1. CS 150: Analysis of Algorithms

2. Goals for this Unit Begin a focus on data structures and algorithms Understand the nature of the performance of algorithms Understand how we measure performance –Used to describe performance of various data structures Begin to see the role of algorithms in the study of Computer Science

3. Algorithm An algorithm is –a detailed step-by-step method for –solving a problem Computer algorithms Properties of algorithms –Steps are precisely stated –Determinism: based on inputs, previous steps –Algorithm terminates –Also: correctness, generality, efficiency

4. C_M_L_X_T_ 1.R O U F O 2.U B I R A 3.O U M A J 4.I P E Y O This is an important CS term for this unit. Can you guess? Choose a the set of letters below (their order is jumbled).

5. What makes an algorithm “better”? Let’s look again at the Fibonacci example Why does one version run much much slower than the other? If recursion is this slow, why ever use it?

6. Why Not Just Time Algorithms? We want a measure of work that gives us a direct measure of the efficiency of the algorithm –independent of computer, programming language, programmer, and other implementation details. –Usually depending on the size of the input –Also often dependent on the nature of the input Best-case, worst-case, average

7. Analysis of Algorithms Use mathematics as our tool for analyzing algorithm performance –Measure the algorithm itself, its nature –Not its implementation or its execution Need to count something –Cost or number of steps is a function of input size n: e.g. for input size n, cost is f(n) –Count all steps in an algorithm? (Hopefully avoid this!)

8. Counting Operations Strategy: choose one operation or one section of code such that –The total work is always roughly proportional to how often that’s done So we’ll just count: –An algorithm’s “basic operation” –Or, an algorithms’ “critical section” Sometimes the basic operation is some action that’s fundamentally central to how the algorithm works –Example: Search a List for a target involves comparing each list-item to the target. –The comparison operation is “fundamental.”

9. Asymptotic Analysis Given some formula f(n) for the count/cost of some thing based on the input size –We’re going to focus on its “order” –f(n) = 2 n ---> Exponential function –f(n) = 100n 2 + 50n + 7 ---> Quadratic function –f(n) = 30 n lg n – 10 ---> Log-linear function –f(n) = 1000n ---> Linear function These functions grow at different rates –As inputs get larger, the amount they increase differs “Asymptotic” – how do things change as the input size n gets larger?

10. Comparison of Growth Rates

11. Comparison of Growth Rates (2)

12. Comparison of Growth Rates (3)

13. Order Classes For a given algorithm, we count something: –f(n) = 100n 2 + 50n + 7 ---> Quadratic function –How different is this than this? f(n) = 20n 2 + 7n + 2 –For large inputs? Order class: a “label” for all functions with the same highest-order term –Label form: O(n 2 ) or Θ (n 2 ) or a few others –“Big-Oh” used most often

14. Growth Notations g  O(f) (“Big-Oh”) g grows no faster than f (upper bound) g   (f) (“Theta”) g grows as fast as f (tight bound) g   (f) (“Omega”) g grows no slower than f (lower bound) Which one would we most like to know?

15. Meaning of O (“big Oh”) g is in O (f) iff: There are positive constants c and n 0 such that g(n) ≤ cf(n) for all n ≥ n 0.

16. O Examples Is n in O (n 2 ) ? Yes, c = 1 and n 0 =1 works. Is 10n in O (n) ? Yes, c =.09 and n 0 =1 works. Is n 2 in O (n) ? No, no matter what c we pick, cn 2 > n for big enough n (n > c) g is in O (f) iff there are positive constants c and n 0 such that g(n) ≤ cf(n) for all n ≥ n 0.

17. Back to Order Classes Order classes group “equivalently” efficient algorithms –O(1) – constant time! Input size doesn’t matter –O(lg n) – logarithmic time. Very efficient. E.g. binary search (after sorting) –O(n) – linear time –O(n lg n) – log-linear time. E.g. best sorting algorithms –O(n 2 ) – quadratic time. E.g. poorer sorting algorithms –O(n 3 ) – cubic time –…. –O(2 n ) – exponential time. Many important problems, often about optimization.

18. Ω (“Omega”): Lower Bound g is in Ω (f) iff there are positive constants c and n 0 such that for all n ≥ n 0. g is in O (f) iff there are positive constants c and n 0 such that g(n) ≤ cf(n) for all n ≥ n 0. g(n) ≥ cf(n)

19. Example: Watch Code Run Two implementations to calculate Fibonacci numbers: F(0) = F(1) = 1 F(n) = F(n-1) + F(n-2) for n > 1 Both correct! Let’s run both for n=8, 16, 32, 64

20. When Does this Matter? Size of input matters a lot! –For small inputs, we care a lot less –But what’s a big input? Hard to know. For some algorithms, smaller than you think!

