proofs

Recommendations

Info

Communicating without errors Chapter 37 In 1956, Claude Shannon, the founder of information theory, posed the following very interesting question: Suppose we want to transmit messages across a channel (where some symbols may be distorted) to a receiver. What is the maximum rate of transmission such that the receiver may recover the original message without errors? Let us see what Shannon meant by “channel” and “rate of transmission.” We are given a set V of symbols, and a message is just a string of symbols from V. We model the channel as a graph G = (V, E), where V is the set of symbols, and E the set of edges between unreliable pairs of symbols, that is, symbols which may be confused during transmission. For example, communicating over a phone in everyday language, we connnect the symbols B and P by an edge since the receiver may not be able to distinguish them. Let us call G the confusion graph. The 5-cycle C 5 will play a prominent role in our discussion. In this example, 1 and 2 may be confused, but not 1 and 3, etc. Ideally we would like to use all 5 symbols for transmission, but since we want to communicate error-free we can — if we only send single symbols — use only one letter from each pair that might be confused. Thus for the 5-cycle we can use only two different letters (any two that are not connected by an edge). In the language of information theory, this means that for the 5-cycle we achieve an information rate of log 2 2 = 1 (instead of the maximal log 2 5 ≈ 2.32). It is clear that in this model, for an arbitrary graph G = (V, E), the best we can do is to transmit symbols from a largest independent set. Thus the information rate, when sending single symbols, is log 2 α(G), where α(G) is the independence number of G. Let us see whether we can increase the information rate by using larger strings in place of single symbols. Suppose we want to transmit strings of length 2. The strings u 1 u 2 and v 1 v 2 can only be confused if one of the following three cases holds: • u 1 = v 1 and u 2 can be confused with v 2 , • u 2 = v 2 and u 1 can be confused with v 1 , or • u 1 ≠ v 1 can be confused and u 2 ≠ v 2 can be confused. In graph-theoretic terms this amounts to considering the product G 1 × G 2 of two graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ). G 1 × G 2 has the vertex Claude Shannon 1 5 4 3 2
242 Communicating without errors set V 1 × V 2 = {(u 1 , u 2 ) : u 1 ∈ V 1 , u 2 ∈ V 2 }, with (u 1 , u 2 ) ≠ (v 1 , v 2 ) connected by an edge if and only if u i = v i or u i v i ∈ E i for i = 1, 2. The confusion graph for strings of length 2 is thus G 2 = G × G, the product of the confusion graph G for single symbols with itself. The information rate of strings of length 2 per symbol is then given by log 2 α(G 2 ) 2 = log 2 √ α(G2 ). Now, of course, we may use strings of any length n. The n-th confusion graph G n = G×G×. . .×G has vertex set V n = {(u 1 , . . .,u n ) : u i ∈ V } with (u 1 , . . .,u n ) ≠ (v 1 , . . . v n ) being connected by an edge if u i = v i or u i v i ∈ E for all i. The rate of information per symbol determined by strings of length n is log 2 α(G n ) n = log 2 n √ α(G n ). What can we say about α(G n )? Here is a first observation. Let U ⊆ V be a largest independent set in G, |U| = α. The α n vertices in G n of the form (u 1 , . . . , u n ), u i ∈ U for all i, clearly form an independent set in G n . Hence and therefore α(G n ) ≥ √ n α(Gn ) ≥ α(G) n α(G), meaning that we never decrease the information rate by using longer strings instead of single symbols. This, by the way, is a basic idea of coding theory: By encoding symbols into longer strings we can make error-free communication more efficient. Disregarding the logarithm we thus arrive at Shannon’s fundamental definition: The zero-error capacity of a graph G is given by n Θ(G) := sup n≥1 √ α(Gn ), and Shannon’s problem was to compute Θ(G), and in particular Θ(C 5 ). The graph C 5 × C 5 Let us look at C 5 . So far we know α(C 5 ) = 2 ≤ Θ(C 5 ). Looking at the 5-cycle as depicted earlier, or at the product C 5 × C 5 as drawn on the left, we see that the set {(1, 1), (2, 3), (3, 5), (4, 2), (5, 4)} is independent in C5 2. Thus we have α(C5) 2 ≥ 5. Since an independent set can contain only two vertices from any two consecutive rows we see that α(C5 2 ) = 5. Hence, by using strings of length 2 we have increased the lower bound for the capacity to Θ(C 5 ) ≥ √ 5. So far we have no upper bounds for the capacity. To obtain such bounds we again follow Shannon’s original ideas. First we need the dual definition of an independent set. We recall that a subset C ⊆ V is a clique if any two vertices of C are joined by an edge. Thus the vertices form trivial
Page 2:
Martin Aigner Günter M. Ziegler Pr
Page 5 and 6:
Prof. Dr. Martin Aigner FB Mathemat
Page 7 and 8:
VI Preface to the Fourth Edition Wh
Page 9 and 10:
VIII Table of Contents 21. A theore
Page 12 and 13:
Six proofs of the infinity of prime
Page 14 and 15:
Six proofs of the infinity of prime
Page 16 and 17:
Bertrand’s postulate Chapter 2 We
Page 18 and 19:
Bertrand’s postulate 9 times. Her
