CMSC 100 Efficiency of Algorithms

1. CMSC 100Efficiency of Algorithms Guest Lecturers: Clay Alberty and Kellie LaFlamme Professor Marie desJardinsTuesday, October 2, 2012 Some material adapted from instructor slides for Schneider & Gerstung

2. Overview • What makes a good algorithm? • Correctness • Ease of understanding • Elegance • Efficiency • Computational efficiency – order of magnitude of running time • Polynomial – time increases (reasonably) slowly as problem size increases  tractable (solvable in reasonable time) • Exponential (or worse!) – time increases explosively as problem size increases  intractable (can’t be solved in practice for big problems) • (We are also sometimes interested in memory or space efficiency) Efficiency of Algorithms

3. Introduction • There are many solutions to any given problem • How can we judge and compare algorithms? • Analogy: Purchasing a car • safety • ease of handling • style • fuel efficiency • Evaluating an algorithm • correctness • ease of understanding • elegance • time/space efficiency Efficiency of Algorithms

4. Attributes of Algorithms • Attributes of interest: correctness, ease of understanding, elegance, and efficiency • Correctness: • Is the problem specified correctly? • Does the algorithm produce the correct result? • Example: pattern matching • Problem specification: “Given pattern p and text t, determine the location, if any, of pattern p occurring in text t” • Correctness: does the algorithm always work? • If p is in t, will it say so? • If the algorithm says p is in t, is it? Efficiency of Algorithms

5. Attributes of Algorithms (continued) • Ease of understanding, useful for: • checking correctness • program maintenance • Elegance: using a clever or non-obvious approach • Example: Gauss’ summing of 1 + 2 + … + 100 • Attributes may conflict: • Elegance often conflicts with ease of understanding • Attributes may reinforce each other: • Ease of understanding supports correctness Efficiency of Algorithms

6. Attributes of Algorithms (continued) • Efficiency: an algorithm’s use of time and space resources • We’ll focus on computational efficiency (time) • Timing an algorithm with a clock is not always useful • Confounding factors: machine speed, size of input • Benchmarking: timing an algorithm on standard data sets • Testing hardware and operating system, etc. • Testing real-world performance limits • Analysis of algorithms: the study of the efficiency of algorithms • Order of magnitude Θ() or just O() (“big O”): • The class of functions that describes how time increases as a function of problem size (more on which later) Efficiency of Algorithms

7. Sequential Search Analysis Efficiency of Algorithms

8. Measuring EfficiencySequential Search • Searching: the task of finding a specific value in a list of values, or deciding it is not there • Solution #1: Sequential search (from Ch. 2): “Given a target value and a random list of values, find the location of the target in the list, if it occurs, by checking each value in the list in turn” Efficiency of Algorithms

9. Sequential Search Algorithm get (NameList, PhoneList, Name)i = 1N = length(NameList)Found = FALSEwhile ( (not Found) and (i <= N) ) { if ( Name == NameList[i] ) { print (Name, “’s phone number is ”, PhoneList[i]) Found = TRUE } i = i+1}if ( not Found ) { print (Name, “’s phone number not found!”) } Efficiency of Algorithms

10. Measuring EfficiencySequential Search (continued) • Central unit of work: operations that occur most frequently • Central unit of work in sequential search: • Comparison of target Name to each name in the list • Also add 1 to i • Typical iteration: two steps (one comparison, one addition) • Given a large input list: • Best case: smallest amount of work algorithm must do • Worst case: greatest amount of work algorithm must do • Average case: depends on likelihood of different scenarios occurring • What are the best, worst, and average cases for sequential search? Efficiency of Algorithms

11. Measuring EfficiencySequential Search (continued) • Best case: target found with the first comparison (1 iteration) • Worst case: target never found or last value (N iterations) • Average case: if each value is equally likely to be searched, work done varies from 1 to N, on average N/2 iterations • Most often we will consider the worst case • Best case is too lucky – can’t count on it • Average case is much harder to compute for many problems (hard to know the distribution of possible solutions) Efficiency of Algorithms

12. Measuring EfficiencyOrder of Magnitude—Order n • Sequential search worst case (N) grows linearly in the size of the problem • 2N steps (one comparison and one addition per loop) • Also some initialization steps... • On the last iteration, we may print something... • After the loop, we test and maybe print... • To simplify analysis, disregard the “negligible” steps (which don’t happen asoften), and ignore the coefficient in 2N • Just pay attention to the dominant term (N) • Order of magnitudeO(N): the class of all linear functions (any algorithm that takes C1N + C2steps for any constants C1 and C2) Efficiency of Algorithms

