Discrete Structures & Algorithms More about sets

1. Discrete Structures & AlgorithmsMore about sets EECE 320 — UBC

2. Reminder: How to specify sets? • Explicit enumeration: {John, Paul, George, Ringo} • Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…} • Using predicates: { x : P(x) is true }, • where P(x) is some predicate that is true for all elements of the set • Example: { x : x is prime } • Recursive definitions • Example: The set of integers divisible by 3 can be specified as. • Basis step: 3 S3 • Recursive step: If x, y S3 then x+y S3

3. Recursive set definitions Assume S is a an alphabet . (for example S={0,1}) What set does this definition specifies? • Basis step:  S* (where  is the notation for the null string) • Recursive step: If w S* and x S then wx  S*

4. Cardinality of sets If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S. If S = {1,2,3}, |S| = 3. If S = {3,3,3,3,3}, |S| = 1. If S = , |S| = 0. |S| = 3. If S = { , {}, {,{}} }, If S = {0,1,2,3,…}, |S| is infinite. (more on this later)

5. ECE320 ECE315 100 + 120 – 50 = 170 Inclusion/Exclusion Example: There are 250 ECE students. 100 are taking ECE 320. 120 are taking ECE 315. 50 are taking both. How many people are taking ECE320 or ECE315? |A  B| = |A| + |B| - |A  B|

6. ECE320 ECE315 250 - (100+120-50) = 80 Inclusion/Exclusion Example: There are 250 ECE students. 100 are taking ECE 320. 150 are taking ECE 315. 50 are taking both. How many are taking neither ECE320 nor ECE315?

7. Generalized Inclusion/Exclusion Suppose we have: B A C And we want to know |A U B U C| |A U B U C| = |A| + |B| + |C| We have counted these intersections twice here - |A  B| - |A  C| - |B  C| + |A  B  C| Therefore, add it back once to ensure it is correctly counted. This intersection has been counted thrice here But has been subtracted thrice here!

8. Wrap up • The principle of inclusion/exclusion allows us to compose sets for smaller sets and to reason about the size of the new set. • Solve examples in the textbook to become comfortable with inclusion/exclusion.

9. Set representations

10. (101011) Sets as bit strings Let U = {x1, x2,…, xn}, and let A  U. Then the characteristic vector of A is the n-vector whose elements, xi, are 1 if xi A, and 0 otherwise. Example: If U = {x1, x2, x3, x4, x5, x6}, and A = {x1, x3, x5, x6}, then the characteristic vector of A is

11. Bit-wise OR Bit-wise AND Sets as bit strings Example: If U = {x1, x2, x3, x4, x5, x6}, A = {x1, x3, x5, x6}, and B = {x2, x3, x6}, then we have a quick way of finding the characteristic vectors of A  B and A  B.

12. Representing sets • Are bit strings always sufficient? • What assumptions do we make when we adopt the bit string representation? • That we know the universe of possible elements. • We also need to keep a list of all possible elements. • If we are storing a set of names using a bit string, we may need to also store information such as “Smith is bit 0, Rogers is bit 2, …”

13. Representing sets • Alternative representations of sets? • Arrays • Linked lists • … • What is the primary operation to be supported? • Set membership • And, how do we implement such an operation? Search.

14. Sets as arrays Smith Rogers Paul Brown … Campbell • What is the primary operation? • Set membership. • How do we test if “Scott” is in the set? • Elements are in arbitrary positions. • What is the maximum number of comparisons needed? • If n is the cardinality (size) of the set? • Maximum comparisons: n • What is the minimum number of comparisons needed? • What is the average number of comparisons needed? • n/2. Why?

15. Complexity of search Smith Rogers Paul Brown … Campbell • If the set is represented as an unordered list, we have to search through all elements. • Let f(n) be the time required to search for an element in a list of size n. • Let t be the time taken per comparison. • Then: f(n) (t)(n), and this is expressed as f(n)=O(n), or the time complexity of sequential search is O(n). • More about the big-O notation in another lecture.

16. Binary search What if the array or the list were kept ordered? Suppose we were searching for “Brown”. Brown Campbell Dali Rogers … Scott • Compare with the middle element of the array. • If “Brown” is smaller (lexicographically) then we need to search only in the left half of the list, else we need to search only in the right half of the list. • We then proceed, iteratively, to make a comparison with the middle element of the search region.

17. How many comparisons? (binary search) • For simplicity, assume N, the size of the array, is 2k. • After each comparison, we reduce the size of our search to a smaller array that is half the size of the initial array. • Therefore: the maximum number of steps needed is … • At worst, we need to bring the size of the search region to 1. • How many steps would that take? • Diminishing array sizes: 2k, 2k-1, …, 20 • k+1 = log2N+1. Brown Campbell Dali Rogers … Scott

18. How many comparisons? (binary search) • If g(n) denotes the time required to search for an element using binary search in an array of size n, then • g(n)  (log2n+1)t, • where t is the time needed per comparison/iteration. • g(n) = O(log2n), is what we say. Brown Campbell Dali Rogers … Scott

19. What do you need to know? (as good engineers) • To understand how to perform computations efficiently, we need to know • How to count! • How to sum a sequence of steps. • Reason about the complexity/efficiency of an algorithm. • Prove the correctness of our analysis. • Logic • Methods of inference • Proof strategies • Induction • … • (and eventually, how to create better algorithms.)