Page 20 and 21:
Bertrand’s postulate 11 by compar
Page 22 and 23:
Binomial coefficients are (almost)
Page 24 and 25:
Binomial coefficients are (almost)
Page 26 and 27:
Representing numbers as sums of two
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
The law of quadratic reciprocity Ch
Page 34 and 35:
The law of quadratic reciprocity 25
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Every finite division ring is a fie
Page 42 and 43:
Every finite division ring is a fie
Page 44 and 45:
Some irrational numbers Chapter 7
Page 46 and 47:
Some irrational numbers 37 and this
Page 48 and 49:
Some irrational numbers 39 F(x) may
Page 50:
Some irrational numbers 41 Now assu
Page 53 and 54:
44 Three times π 2 /6 v 1 1 2 y v
Page 55 and 56:
46 Three times π 2 /6 For m = 1, 2
Page 57 and 58:
48 Three times π 2 /6 1 f(t) = 1 t
Page 59 and 60:
50 Three times π 2 /6 (3) It has b
Page 62 and 63:
Hilbert’s third problem: decompos
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70:
Page 73 and 74:
64 Lines in the plane, and decompos
Page 75 and 76:
66 Lines in the plane, and decompos
Page 78 and 79:
The slope problem Chapter 11 Try fo
Page 80 and 81:
The slope problem 71 • Every move
Page 82:
The slope problem 73 For the second
Page 85 and 86:
76 Three applications of Euler’s
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
82 Cauchy’s rigidity theorem Q: Q
Page 93 and 94:
84 Cauchy’s rigidity theorem A be
Page 95 and 96:
86 Touching simplices l l l ′ Pr
Page 97 and 98:
88 Touching simplices For our examp
Page 99 and 100:
90 Every large point set has an obt
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
96 Borsuk’s conjecture Theorem. L
Page 107 and 108:
98 Borsuk’s conjecture On the oth
Page 109 and 110:
100 Borsuk’s conjecture To obtain
Page 112 and 113:
Sets, functions, and the continuum
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
In praise of inequalities Chapter 1
Page 130 and 131:
In praise of inequalities 121 where
Page 132 and 133:
In praise of inequalities 123 Proo
Page 134:
In praise of inequalities 125 Seco
Page 137 and 138:
128 The fundamental theorem of alge
Page 140 and 141:
One square and an odd number of tri
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
A theorem of Pólya on polynomials
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
On a lemma of Littlewood and Offord
Page 156 and 157:
On a lemma of Littlewood and Offord
Page 158 and 159:
Cotangent and the Herglotz trick Ch
Page 160 and 161:
Cotangent and the Herglotz trick 15
Page 162 and 163:
Cotangent and the Herglotz trick 15
Page 164 and 165:
Buffon’s needle problem Chapter 2
Page 166 and 167:
Buffon’s needle problem 157 The c
Page 168:
Combinatorics 25 Pigeon-hole and do
Page 171 and 172:
162 Pigeon-hole and double counting
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 182 and 183:
Tiling rectangles Chapter 26 Some m
Page 184 and 185:
Tiling rectangles 175 Third proof.
Page 186:
Tiling rectangles 177 References [1
Page 189 and 190:
180 Three famous theorems on finite
Page 191 and 192:
182 Three famous theorems on finite
Page 194 and 195:
Shuffling cards Chapter 28 How ofte
Page 196 and 197:
Shuffling cards 187 Indeed, if A i
Page 198 and 199: Shuffling cards 189 Let T be the nu
Page 200 and 201: Shuffling cards 191 Theorem 1. Let
Page 202 and 203: Shuffling cards 193 Proof. (1) We
Page 204 and 205: Lattice paths and determinants Chap
Page 206 and 207: Lattice paths and determinants 197
Page 208 and 209: Lattice paths and determinants 199
Page 210 and 211: Cayley’s formula for the number o
Page 216 and 217: Identities versus bijections Chapte
Page 218 and 219: Identities versus bijections 209 Pr
Page 220 and 221: Identities versus bijections 211 As
Page 222 and 223: Completing Latin squares Chapter 32
Page 224 and 225: Completing Latin squares 215 Proof
Page 226 and 227: Completing Latin squares 217 be dis
Page 228: Graph Theory 33 The Dinitz problem
Page 231 and 232: 222 The Dinitz problem In 1976 Vizi
Page 233 and 234: 224 The Dinitz problem X A bipartit
Page 235 and 236: 226 The Dinitz problem G : L(G) : a
Page 237 and 238: 228 Five-coloring plane graphs From
Page 239 and 240: 230 Five-coloring plane graphs is s
Page 241 and 242: 232 How to guard a museum The follo
Page 243 and 244: 234 How to guard a museum “Museum
Page 245 and 246: 236 Turán’s graph theorem Let us
Page 247 and 248: 238 Turán’s graph theorem Thus b
Page 252 and 253: Communicating without errors 243 cl
Page 254 and 255: Communicating without errors 245 Le
Page 256 and 257: Communicating without errors 247 On
Page 258 and 259: Communicating without errors 249 No
Page 260 and 261: The chromatic number of Kneser grap
Page 262 and 263: The chromatic number of Kneser grap
Page 264: The chromatic number of Kneser grap
Page 267 and 268: 258 Of friends and politicians w 1
Page 270 and 271: Probability makes counting (sometim
Page 278 and 279: Proofs from THE BOOK 269
Page 280 and 281: Index acyclic directed graph, 196 a
Page 282 and 283: Index 273 matrix-tree theorem, 203
show all

proofs

Create successful ePaper yourself

Delete template?

Save as template?