internet engineering czesław smutnicki discrete mathematics – combinatorics
TRANSCRIPT
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Internet EngineeringInternet Engineering
Czesław SmutnickiCzesław Smutnicki
Discrete Mathematics Discrete Mathematics – – CombinatorCombinatoricsics
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CONTENTS
• Functions and distributions• Combinatorial objects• K-subsets• Subsets• Sequences• Set partitions• Number partitions• Stirling numbers• Bell numbers• Permutations • Set on/off rule
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FUNCTIONS AND DISTRIBUTIONS
X Y
X elements, Y boxes
Element can be packed to any box: n-length sequence Each box contains at most one element: set partitionBox contains exactly one element: permutation
mYnXYXf ,,:
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K-SUBSETS
Generate all subsets with k-elements from the set of n elements
k
n
123412351236124512461256134513461356145623452346235624563456
ss CCCC ,,...,, 121
1,1,...,1,1,0,0,...,0,0 121121 CCCCCCCC ssss
)1,0,...,1,0,0,1(iC
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SUBSETS
0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001
10 101011 101112 110013 110114 111015 1111
ss CCCC ,,...,, 121
1,1,...,1,1,0,0,...,0,0 121121 CCCCCCCC ssss
)1,0,...,1,0,0,1(iC
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SET PARTITION
(1)
(1,2) (1)(2)
(1,2,3) (1,2)(3) (1,3)(2) (1)(2,3) (1)(2)(3)
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NUMBER PARTITION
76 15 25 1 14 34 2 14 1 1 13 3 13 2 1 13 1 1 1 12 2 2 12 2 1 1 12 1 1 1 1 11 1 1 1 1 1 1
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SECOND TYPE STIRLING NUMBERS
0,0)0,(
0,1),(
0,),1()1,1(),(
nnS
nnnS
nkknkSknSknS
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BELL NUMBERS
nkn knSB 0 ),(
n Bn
0 1
1 1
2 2
3 5
4 15
5 52
6 203
7 877
8 4140
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PERMUTATIONS. INTRODUCTION
• Permutation• Inverse permutation• Id permutation• Composition• Inversions• Cycles• Sign
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4
)1( nn
72
)52)(1( nnn
)( 2nO )(nO )log( nnO
)2(nn HH
nHn
)( 3
1
n
nn 2
complexity
variance
mean
receiptnumber of inversionin -1 o
n minus the numberof cycles in -1 o
n minus the lenght of the maximal increasing subsequence in -1 o
measure DA (, ) DS (, ) DI (, )
Move type API NPI INS
PERMUTATIONS. DISTANCE MEASURES
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GENERATING PERMUTATIONS. IN ANTYLEXICOGRAPHICAL ORDER
void swap(int& a, int& b) { int c=a; a=b; b=c; }void reverse(int m) { int i=1,j=m; while (i<j) swap(pi[i++],pi[j--]); }
void antylex(int m){ int i;
if (m==1){ for (i=1;i<=m;i++) cout << pi[i] << ' '; cout << endl; }
elsefor (i=1;i<=m;i++)
{ antylex(m-1); if (i<m) { swap(pi[i],pi[m]); reverse(m-1); }}}
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GENERATING PERMUTATIONS. MINIMAL NUMBER OF TRANSPOSITIONS
void swap(int& a, int& b) { int c=a; a=b; b=c; }
int B(int m, int i) { return (!(m%2)&&(m>2))?(i<(m-1)?i:m-2):m-1; }
void perm(int m){ int i;
if (m==1) { for (i=1;i<=n;i++) cout << pi[i] << ' '; cout << endl; }else for (i=1;i<=m;i++) { perm(m-1); if (i<m) swap(pi[B(m,i)],pi[m]); }
}
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GENERATING PERMUTATIONS. MINIMAL NUMBER OF ADJACENT SWAPS
void swap(int& a, int& b) { int c=a; a=b; b=c; }
void permtp(int m){ int i,j,x,k;
int *c=new int[m+1],*pr=new int[m+1];
for (i=1;i<m;i++) { pi[i]=i; c[i]=1; pr[i]=1; }c[m]=0; for (j=1;j<=n;j++) cout << pi[j] << ' '; cout << endl;i=1;while (i<m){ i=1; x=0;
while (c[i]==(m-i+1)) { pr[i]=!pr[i]; c[i]=1; if (pr[i]) x++; i++; }if (i<m){ k=pr[i]?c[i]+x:m-i+1-c[i]+x; swap(pi[k],pi[k+1]); c[i]++; for (j=1;j<=n;j++) cout << pi[j] << ' '; cout << endl;}
}delete[] c; delete[] pr;
}
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Thank you for your attention
DISCRETE MATHEMATICSCzesław Smutnicki