bitvector: polynomial ring over GF(2) operations
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148
bitvector.cc
148
bitvector.cc
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@ -103,7 +103,7 @@ bitvector& bitvector::operator&=(const bitvector &rhs)
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{
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if(size()<rhs.size()) resize(rhs.size(),true);
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copyonwrite();
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for(int i=0; i<nn; ++i) v[i] &= rhs.v[i];
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for(int i=0; i<nn; ++i) v[i] &= (i>=rhs.nn? 0 : rhs.v[i]);
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return *this;
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}
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@ -111,7 +111,7 @@ bitvector& bitvector::operator|=(const bitvector &rhs)
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{
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if(size()<rhs.size()) resize(rhs.size(),true);
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copyonwrite();
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for(int i=0; i<nn; ++i) v[i] |= rhs.v[i];
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for(int i=0; i<nn && i<rhs.nn; ++i) v[i] |= rhs.v[i];
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return *this;
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}
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@ -119,7 +119,7 @@ bitvector& bitvector::operator^=(const bitvector &rhs)
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{
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if(size()<rhs.size()) resize(rhs.size(),true);
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copyonwrite();
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for(int i=0; i<nn; ++i) v[i] ^= rhs.v[i];
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for(int i=0; i<nn && i<rhs.nn; ++i) v[i] ^= rhs.v[i];
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return *this;
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}
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@ -136,7 +136,7 @@ x+= (x>>16);
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return x&0x3f;
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}
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#else
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//@@@@ use an efficient trick
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//@@@@ use an efficient trick too
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static unsigned int word_popul(unsigned long x)
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{
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unsigned int s=0;
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@ -222,7 +222,7 @@ if(modulo)
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return s+word_popul(a);
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}
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unsigned int bitvector::operator%(const bitvector &y) const
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unsigned int bitvector::bitdiff(const bitvector &y) const
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{
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if(nn!=y.nn) laerror("incompatible size in bitdifference");
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@ -236,6 +236,143 @@ if(modulo)
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a &= ~mask;
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}
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return s+word_popul(a);
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}
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static unsigned int nlz64(uint64_t x0)
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{
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int64_t x=x0;
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uint64_t y;
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unsigned int n;
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n=0;
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y=x;
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L: if ( x<0) return n;
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if(y==0) return 64-n;
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++n;
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x<<=1;
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y>>=1;
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goto L;
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}
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static unsigned int ntz64(uint64_t x)
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{
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unsigned int n;
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if(x==0) return 64;
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n=1;
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if((x&0xffffffff)==0) {n+=32; x>>=32;}
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if((x&0xffff)==0) {n+=16; x>>=16;}
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if((x&0xff)==0) {n+=8; x>>=8;}
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if((x&0xf)==0) {n+=4; x>>=4;}
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if((x&0x3)==0) {n+=2; x>>=2;}
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return n-(x&1);
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}
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unsigned int bitvector::nlz() const
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{
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int leadblock=nn-1;
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unsigned int n=0;
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while(leadblock>0 && v[leadblock] == 0)
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{
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--leadblock;
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n+=blockbits;
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}
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n+= nlz64(v[leadblock]);
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if(modulo) n-= blockbits-modulo;
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return n;
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}
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unsigned int bitvector::ntz() const
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{
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int tailblock=0;
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unsigned int n=0;
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if(iszero()) return size();
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while(tailblock<nn-1 && v[tailblock] == 0)
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{
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++tailblock;
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n+=blockbits;
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}
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n+= ntz64(v[tailblock]);
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return n;
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}
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//NOTE: naive algorithm, just for testing
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//does not perform modulo irreducible polynomial, is NOT GF(2^n) multiplication
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bitvector bitvector::operator*(const bitvector &rhs) const
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{
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bitvector r(size()+rhs.size());
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r.clear();
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bitvector tmp(rhs);
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tmp.resize(size()+rhs.size(),true);
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for(int i=0; i<=degree(); ++i)
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{
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if((*this)[i]) r+= tmp;
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tmp.leftshift(1,false);
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}
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return r;
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}
