Italian Folk Multiplication Why Parallelization Algorithm Is Indeed - PowerPoint PPT Presentation

How We Learn to . . . Ethnic Multiplication . . . Italian Folk Algorithm . . . How Are Ethnic . . . Italian Folk Multiplication Why Parallelization Algorithm Is Indeed Better: Which Algorithm Is . . . What Can Be . . . It Is More Parallelizable Conclusion Caution Martine Ceberio 1 , Olga Kosheleva 2 , Home Page and Vladik Kreinovich 1 Title Page Departments of 1 Computer Science and ◭◭ ◮◮ 2 Teacher Education University of Texas at El Paso ◭ ◮ 500 W. University, El Paso, TX 79968, USA mceberio@utep.edu, olgak@utep.edu, Page 1 of 22 vladik@utep.edu Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 1. How We Learn to Multiply Numbers Italian Folk Algorithm . . . • How students learn multiplication is school? How Are Ethnic . . . Why Parallelization • First, they memorize the multiplication table – which Which Algorithm Is . . . enables them to multiply 1-digit numbers. What Can Be . . . • Then, they learn how to multiply a multi-digit number Conclusion by a digit. Caution Home Page • Finally, they learn how to multiply two multi-digit numbers. Title Page • Let us recall how this is taught in school. ◭◭ ◮◮ • Suppose that we want to multiply a multi-digit number ◭ ◮ by a digit, e.g., multiply 23 by 4: Page 2 of 22 23 Go Back X 4 Full Screen --- ? Close Quit

How We Learn to . . . Ethnic Multiplication . . . 2. How We Learn to Multiply (cont-d) Italian Folk Algorithm . . . • We start with the lowest digit – in this case, with 3, How Are Ethnic . . . and multiply it by 4. Why Parallelization Which Algorithm Is . . . • From the multiplication table, we know that the result What Can Be . . . is 12, so: Conclusion – we place 2 in the corresponding digit of a product, Caution and Home Page – remember 1 as a carry , to be added to the next Title Page digit: ◭◭ ◮◮ 23 ◭ ◮ X 4 Page 3 of 22 --- ?2 Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 3. How We Learn to Multiply (cont-d) Italian Folk Algorithm . . . • Then, we multiply the next digit (in this case, 2) by 4, How Are Ethnic . . . getting 8. Why Parallelization Which Algorithm Is . . . • We add the carry (in this case, 1) to this product, What Can Be . . . getting 9: Conclusion 23 Caution X 4 Home Page --- 92 Title Page • Similarly, if we multiply 23 by 6, we: ◭◭ ◮◮ – get 3 · 6 = 18, so the carry is 1, and ◭ ◮ – then compute 2 · 6 + 1 = 13, so the result is: Page 4 of 22 23 Go Back X 6 Full Screen --- Close 138 Quit

How We Learn to . . . Ethnic Multiplication . . . 4. How We Learn to Multiply (cont-d) Italian Folk Algorithm . . . • First, the students master the art of multiplying a How Are Ethnic . . . multi-digit number by a digit. Why Parallelization Which Algorithm Is . . . • Then, they learn to multiply multi-digit numbers: What Can Be . . . – first, we multiply the first number by each digit of Conclusion the second number, and Caution – then, we add up all the resulting products. Home Page • For example, to multiply 23 by 64, we first perform the Title Page above two multiplications, and then add the results: ◭◭ ◮◮ 23 ◭ ◮ X 64 ---- Page 5 of 22 92 Go Back + 138 Full Screen ------ 1472 Close Quit

How We Learn to . . . Ethnic Multiplication . . . 5. Ethnic Multiplication Algorithms Italian Folk Algorithm . . . • In the past, different ethic groups used different algo- How Are Ethnic . . . rithms for multiplication. Why Parallelization Which Algorithm Is . . . • Probably the most well known is the Russian multipli- What Can Be . . . cation algorithm. Conclusion • In this algorithm, to compute the product a · b , we, in Caution effect: Home Page – translate the second number b into the binary code, Title Page i.e., represent it as ◭◭ ◮◮ b = 2 i 1 + 2 i 2 + . . . + 2 i k for some i 1 > i 2 > . . . > i k ; ◭ ◮ – then we consequently double the first number a , to Page 6 of 22 get the values a, 2 1 · a, 2 2 · a, . . . , 2 i 1 · a ; Go Back – after this, we add the products corresponding to the powers of 2 that form b : Full Screen a · b = 2 i 1 · a + 2 i 2 · a + . . . + 2 i k · a. Close Quit

