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

2. Map

3. Map Interface signature type map type key type value map newMap (); void mapInsert (map m, key k, value v); value mapLookup (map m, key k); void mapDelete (map m, key k); … end

4. Interface in C #ifndef MAP_H #define MAP_H typedef struct map *map; // type map typedef void *poly; // type key, value map newMap (); void mapInsert (map m, poly k, poly v); poly mapLookup (map m, poly k); void mapDelete (map m, poly k); … #endif

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

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

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

8. Sample Impl ’ Using Linked List // functions void mapInsert (map m, poly k, poly v) { linkList list = m->list; linkedListInsert (list, newTuple (k, v)); return; } list m

9. Sample Impl ’ Using Linked List // functions poly mapLookup (map m, poly k) { linkList list = s->list; while(…) { // scan the list and lookup key k // return v, if found } return NULL; } list m

10. Set

11. Set Interface signature type set // set type type t // element type set newSet (); int setSize (set s); void setInsert (set s, t x); set setUnion (set s1, set s2); … end

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

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

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

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

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

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

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

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

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

21. Summary So Far set

22. Set in Java

23. Interface in Java public interface SetInter // the type “set” { int size (); // “Object” is very polymorphic… void insert (Object x); SetInter union (SetInter s); … } // Follow this, all the stuffs are essentially // same with those in C

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

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

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

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

28. 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; }

29. 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); }

30. Bit-Vector

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

32. 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

33. Implementation in C #include “bitArray.h” // a not-so efficient one struct bitArray { int *array; int size; };

34. 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; }

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

36. Bit-Array in Java

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

38. 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; }

39. Bit-Vector-based Set Representation

40. Big Picture Universe set

41. Client Code int main () { // cook a universe set set universe = newSet (); // cook sets s1 and s2 set s1 = newSet (); set s2 = newSet (); setUnion (universe, s1, s2); }

42. 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 set element? 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.

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

44. 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”}

45. 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”}

46. 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.

47. How to Store the Universe? // Method 1: stored in separate memory int main () { // cook a universe set set universe = newSet (); // cook two sets s1 and s2 set s1 = newSet (); set s2 = newSet (); setUnion (universe, s1, s2); // ugly }

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

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

50. Client Code int main () { // cook a universe set set universe = newUniverse (); // cook two sets s1 and s2 set s1 = newSet (); set s2 = newSet (); setUnion (s1, s2); // hummm, no universe AT ALL! }