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Bubble sort starts by comparing 6 and 7. They are in the correct order, so it then compares 7 and 3. They are in inverse order, so bubble sort exchanges 7 and 3, and then compares 7 and 1. The numbers 7 and 1 are in reverse order, so bubble sort swaps them, and then compares 7 and 4. Once again, the order is incorrect, so it swaps 7 and 4. End of scan 1: 63147 Scanning left to right again results in: 31467 Scanning left to right again results in a correct ordering: 13467 Write pseudocode for the bubble sort. 2.8 What is the bubble sort T 2.9 How will the bubble sort compare for speed with the merge sort when the task is to sort 1,000,000 social security numbers which initially are in random order
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VON NEUMANN ARCHITECTURE Most computers today operate according to the von Neumann architecture. The main idea of the von Neumann architecture is that the program to be executed resides in the computer s memory, along with the program s data. John von Neumann published this idea in 1945. Today this concept is so familiar it seems self-evident, but earlier computers were usually wired for a certain function. In effect, the program was built into the construction of the computer. Think of an early calculator; for example, imagine an old hand-cranked mechanical calculator. The machine was built to do one well-defined thing. In the case of an old hand-cranked calculator, it was built only to add. Put a number in; crank it; get the new sum. To subtract, the operator needed to know how to do complementary subtraction, which uses addition to accomplish subtraction. Instead of offering a subtract function, the old calculator required the operator to add the ten s complement of the number to be subtracted. You can search for ten s complement on Google to learn more, but the point for now is that early computing devices were built for certain functions only. One could never, for instance, use an old adding machine to maintain a list of neighbors phone numbers! The von Neumann architecture is also called the stored program computer. The program steps are stored in the computer s memory, and the computation cycle of the machine retrieves the next step (instruction to be executed) from memory, completes that computation, and then retrieves the next step. This cycle simply repeats until the computer retrieves an instruction to halt. There are three primary units in the von Neumann computer. Memory is where both programs and data are stored. The central processing unit (CPU) accesses the program and data in memory and performs the calculations. The I / O unit provides access to devices for data input and output. DATA REPRESENTATION We re used to representing numbers in base 10. Presumably this number base makes sense to us because we have 10 fingers. If our species had evolved with 12 fingers, we would probably have 2 more digits among the set of symbols we use, and we would find it quite natural to compute sums in base 12. However, we have only 10 fingers, so let s start with base 10. Remember what the columns mean when we write a number like 427. The seven means we have 7 units, the two means we have 2 tens, and the four means we have 4 hundreds. The total quantity is 4 hundreds, plus
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2 tens, plus 7. The column on the far right is for units (which you can also write as 100), the next column to the left is for 10s (which you can also write as 101), and the next column is for 100s (which you can write as 102). We say that we use base 10 because the columns correspond to powers of 10 100, 101, 102, etc. Suppose that we had evolved with 12 fingers and were more comfortable working in base 12, instead. What would the meaning of 427 be The seven would still mean 7 units (120 is also equal to 1), but now the two would mean 2 dozen (121 equals 12), and the four would mean 4 gross (122 equals 144). The value of the number 427 in base 12 would be 4 gross, plus 2 dozen, plus 7, or 607 in our more familiar base-10 representation. Some people say we would be better off using base 12, also known as the duodecimal or dozenal system. For example, you can readily find a sixth, a third, a quarter, or a half in base 12, whereas you can only find a half easily in base 10. Twelve is also a good match for our calendar, our clock, and even our compass. Ah well, the decision to use base 10 in daily life was made long ago! The point of this discussion is to show that base 10 is simply one number system of many. One can compute in base 10, or base 12, or base-any-other-number. Our choice of number system can be thought of as arbitrary we ve got 10 fingers, so let s use base 10. We could compute just as competently in base 7, or base 12, or base 2. Computers use base 2, because it s easy to build hardware that computes based on only two states on and off, one and zero. Base 2 is also called the binary number system, and the columns in a base-2 number work the same way as in any other base. The rightmost column is for units (20), the next column to the left is for twos (21), the next is for fours (22 = 4), the next is for eights (23 = 8), the next is for sixteens (24 = 16), etc. What is the base-10 value of the binary number 10011010 The column quantities from right to left are 128 (27), 64 (26), 32 (25), 16 (24), 8 (23), 4 (22), 2 (21), 1 (20). So, this number represents 128, plus 16, plus 8, plus 2 154 in base 10. We can calculate in base 2 after learning the math facts for binary math. You learned the math facts for base 10 when you studied your addition, subtraction, and multiplication tables in elementary school. The base-2 math facts are even simpler: 0 + 0 = 0 0 + 1 = 1 1 + 1 = 10 (remember, this means 2; and also 0 carry 1 to the next column) Let s add the binary value of 1100 to 0110: 1100 (12 in base 10) 0110 (6 in base 10) 10010 (18 in base 10) rightmost digit: next rightmost: next rightmost: next rightmost: last digit: 0+0=0 0+1=1 1 + 1 = 10 (or 0 carry 1) carried 1 + 1 + 0 = 10 (or 0 carry 1) 1 (from the carry)
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So, any kind of addition can be carried out using the binary number system, and the result will mean the same quantity as the result from using base 10. The numbers look different, but the quantities mean the same value. COMPUTER WORD SIZE Each computer deals with a certain number of bits at a time. The early hobbyist computers manipulated 8 bits at a time, and so were called 8-bit computers. Another way to say this was that the computer word size was 8 bits. The computer might be programmed to operate on more than 8 bits, but its basic operations dealt with 8 bits at a time.
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