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void bitvector::resize(const unsigned int n, bool preserve)
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{
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int old=size();
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NRVec<bitvector_block>::resize((n+blockbits-1)/blockbits,preserve);
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modulo=n%blockbits;
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if(preserve) //clear newly allocated memory
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{
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for(int i=old; i<nn*blockbits; ++i) this->reset(i);
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}
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else clear();
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}
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bitvector bitvector::division(const bitvector &rhs, bitvector &remainder) const
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{
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if(rhs.is_zero()) laerror("division by zero binary polynomial");
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if(is_zero() || rhs.is_one()) {remainder.clear(); return *this;}
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bitvector r(size());
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r.clear();
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remainder= *this;
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remainder.copyonwrite();
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int rhsd = rhs.degree();
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int d;
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while((d=remainder.degree()) >= rhsd)
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{
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unsigned int pos = d-rhsd;
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r.set(pos);
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remainder -= (rhs<<pos);
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}
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return r;
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}
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bitvector bitvector::gcd(const bitvector &rhs) const
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{
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bitvector big,small;
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if(degree()>=rhs.degree())
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{big= *this; small=rhs;}
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else
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{big=rhs; small= *this;}
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if(big.is_zero())
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{
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if(small.is_zero()) laerror("two zero arguments in gcd");
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return small;
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}
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if(small.is_zero()) return big;
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if(small.is_one()) return small;
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if(big.is_one()) return big;
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do {
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bitvector help=small;
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small= big%small;
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big=help;
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}
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while(! small.is_zero());
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return big;
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}
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void bitvector::read(int fd, bool dimensions, bool transp)
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@ -260,4 +397,5 @@ NRVec<bitvector_block>::put(fd,dimensions,transp);
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}//namespace
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23
bitvector.h
23
bitvector.h
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@ -25,7 +25,7 @@
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namespace LA {
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//compressed storage of large bit vectors
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//let's now use 64-bit blocks exclusively
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//let's now use 64-bit blocks exclusively for simplicity
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typedef uint64_t bitvector_block;
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@ -48,10 +48,10 @@ public:
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//operator= seems to be correctly synthetized by the compiler
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//override dereferencing to address single bits, is however possible
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//only in the const context (otherwise we would have to define a type which, when assigned to, changes a single bit - possible but probably inefficient)
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void resize(const unsigned int n, bool preserve=false) {NRVec<bitvector_block>::resize((n+blockbits-1)/blockbits,preserve); modulo=n%blockbits;};
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void resize(const unsigned int n, bool preserve=false); //preserve data or clear
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unsigned int size() const {return (nn*blockbits)-blockbits+(modulo?modulo:blockbits);};
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//arguments must be unsigned to keep the resulting assembly code simple and efficient
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const bool operator[](const unsigned int i) const {return (v[i/blockbits] >>(i%blockbits))&1UL;};
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const bool operator[](const unsigned int i) const {return (v[i/blockbits] >>(i%blockbits))&1ULL;};
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const bool get(const unsigned int i) const {return (*this)[i];};
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bitvector_block getblock(const unsigned int i) const {return v[i];}; //integer interpretation
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void setblock(const unsigned int i, const bitvector_block b) {v[i]=b;};
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@ -62,6 +62,8 @@ public:
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void clear() {copyonwrite(true); memset(v,0,nn*sizeof(bitvector_block));};
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void fill() {memset(v,0xff,nn*sizeof(bitvector_block));};
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bool iszero() const {for(int i=0; i<nn; ++i) if(v[i]) return false; return true;};
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bool is_zero() const {return iszero();};
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bool is_one() const {if(v[0]!=1) return false; for(int i=1; i<nn; ++i) if(v[i]) return false; return true;};
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void randomize();
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bool operator!=(const bitvector &rhs) const;
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bool operator==(const bitvector &rhs) const {return !(*this != rhs);};
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@ -80,15 +82,24 @@ public:
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bitvector operator^(const bitvector &rhs) const {return bitvector(*this) ^= rhs;};
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bitvector operator+(const bitvector &rhs) const {return *this ^ rhs;}; //addition modulo 2
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bitvector operator-(const bitvector &rhs) const {return *this ^ rhs;}; //subtraction modulo 2
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unsigned int operator%(const bitvector &y) const; //number of differing bits
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bitvector operator*(const bitvector &rhs) const; //multiplication of polynomials over GF(2) NOTE: naive algorithm, does not employ CLMUL nor fft-like approach, only for short vectors!!!
