1. Set & Bit-Vector Discrete Mathematics and Its Applications Baojian Hua bjhua@ustc.edu.cn

2. Set

3. Set Interface signature type set type X set newSet (); int setSize (set s); void setInsert (set s, X x); set setUnion (set s1, set s2); … end

4. Interface in C #ifndef SET_H #define SET_H typedef struct set *set; // type set typedef void *poly; // type X set newSet (); int setSize (set s); void setInsert (set s, poly x); set setUnion (set s1, set s2); … #endif

5. Implementation in C #include “set.h” struct set { // your favorite concrete representation }; set newSet () { // real code goes here } …

6. Sample Impl ’ Using Linked List #include “linkedList.h” #include “set.h” struct set { linkedList list; };

7. Sample Impl ’ Using Linked List // functions set newSet () { set s = malloc (sizeof (*s)); s->list = newLinkedList (); return s; } list s

8. Sample Impl ’ Using Linked List // functions int setSize (set s) { linkList l = s->list; int size = linkedListSize (l); return size; } list s

9. Sample Impl ’ Using Linked List // functions void setInsert (set s, poly x) { if (setExists (s, x)) return; linkedListInsert (s->list, x); return; }

10. Sample Impl ’ Using Linked List // functions int setExist (set s, poly x) { return linkedListExists (s->list, x); }

11. Equality Testing How to do equality testing on “polymorphic” data? 1. Extra “equals” function pointer as argument. int linkedListExist (linkedList l, poly x, tyEq equals); 2. Extra “equals” function pointers in data. int linkedListExist (linkedList l, poly x) { for (…p…) (p->data)->equals (p->data, x); } // As we can see next in Java.

12. Client Code int main () { set s1 = newSet (); set s2 = newSet (); for (…) setInsert (s1, …); for (…) setInsert (s1, …); set s3 = setUnion (s1, s2); }

13. Summary So Far set

14. Set in Java

15. Interface in Java public interface SetInter // the type “set” { int size (); // “Object” is the most polymorphic type void insert (Object x); SetInter union (SetInter s); … }

16. Or Using Generic // Type “set”, with type argument “X” public interface SetInter { int size (); void insert (X x); SetInter union (SetInter s); … } // We’ll use this strategy in following slides

17. Implementation in Java public class Set implements SetInter { // any concrete internal representation Set () // constructor { // code goes here } int size () { // code goes here } … }

18. Sample Impl ’ Using Linked List import ….linkedList; public class Set implements SetInter { private linkedList list; Set () { this.list = new LinkedList (); }

19. Sample Impl ’ Using Linked List import ….linkedList; public class Set implements SetInter { private linkedList list; int size () { return this.list.size (); }

20. Sample Impl ’ Using Linked List import ….linkedList; public class Set implements SetInter { private linkedList list; void insert (X x) { if (exists (x)) // equality testing? return; this.list.insert (x); return; }

21. Client Code import ….Set; public class Main { public static void main (String[] args) { SetInter s1 = new Set (); SetInter s2 = new Set (); s1.size (); s1.union (s2); }

22. Bit-Vector

23. Bit-Vector Interface interface type bitArray bitArray newBitArray (int size); void assignOne (bitArray ba, int index); bitArray and (bitArray ba1, bitArray ba2); … end

24. Interface in C #ifndef BITARRAY_H #define BITARRAY_H typedef struct bitArray *bitArray; bitArray newBitArray (int size); void assignOne (bitArray ba, int index); bitArray and (bitArray ba1, bitArray ba2); … #endif

25. Implementation in C #include “bitArray.h” struct bitArray { int *array; int size; };

26. Operations bitArray newBitArray (int i) { bitArray ba = malloc (sizeof (*ba)); ba->array = malloc (sizeof (*(ba->array)) * i); for (int k=0; k<i; k++) (ba->array)[i] = 0; ba->size = i; return ba; }

27. Operations bitArray and (bitArray ba1, bitArray ba2) { if (ba1->size != ba2->size) error (…); bitArray ba = newBitArray (); for (…) assignOne (ba, …); return ba; }

28. Bit-Array in Java

29. In Java // I omit the interface for simplicity public class BitArray { private int[] array; BitArray (int size) { this.array = new int[size]; }

30. Other Methods public class BitArray { private int[] array; BitArray and (BitArray ba2) { if (this.size () != ba2.size ()) throw new Error (…); BitArray ba = new BitArray (this.size()); … return ba; }

31. Re-implement Set using Bit-Vector

32. Big Picture Universe set

33. Client Code int main () { set universe = newSet (); // insert elements into universe set s1 = newSet(); set s2 = newSet (); // insert elements into s1 and s2 setUnion (universe, s1, s2); }

34. What does the Universe Look Like? Universe is a set of (element, index) tuple. For instance: Universe = {(“a”, 0), (“b”, 3), (“c”, 1”), (“d”, 2)} Question: How to build such kind of universe from the input strings? Answer: associate every set element e a unique (and continuous) integer i (i will be used as an index in the bit-vector. Details leave to you.

35. Big Picture 1. Build the bit-array from the universe {( “ a ”, 0), ( “ b ”, 3), ( “ c ”, 1 ” ), ( “ d ”, 2)} {“a”}{“a”}{“d”}{“d”}

36. Big Picture 1. Build the bit-array from the universe baSet1 = [0, 0, 0, 0] baSet2 = [0, 0, 0, 0] {( “ a ”, 0), ( “ b ”, 3), ( “ c ”, 1 ” ), ( “ d ”, 2)} {“a”}{“a”}{“d”}{“d”}

37. Big Picture 1. Build the bit-array from the universe baSet1 = [0, 0, 0, 0] baSet2 = [0, 0, 0, 0] 2. Build bit-array from set baSet1 = [1, 0, 0, 0] baSet2 = [0, 0, 1, 0] {( “ a ”, 0), ( “ b ”, 3), ( “ c ”, 1 ” ), ( “ d ”, 2)} {“a”}{“a”}{“d”}{“d”}

38. Big Picture 1. Build the bit-array from the universe baSet1 = [0, 0, 0, 0] baSet2 = [0, 0, 0, 0] 2. Build bit-array from set baSet1 = [1, 0, 0, 0] baSet2 = [0, 0, 1, 0] {( “ a ”, 0), ( “ b ”, 3), ( “ c ”, 1 ” ), ( “ d ”, 2)} {“a”}{“a”}{“d”}{“d”} 3. Bit-vector operations baSet3 = or (baSet1, baSet2) baSet3 = [1, 0, 1, 0] 4. Turn baSet3 to ordinary set How? Leave it to you.

39. How to Store the Universe? // Method 1: stored in separate space int main () { set universe = newSet (); // insert elements into universe set s1 = newSet(); set s2 = newSet (); // insert elements into s1 and s2 setUnion (universe, s1, s2); // ugly }

40. How to Store the Universe? // Method 2: shared Universe set

41. How to Make Things Shared? In C: extern variables In C++ or Java: static fields What ’ s the pros and cons?

42. Client Code int main () { set universe = newUniverse (); // insert elements into universe set s1 = newSet(); set s2 = newSet (); // insert elements into s1 and s2 setUnion (s1, s2); // hummm, no universe AT ALL! }