21. Two Important Problems Search –Given a list of items and a target value –Find if/where it is in the list Return special value if not there (“sentinel” value) –Note we’ve specified this at an abstract level Haven’t said how to implement this Is this specification complete? Why not? Sorting –Given a list of items –Re-arrange them in some non-decreasing order With solutions to these two, we can do many useful things! What to count for complexity analysis? For both, the basic operation is: comparing two list-elements

22. Sequential Search Algorithm Sequential Search –AKA linear search –Look through the list until we reach the end or find the target –Best-case? Worst-case? –Order class: O(n) –Advantages: simple to code, no assumptions about list

23. Binary Search Algorithm Binary Search –Input list must be sorted –Strategy: Eliminate about half items left with one comparison Look in the middle If target larger, must be in the 2 nd half If target smaller, must be in the 1 st half Complexity: O(lg n) Must sort list first, but… Much more efficient than sequential search –Especially if search is done many times (sort once, search many times) Note: Java provides static binarySearch()

24. Class Activity We’ll give you some binary search inputs –Array of int values, plus a target to find –You tell us what index is returned, and the sequence of index values that were compared to the target to get that answer Work in two’s or three’s, and we’ll call on groups to explain

25. Binary Search Example #1 Input: -1 4 5 11 13 and target 4 Index returned is? Vote on indices (positions) compared to target-value: 1.-1 4 2.0 1 3.5 -1 4 4.2 0 1 5.other

26. Binary Search Example #2 Input: -5 -2 -1 4 5 11 13 14 17 18 and target 3 Index returned is? Vote on indices compared below: 1.5 2 3 4 2.4 1 2 3 3.4 2 3 4 4.other

27. Binary Search Example #3 Input: -1 4 5 11 13 and target 13 Index returned is? Vote on indices compared below: 1.0 1 2 3 4 2.2 4 3.2 3 4 4.other

28. Note on last examples Input size doubled from 5 to 10 How did the worst-case number of comparisons change? –Also double? –Something else Note binary search is O(lg n) –What does this imply here?

29. How to Sort? Many sorting algorithms have been found! –Problem is a case-study in algorithm design Some “straightforward” sorts –Insertion Sort, Selection Sort, Bubble Sort –O(n 2 ) More efficient sorts –Quicksort, mergesort, heapsort –O(n lg n) Note: these are for sorting in memory, not on disk

30. Reminder Slide: Order Classes For a given algorithm, we count something: –f(n) = 100n 2 + 50n + 7 ---> Quadratic function –How different is this than this? f(n) = 20n 2 + 7n + 2 –For large inputs? Order class: a “label” for all functions with the same highest-order term –Label form: O(n 2 ) or Θ (n 2 ) or a few others –“Big-Oh” used most often

31. Order Classes Details What does the label mean? O(n 2 ) –Set of all functions that grow at the same rate as n 2 or more slowly –I.e. as efficient as any “n 2 ” or more efficient, but no worse –So this is an upper-bound on how inefficient an algorithm can be Usage: We might say: Algorithm A is O(n 2 ) –Means Algorithm A’s efficiency grows like a quadratic algorithm or grows more slowly. (As good or better) What about that other label, Θ (n 2 )? –Set of all functions that grow at exactly the same rate –A more precise bound

32. Reminders: Input and Performance Input matters: –First, we said for small size of inputs, often no difference Also, often focus on the worst-case input –Why? Sometimes interested in the average-case input –Why? Also the best-case, but do we care?

33. Reminder: What to Count Often count some “basic operation” Or, we count a “critical section” Examples: –The block of code most deeply nested in a nested set of loops –An operation like comparison in sorting –An expensive operation like multiplication or database query

34. Back to Order Classes Order classes group “equivalently” efficient algorithms –O(1) – constant time! Input size doesn’t matter –O(lg n) – logarithmic time. Very efficient. E.g. binary search (after sorting) –O(n) – linear time –O(n lg n) – log-linear time. E.g. best sorting algorithms –O(n 2 ) – quadratic time. E.g. poorer sorting algorithms –O(n 3 ) – cubic time –…. –O(2 n ) – exponential time. Many important problems, often about optimization.

35. Discussion Question Binary search is faster than sequential search –But extra cost! Must sort the list first! When do you think it’s worth it?

36. Bigger Issues in Complexity Many practical problems seem to require exponential time – Θ (2 n ) –This grows much faster more quickly than any polynomial –Such problems are called intractable problems –A famous subset of these are called NP-complete problems Most famous theoretical question in CS: –Is it really impossible to find a polynomial algorithm for the NP-complete problems? (Does P=NP?)

37. Example Weighted-graph problems Find the cheapest way to connect things? –O(n 2 ) Find the shortest path between two points? All pairs of points? –O(n 2 ) or better But, find the shortest path that visits all points in the graph? –Best we can do is exponential, Θ (2 n ) or worse –Is it impossible to do better? No one knows!

38. Summary and Major Points When we measure algorithm complexity: –Base this on size of input –Count some basic operation or how often a critical section is executed –Get a formula f(n) for this –Then we think about it in terms of its “label”, the order class O( f(n) ) “Big-Oh” means as efficient as “class f(n) or better –We usually use order-class to compare algorithms –We can measure worst-case, average-case