How We Learn to . . . Ethnic Multiplication . . . 6. Ethnic Multiplication Algorithms (cont-d) Italian Folk Algorithm . . . • If we only have two numbers a and b , this algorithm How Are Ethnic . . . takes too many steps. Why Parallelization Which Algorithm Is . . . • It starts making sense if we have to multiply the same What Can Be . . . number a by different values b . Conclusion • For example – and this is where this algorithm origi- Caution nated: Home Page – a merchant is selling some material by yards, and Title Page – he (in the old days, it was usually he) needs to find ◭◭ ◮◮ the price of different amounts of material. ◭ ◮ • In this case, a – the price per yard – remains the same, Page 7 of 22 while the length b changes. Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 7. Ethnic Multiplication Algorithms (cont-d) Italian Folk Algorithm . . . • The advantage is that in this case, we perform all the How Are Ethnic . . . doublings only once – as result, we only need: Why Parallelization Which Algorithm Is . . . – to translate into binary code – and for this, it is What Can Be . . . sufficient to divide by 2, and then Conclusion – to add the corresponding products 2 i · a . Caution • Different ethnic groups had different algorithms. Home Page • For example, in the traditional Italian folks multipli- Title Page cation algorithm, we: ◭◭ ◮◮ – multiply each digit of the first number by each digit ◭ ◮ of the second number, and then Page 8 of 22 – add the results. Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 8. Italian Folk Algorithm (cont-d) Italian Folk Algorithm . . . • For example, in this algorithm, the multiplication of How Are Ethnic . . . 23 by 64 takes the following form: Why Parallelization Which Algorithm Is . . . 23 What Can Be . . . X 64 Conclusion ---- Caution 12 = 3 x 4 Home Page 18 = 3 x 6 Title Page 8 = 2 x 4 ◭◭ ◮◮ + 12 = 2 x 6 ------- ◭ ◮ 1472 Page 9 of 22 Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 9. How Are Ethnic Algorithms Viewed Now Italian Folk Algorithm . . . • At present, the ethnic algorithms are studied mostly How Are Ethnic . . . by historians of science and by pedagogues. Why Parallelization Which Algorithm Is . . . • To pedagogues, such algorithms are an interesting way What Can Be . . . to make arithmetic more exciting to the kids. Conclusion • In general, studying different algorithms raises the stu- Caution dents’ curiosity level. Home Page • Also, studying algorithms of one’s own ethnic group is Title Page enhanced by the students’ patriotic feelings. ◭◭ ◮◮ • Although, strangely enough, OK and VK, ◭ ◮ – when studying arithmetic in Russia, Page 10 of 22 – never heard of the Russian multiplication algorithm. Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 10. What We Show in This Talk Italian Folk Algorithm . . . • Our goal is to show that ethnic algorithms actually How Are Ethnic . . . make sense. Why Parallelization Which Algorithm Is . . . • Many of them are, in some sense, better than the al- What Can Be . . . gorithm that we learn at school. Conclusion • We have already mentioned this for the Russian mul- Caution tiplication algorithm. Home Page • In this talk, we show that the Italian folk multiplication Title Page algorithm also has its advantages. ◭◭ ◮◮ • Namely, we show that the Italian algorithm is easier to ◭ ◮ parallelize. Page 11 of 22 Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 11. Why Parallelization Italian Folk Algorithm . . . • Nowadays, most multiplication is performed by com- How Are Ethnic . . . puters. Why Parallelization Which Algorithm Is . . . • Computers have no problem multiplying large num- What Can Be . . . bers. Conclusion • However, in the past, multiplication was not easy. Caution Home Page • In the Middle Ages, when even literacy was rather an exception, Title Page – those who could multiply never needed to do a ◭◭ ◮◮ back-breaking menial work: ◭ ◮ – they could easily find employment as assistants to Page 12 of 22 merchants. Go Back Full Screen Close Quit

How We Learn to . . . Ethnic Multiplication . . . 12. Why Parallelization (cont-d) Italian Folk Algorithm . . . • If one needs to multiply two large numbers, and the How Are Ethnic . . . result is important – a natural idea is: Why Parallelization Which Algorithm Is . . . – to ask for help, to divide the job, so that What Can Be . . . – two specialists in this complex art of multiplication Conclusion could perform some operations at the same time Caution (“in parallel”) and thus, speed up the process. Home Page • In a nutshell, this is the same reason why modern Title Page computer-based computations use parallelization: ◭◭ ◮◮ – if a computation takes too long on a single proces- ◭ ◮ sor, Page 13 of 22 – a reasonable idea is to have several processors work- ing in parallel. Go Back Full Screen Close Quit