13. Binary Search • Solution #2: Binary search • Assume list is sorted • Split the list in half on each iteration • On each iteration: • If we’ve run out of things to look at, quit • Is the centerpoint of the list the name we’re looking for? • If so, we’re done! • If not, check whether the name is alphabetically after or before the centerpoint of the list • If it’s after, take the second half of the list and continue looping • If it’s before, take the first half of the list and continue looping Efficiency of Algorithms

14. Binary Search: Algorithm get (NameList, PhoneList, Name)N = length(NameList)upper = Nlower = 1Found = FALSEwhile ( (not Found) and (lower <= upper) ) { if ( Name == NameList[(lower+upper)/2] ) { // is it in the center? print (Name, “’s phone number is ”, PhoneList[lower+upper/2]) Found = TRUE } else if ( Name < NameList[(lower+upper)/2] ) { upper = ((lower+upper)/2) – 1 // keep looking AFTER center } else if (Name > NameList[(lower+upper)/2] ) { lower = ((lower+upper)/2) + 1 // keep looking BEFORE center}if ( not Found ) { print (Name, “’s phone number not found!”) } Efficiency of Algorithms

15. Measuring EfficiencyBinary Search • Best case: target found with the first comparison (1 iteration) • Worst case: target never found or last value (split the list in half repeatedly until only one item to examine) • For a list of length 2, test twice • For a list of length 4, test three times (split twice) • For a list of length 8, only test four times! (split three times) • For a list of length 2k, how many times to test? • For a list of length N, how many times to test? • Average case: harder than you would think to analyze... • and surprisingly, the average case is only one step better than the worst case • Why? Hint: Try drawing a tree of all of the cases (left side, right side after each time the list is split in half) Efficiency of Algorithms

16. Orders of Magnitude: log N • Binary search has order of magnitude O(log2 N): grows very slowly • Here: log2 (base 2) but other bases behave similarly Efficiency of Algorithms

17. Sorting Algorithms and Analysis http://www.youtube.com/watch?v=k4RRi_ntQc8 http://www.youtube.com/watch?v=2HjspVV0jK4 Efficiency of Algorithms

18. Measuring EfficiencySelection Sort • Sorting: The task of putting a list of values into numeric or alphabetical order • Selection sort: • Pass repeatedly over the unsorted portion of the list • On each pass, select the largest remaining value • Move that value immediately after the other unsorted values • Accumulate the largest values, in reverse order, at the end of the list • After each iteration, the number of sorted values grows by one and the number of unsorted values shrinks by one Efficiency of Algorithms

19. How long does this step take?(Hint: do we use sequential search or binary search? Efficiency of Algorithms

20. Measuring EfficiencySelection Sort (continued) • Central unit of work: hidden in “find largest” step • Work done to find largest changes as unsorted portion shrinks • (N-1) + (N-2) + … + 2 + 1 = N (N-1) / 2 Efficiency of Algorithms

21. Measuring EfficiencyOrder of Magnitude—Order N2 • Selection sort takes N(N-1)/2 steps = ½ N2 – N/2 • Order of magnitude:ignore all but thedominant (highest-order)term; ignore thecoefficient • Order O(N2): the set of functions whose growth is on the order of N2 Efficiency of Algorithms

22. Measuring EfficiencyOrder of Magnitude—Order N2 (continued) Eventually, every function with order N2 has greater values than any function with order N Efficiency of Algorithms

23. Efficiency of Algorithms

24. Quicksort • Quicksort is a faster searching algorithm • Divide and conquer approach: • Pick a point in the list (the pivot) • Toss everything smaller than the pivot to the left and everything larger to the right • Separately sort those two sublists (using quicksort! This is an example of a recursive algorithm, which we’ll talk about more later in the semester...) • Quicksort is O(N log N) on average (but can be O(N2) in the worst case...) • O(N log N) is slower than linear but faster than quadratic Efficiency of Algorithms

25. Getting Out of Control • Polynomially bounded: an algorithm that does work on the order of O(Nk) (or less) • linear, log, quadratic, ... degree k polynomial • Most common problems are polynomially bounded (in “P”) • Hamiltonian circuit (“traveling salesman problem”): no known polynomial solution—it is in “NP” • Given a graph, find a path that passes through each vertex exactly once and returns to its starting point Efficiency of Algorithms

26. # possible circuits grows exponentially in the number of cities  takes a long time to find the best one! Efficiency of Algorithms

27. Efficiency of Algorithms

28. Summary • We must evaluate the quality of algorithms, and compare competing algorithms to each other • Attributes: correctness, efficiency, elegance, and ease of understanding • Compare competing algorithms for time and space efficiency (time/space tradeoffs are common) • Orders of magnitude capture work as a function of input size: O(log N), O(N), O(N2), O(2N) • Problems with only exponential algorithms are intractable Efficiency of Algorithms

29. Summary of Complexity Classes Efficiency of Algorithms