (e.g. n 2048)? x 1 x 2 x 3 x 4 ! x n 1 x n + y 1 y 2 y 3 y 4 ! y n 1 y n z 0 z 1 z 2 z 3 z 4 ! z n 1 z n 1 1 + 1 0 1 + 1 0 0 + 1 1 1 + 0 1 0 + 1 1 0 + 0 0 0 + 0 n-bit # + n-bit # (n+1)-bit # O(n) bit operations thus O(n) time
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Binary Addition* (cont) To add two n-bit number we needed to be able to read them and write down the answer. Each bit in the sum took a constant amount of time to compute So, our addition algorithm is optimal up to multiplicative contents! We will usually use N to denote the number and n to denote the number of bits in the binary representation of N N = ( 1 0 0 1 1 ) 2 16’s 8’s 4’s 2’s 1’s n = 5 10 = 19 10 *Of course, assumptions on the number and the data structure could change the analysis
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We now can use addition as a subroutine in other algorithms For n = log 2 (N+1) , adding two n-bit binary numbers takes O(n) time
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X*2, X/2 We are going to do multiplication and division next. In order to do this, we need two lower level operations. The ability to double X, and the ability to halve X (With flooring) We accomplish this by the use of bit shift operations. In C this is represented by << and >> These shift a binary number “over”.
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Bit shift example Doubling: 0010 ShiftLeft 1 = 0100 (I.e. 2 ShiftLeft 1 = 4) 0010 ShiftRight 1 = 0001 (I.e. 2 shiftRight 1 = 1) Note that 0011 ShiftRight 1 = 0001 (I.e. 3 ShiftRight 1 = 1) This means that if you use shift to quickly compute division by two you will get the floor
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Bit Shift Complexity As you might have guessed, bit shifting takes each bit and moves it over by one. This is O(n) in the bit model
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How can you multiply two very large binary numbers?
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