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bitvector division(const bitvector &rhs, bitvector &remainder) const;
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bitvector operator/(const bitvector &rhs) const {bitvector rem(rhs.size()); return division(rhs,rem);};
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bitvector operator%(const bitvector &rhs) const {bitvector rem(rhs.size()); division(rhs,rem); return rem;};
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bitvector gcd(const bitvector &rhs) const;
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bitvector lcm(const bitvector &rhs) const {return (*this)*rhs/this->gcd(rhs);};
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unsigned int bitdiff(const bitvector &y) const; //number of differing bits
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unsigned int population(const unsigned int before=0) const; //number of 1's
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unsigned int nlz() const; //number of leading zeroes
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unsigned int degree() const {if(iszero()) return 0; else return size()-nlz()-1;}; //interprested as a polynomial over GF(2)
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unsigned int ntz() const; //number of trailing zeroes
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//extended, truncated const i.e. not on *this but return new entity, take care of modulo's bits
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//logical shifts
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bitvector& operator>>=(unsigned int i);
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bitvector& leftshift(unsigned int i, bool autoresize=false);
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bitvector& operator<<=(unsigned int i) {return leftshift(i,true);};
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bitvector operator>>(unsigned int i) {bitvector r(*this); return r>>=i;};
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bitvector operator<<(unsigned int i) {bitvector r(*this); return r<<=i;};
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bitvector operator>>(unsigned int i) const {bitvector r(*this); return r>>=i;};
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bitvector operator<<(unsigned int i) const {bitvector r(*this); return r<<=i;};
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//logical rotations not implemented yet
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//unformatted file IO
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void read(int fd, bool dimensions=1, bool transp=0);
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@ -69,7 +69,7 @@ public:
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}
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Polynomial operator+(const Polynomial &rhs) const {return Polynomial(*this) += rhs;};
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Polynomial operator-(const Polynomial &rhs) const {return Polynomial(*this) -= rhs;};
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Polynomial operator*(const Polynomial &rhs) const //for very long polynomials FFT should be used
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Polynomial operator*(const Polynomial &rhs) const //NOTE: naive implementation! For very long polynomials FFT-based methods should be used
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{
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NOT_GPU(*this);
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Polynomial r(degree()+rhs.degree());
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74
t.cc
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t.cc
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@ -2852,7 +2852,7 @@ cout <<endl<<"Inverse via svd\n"<<ainv2<<endl;
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cout <<"Difference of inverses = "<<(ainv-ainv2).norm()<<endl;
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}
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if(1)
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if(0)
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{
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int seed;
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int f=open("/dev/random",O_RDONLY);
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@ -2864,17 +2864,79 @@ int n;
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cin >>n;
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bitvector v(n);
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v.randomize();
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//do{
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// cout <<v <<endl;
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// v>>=1;
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//}while(!v.iszero());
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do{
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cout <<v << " NLZ "<<v.nlz()<<" DEG "<<v.degree()<<" NTZ "<<v.ntz()<<endl;
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v>>=1;
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}while(!v.iszero());
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cout <<v << " NLZ "<<v.nlz()<< " DEG "<<v.degree()<<" NTZ "<<v.ntz()<<endl;
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v.randomize();
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for(int i=0; i<n; ++i)
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{
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cout <<v <<endl;
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cout <<v << " size "<<v.size()<<" NLZ "<<v.nlz()<< " DEG "<<v.degree()<<" NTZ "<<v.ntz()<<endl;
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v<<=1;
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}
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cout <<v << " size "<<v.size()<<" NLZ "<<v.nlz()<< " DEG "<<v.degree()<<" NTZ "<<v.ntz()<<endl;
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v.randomize();
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for(int i=0; i<v.size(); ++i) cout <<(v<<i)<<endl;
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}
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if(1)
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{
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int seed;
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int f=open("/dev/random",O_RDONLY);
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if(sizeof(int)!=read(f,&seed,sizeof(int))) laerror("cannot read /dev/random");
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close(f);
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srand(seed);
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int n;
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cin >>n;
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bitvector v(n),u(n);
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u.randomize();
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v.randomize();
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bitvector w=u*v;
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bitvector z=v*u;
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cout <<u<<endl;
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cout <<v<<endl;
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cout <<w<<endl;
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if(w!=z) laerror("error in bitvector multiplication");
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if(w.degree()!=v.degree()+u.degree()) laerror("error in degree or multiplication");
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cout <<w/u-v<<endl;
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cout <<w%u<<endl;
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cout <<w/v-u<<endl;
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cout <<w%v<<endl;
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if(!(w/u-v).is_zero()) laerror("error in division");
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if(!(w/v-u).is_zero()) laerror("error in division");
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if(!(w%u).is_zero()) laerror("error in division");
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if(!(w%v).is_zero()) laerror("error in division");
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cout <<w.gcd(u)-u<<endl;
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cout <<w.gcd(v)-v<<endl;
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if(!(w.gcd(u)-u).is_zero()) laerror("error in gcd");
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if(!(w.gcd(v)-v).is_zero()) laerror("error in gcd");
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u.randomize();
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v.randomize();
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bitvector g=u.gcd(v);
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bitvector l=u.lcm(v);
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cout <<u<<endl;
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cout <<v<<endl;
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cout <<g<<endl;
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cout <<l<<endl;
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cout <<l/u - v/g<<endl;
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cout <<l/v - u/g<<endl;
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cout <<l%u<<endl;
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cout <<l%v<<endl;
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cout <<u%g<<endl;
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cout <<v%g<<endl;
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if(!(l/u - v/g).is_zero()) laerror("error in gcd");
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if(!(l/v - u/g).is_zero()) laerror("error in gcd");
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if(!(l%u).is_zero()) laerror("error in gcd");
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if(!(l%v).is_zero()) laerror("error in gcd");
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if(!(u%g).is_zero()) laerror("error in gcd");
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if(!(v%g).is_zero()) laerror("error in gcd");
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}
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}
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