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Set & Bit-Vector. Discrete Mathematics and Its Applications Baojian Hua [email protected]. Set. 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. - PowerPoint PPT Presentation
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Set
Set Interfacesignature
type set
type X
set newSet ();
int setSize (set s);
void setInsert (set s, X x);
set setUnion (set s1, set s2);
…
end
Interface in C#ifndef SET_H#define SET_H typedef struct set *set; // type settypedef void *poly; // type X
set newSet ();int setSize (set s);void setInsert (set s, poly x);set setUnion (set s1, set s2);…
#endif
Implementation in C#include “set.h” struct set{ // your favorite concrete representation};
set newSet (){ // real code goes here}
…
Sample Impl’ Using Linked List#include “linkedList.h”#include “set.h” struct set{ linkedList list;};
Sample Impl’ Using Linked List// functions set newSet (){ set s = malloc (sizeof (*s)); s->list = newLinkedList (); return s;}
lists
Sample Impl’ Using Linked List// functions int setSize (set s){ linkList l = s->list; int size = linkedListSize (l); return size;}
lists
Sample Impl’ Using Linked List// functions void setInsert (set s, poly x){ if (setExists (s, x)) return;
linkedListInsert (s->list, x); return;}
Sample Impl’ Using Linked List// functions int setExist (set s, poly x){ return linkedListExists (s->list, x);}
Equality TestingHow 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.
Client Codeint main (){ set s1 = newSet (); set s2 = newSet (); for (…) setInsert (s1, …);
for (…) setInsert (s1, …); set s3 = setUnion (s1, s2);}
Summary So Far
set
set
set
set
set
Set in Java
Interface in Javapublic interface SetInter // the type “set”
{
int size ();
// “Object” is the most polymorphic type
void insert (Object x);
SetInter union (SetInter s);
…
}
Or Using Generic// Type “set”, with type argument “X”
public interface SetInter<X>
{
int size ();
void insert (X x);
SetInter<X> union (SetInter<X> s);
…
}
// We’ll use this strategy in following slides
Implementation in Javapublic class Set<X> implements SetInter<X>{ // any concrete internal representation
Set () // constructor { // code goes here }
int size () { // code goes here } …}
Sample Impl’ Using Linked Listimport ….linkedList;
public class Set<X> implements SetInter<X>{ private linkedList<X> list;
Set () { this.list = new LinkedList<X> (); }
}
Sample Impl’ Using Linked Listimport ….linkedList;
public class Set<X> implements SetInter<X>{ private linkedList<X> list;
int size () { return this.list.size (); }
}
Sample Impl’ Using Linked Listimport ….linkedList;
public class Set<X> implements SetInter<X>{ private linkedList<X> list;
void insert (X x) { if (exists (x)) // equality testing? return; this.list.insert (x); return; }}
Client Codeimport ….Set;
public class Main{ public static void main (String[] args) { SetInter<String> s1 = new Set<String> (); SetInter<String> s2 = new Set<String> ();
s1.size (); s1.union (s2); }}
Bit-Vector
Bit-Vector Interfaceinterface type bitArray bitArray newBitArray (int size); void assignOne (bitArray ba, int index); bitArray and (bitArray ba1, bitArray ba2); …end
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
Implementation in C#include “bitArray.h”
struct bitArray{ int *array; int size;};
OperationsbitArray 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;}
OperationsbitArray and (bitArray ba1, bitArray ba2){ if (ba1->size != ba2->size) error (…);
bitArray ba = newBitArray (); for (…) assignOne (ba, …);
return ba;}
Bit-Array in Java
In Java// I omit the interface for simplicitypublic class BitArray{ private int[] array;
BitArray (int size) { this.array = new int[size]; }}
Other Methodspublic 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;
}}
Re-implement Set using Bit-Vector
Big Picture
Universe
set
setset
set
Client Codeint main ()
{
set universe = newSet ();
// insert elements into universe
set s1 = newSet(); set s2 = newSet ();
// insert elements into s1 and s2
setUnion (universe, s1, s2);
}
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.
Big Picture
1. Build the bit-array from the universe
{(“a”, 0), (“b”, 3), (“c”, 1”), (“d”, 2)}
{“a”} {“d”}
Big Picture
1. Build the bit-array from the universebaSet1 = [0, 0, 0, 0]baSet2 = [0, 0, 0, 0]
{(“a”, 0), (“b”, 3), (“c”, 1”), (“d”, 2)}
{“a”} {“d”}
Big Picture
1. Build the bit-array from the universebaSet1 = [0, 0, 0, 0]baSet2 = [0, 0, 0, 0]2. Build bit-array from setbaSet1 = [1, 0, 0, 0]baSet2 = [0, 0, 1, 0]
{(“a”, 0), (“b”, 3), (“c”, 1”), (“d”, 2)}
{“a”} {“d”}
Big Picture
1. Build the bit-array from theuniversebaSet1 = [0, 0, 0, 0]baSet2 = [0, 0, 0, 0]2. Build bit-array from setbaSet1 = [1, 0, 0, 0]baSet2 = [0, 0, 1, 0]
{(“a”, 0), (“b”, 3), (“c”, 1”), (“d”, 2)}
{“a”} {“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.
How to Store the Universe?// Method 1: stored in separate spaceint 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}
How to Store the Universe?// Method 2: shared
Universe
set
setset
set
How to Make Things Shared?
In C: extern variables In C++ or Java: static fields What’s the pros and cons?
Client Codeint 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!}
