CS10: The Beauty and Joy of Computing Lecture #7: Algorithm Complexity TA Jon Kotker (2010-09-27) LEDs + Math = Art Leo Villareal combines modern LED control.

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CS10: The Beauty and Joy of Computing Lecture #7: Algorithm Complexity TA Jon Kotker ( ) LEDs + Math = Art Leo Villareal combines modern LED control systems to produce contemporary modern art. The exhibit is on display at the San Jose Museum of Art _ html

BIG IDEA Many ways to do the same thing = many algorithms to accomplish the same task. Example: Distributing candy! Example: Searching through a list of numbers to find a specific number. ◦ Linear search (list is unsorted): Go through the list number by number and check if each number is The One. ◦ Binary search (list is sorted): Look at the middle of the list. If it is not The One, break the list into two smaller halves and ignore the irrelevant half. ◦ Any other algorithms? UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity2

MAKING A DECISION How do we decide which algorithm to use? Which is easier to implement? Which takes less time? Which uses up less space (memory)? Which gives a more precise answer? Which of the above questions even matter? UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity3

WHAT DO YOU THINK? Which of the factors below is most important in making a choice between two algorithms? A. Which is easier to implement? B. Which takes less time? C. Which uses up less space (memory)? D. Which gives a more precise answer? E. I don’t know / I don’t have an opinion. UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity4

RUNTIME ANALYSIS One commonly used criterion in making a decision is runtime – how much time does the algorithm take to run and finish its task? Computers are most useful for large inputs, so find the runtime of the algorithm on large inputs. How do we do that? UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity5

RUNTIME ANALYSIS Time the algorithm with a stopwatch! But… Different computers will have different runtimes.  Same computer can have different runtime on the same input.  Need to implement the algorithm first so that we can run it. o_O; Solution: Need to somehow abstract the computer away from the algorithm. UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity6

RUNTIME ANALYSIS Idea: Do not focus on how long the algorithm takes on one input. Instead, focus on how the worst-case runtime of the algorithm scales as we scale the input. Why? Abstracts the computer out. A good algorithm should work well, no matter what the computer = a good recipe should produce a great dish, no matter what the kitchen. UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity7

RUNTIME ANALYSIS Idea: Do not focus on how long the algorithm takes on one input. Instead, focus on how the worst-case runtime of the algorithm scales as we scale the input. Why? Computers are mainly used for large sets of data. The runtime of an algorithm should scale “reasonably” as we make the dataset even larger, or else we need to improve/discard that algorithm. UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity8

A LOT OF APPROXIMATION AHEAD! UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity9 Dangerous for Math majors. Courtesy

ARITHMETIC OPERATIONS Key Idea: As the input scales, arithmetic operations take approximately the same time. Arithmetic operations are constant-time operations. Another Key Idea: We only care about how the runtime of the block scales as the input scales! UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity10

LAUNDRY It Must Be Done UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity11

WASHING LOADS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity12 If each load takes about the same time to launder, how does the runtime scale as the number of loads doubles? Triples?

WASHING LOADS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity13 Key Idea: The runtime of the algorithm scales by the same amount as the size of its input scales. Doing laundry is a linear-time operation in the number of loads.

FINDING CLOTHES TO WEAR UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity14 How does the worst-case total time to find a good pair scale as the number of shirts and pants doubles? 1. Stays the same. 2. Doubles. 3. Halves. 4. Quadruples. 5. Buh? Hint: For each shirt that I own, how many pants do I have to match against it?

FINDING CLOTHES TO WEAR UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity15 Key Idea: If I have 3 shirts and pants, there are 9 different combinations that I can try, because for each shirt, I can try on 3 pants to match with it. Double it: If I have 6 shirts and pants, there are 36 different combinations that I can try. If I double the number of shirts and pants that I have, then the number of different combinations that I can try quadruples.

FINDING CLOTHES TO WEAR UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity16 Key Idea: The runtime of the algorithm scales by the square of the amount that the input scales by. Finding a good pair of clothes to wear is a quadratic-time algorithm in the number of shirts and pants.

LAUNDRY It Has Been Done UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity17

RUNTIME ANALYSIS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity18 What is the runtime of this script? 1. Constant in num. 2. Linear in num. 3. Quadratic in num. 4. Buh?

RUNTIME ANALYSIS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity19 What is the runtime of this script? 1. Constant in num. 2. Linear in num. 3. Quadratic in num. 4. Buh?

RUNTIME ANALYSIS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity20 What is the runtime of this script? 1. Constant in num. 2. Linear in num. 3. Quadratic in num. 4. Buh?

RUNTIME ANALYSIS UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity21 What is the runtime of this script? 1. Constant in num. 2. Linear in num. 3. Quadratic in num. 4. Buh?

IT’S ALL APPROXIMATE! UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity22 Which is better: a linear-time algorithm or a quadratic-time algorithm? As the input size increases, the quadratic-time algorithm takes so much more time than the linear-time algorithm that the linear-time algorithm is negligible in comparison. Input Size (N) Linear (msec) C10C100C1000C10000C Quadratic (msec) C100C10000C10 6 C10 8 C

IT’S ALL APPROXIMATE! UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity23 Which is better: a linear-time algorithm or a quadratic-time algorithm? Since we only consider large sized inputs, expressions like N 2 – N or N 2 + N are considered approximately equal to N and thus quadratic-time; the linear-time part is ignored. Input Size (N) Linear (msec) C10C100C1000C10000C Quadratic (msec) C100C10000C10 6 C10 8 C

RUNTIME ANALYSIS (EXTRA) UC Berkeley CS10 "The Beauty and Joy of Computing" : Algorithm Complexity24 What is the runtime of this algorithm to find the factorial of a number? 1. Constant in the number. 2. Linear in the number. 3. Quadratic in the number. 4. Buh?