Concrete mathematics : a foundation for computer science

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408 DISCRETE PROBABILITY Complicated, yes; but everything simplifies greatly when we set z = 1. For example, we have t n30 Wk B;(l)y zz m,+m ‘)y k>2 / 2(k - 2)! and it follows that = me{” ‘)w wkf2 - - lx k>O 2k! ,,z,im llw 1Y E--e = 2 ( mw)n+2 n(n-l)m”wn 2mn! = IL 2mn! ’ x TX30 II30 The expression for EA in (8.!3g) now gives EA = 1 + (n- 1)/2m, in agreement with (8.96). The formula for Bz (1) involves the similar sum 5 (;) (G>-,) g = f & (k+ ‘)k(k;;)(-“ / , hence we find that = mewm(+mw4 + w”) ; B;(l) = (;:)((1) -l>s* (8.101) Now we can put all the pieces together and evaluate the desired variance VA. Massive cancellation occurs, and the result is surprisingly simple: B”(1) + B;(l) - B;(1)2 VA+L n2 =-- (m- l)i:n- 1) =- 2mln (n+l)(n-2) +-m-n(n-1) 4 2 4 (8.102)
8.5 HASHING 409 When such “coincidences” occur, we suspect that there’s a mathematical reason; there might be another way to attack the problem, explaining why the answer has such a simple form. And indeed, there is another approach (in exercise 60), which shows that the variance of the average successful search has the general form VA = k=l (8.103) when sk is the probability that the kth-inserted element is being sought. Equation (8.102) is the special case sk = l/n for 1 < k 6 n. Besides the variance of the average, we might also consider the average of the variance. In other words, each sequence (hl , . . . , hn) that defines a hash table also defines a probability distribution for successful searching, and the variance of this probability distribution tells how spread out the number of probes will be in different successful searches. For example, let’s go back to Where have 1 seen the case where we inserted n = 16 things into m = 10 lists: that pattern before? Where have 1 seen h , . . . ,h16)=3 141592653589793 that grafito before? IqvP, . Pl, * *.,P,6)=1112111122312133 A successful search in the resulting hash table has the pgf G(3,1,4,1,... ,3) = f SkZP(3,1,4,1,...,3;k) k=l = SlZ+S2Z+S3Z+S4Z2+...+S~~Z3. We have just considered the average number of probes in a successful search of this table, namely A(3,1,4,1,. . . ,3) = Mean(G(3,1,4,1,. . . ,3)). We can also consider the variance, This variance is a random variable, depending on (hl , . . . , h,), so it is natural to consider its average value. In other words, there are three natural kinds of variance that we may wish to know, in order to understand the behavior of a successful search: The overuZZ variance of the number of probes, taken over all (h1,. , . , h,,) and k; the variance of the average number of probes, where the average is taken over all k and the variance is then taken over all (h, , . . . , h,,); and the average of the variance of the number of the probes, where the variance is taken over
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CONCRETE MATHEMATICS Dedicated to L
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CONCRETE MATHEMATICS Ronald L. Grah
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“A odience, level, and treatment
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‘I a concrete life preserver thro
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I’m unaccustomed to this face. De
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n [ n-l 1 n {I m Prestressed concre
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CONTENTS xiii 5.3 Tricks of the Tra
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2 RECURRENT PROBLEMS It’s not imm
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4 RECURRENT PROBLEMS Of course the
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6 RECURRENT PROBLEMS through any of
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8 RECURRENT PROBLEMS From these sma
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10 RECURRENT PROBLEMS But what abou
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12 RECURRENT PROBLEMS that is, in t
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14 RECURRENT PROBLEMS by separating
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16 RECURRENT PROBLEMS For example,
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18 RECURRENT PROBLEMS Homework exer
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20 RECURRENT PROBLEMS 21 Suppose th
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22 SUMS The three-dots notation has
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24 SUMS A slightly modified form of
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26 SUMS where A(n), B(n), and C(n)
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28 SUMS Let’s apply these ideas t
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30 SUMS 2.3 MANIPULATION OF SUMS No
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32 SUMS Typically we use rule (2.20
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34 SUMS because the derivative of a
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36 SUMS instead of summing over all
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38 SUMS The second sum here is zero
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40 SUMS The normal way to evaluate
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42 SUMS First, as usual, we look at
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44 SUMS order to get an equation fo
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46 SUMS One way to use this fact is
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48 SUMS definition where the factor
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50 SUMS Let’s try to recap this;
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52 SUMS we need an appropriate defi
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54 SUMS This formula indicates why
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56 SUMS u(x) = x and Av(x) = 2’;
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58 SUMS The definition in the previ
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60 SUMS In fact, our definition of
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62 SUMS tCj,kiEF a. J, k < A for al
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64 SUMS 18 Let 9%~ and Jz be the re
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66 SUMS 34 35 36 Prove that if the
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68 INTEGER FUNCTIONS below the line
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70 INTEGER FUNCTIONS same inequalit
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72 INTEGER FUNCTIONS An important s
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74 INTEGER FUNCTIONS By the way, th
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76 INTEGER FUNCTIONS (Previously K
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78 INTEGER FUNCTIONS This derivatio
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80 INTEGER FUNCTIONS We’ve got mo
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82 INTEGER FUNCTIONS division, and
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84 INTEGER FUNCTIONS more than one
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86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
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88 INTEGER FUNCTIONS For this purpo
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90 INTEGER FUNCTIONS it comes out i
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92 INTEGER FUNCTIONS To summarize,
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94 INTEGER FUNCTIONS Also, as we gu
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96 INTEGER FUNCTIONS Basics 10 Show
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98 INTEGER FUNCTIONS 31 Prove or di
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100 INTEGER FUNCTIONS 43 Find an in
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4 Number Theory INTEGERS ARE CENTRA
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104 NUMBER THEORY The left side can
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106 NUMBER THEORY product of primes
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108 NUMBER THEORY only finitely man
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110 NUMBER THEORY But 2P - 1 isn’
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112 NUMBER THEORY arrangements in a
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114 NUMBER THEORY The binary repres
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116 NUMBER THEORY A fraction m/n is
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118 NUMBER THEORY A mediant fractio
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120 NUMBER THEORY This representati
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122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
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126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
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130 NUMBER THEORY Now we must show
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132 NUMBER THEORY And it’s possib
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134 NUMBER THEORY For example, (~(1
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136 NUMBER THEORY Conversely, if f(
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138 NUMBER THEORY (We must add 1 to
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140 NUMBER THEORY The problem of co
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142 NUMBER THEORY n is odd, n4 and
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144 NUMBER THEORY But the inner sum
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146 NUMBER THEORY 22 The number 111
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148 NUMBER THEORY 40 If the radix p
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150 NUMBER THEORY 57 Let S(m,n) be
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152 NUMBER THEORY 73 If the 0(n) +
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154 BINOMIAL COEFFICIENTS For examp
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156 BINOMIAL COEFFICIENTS And now t
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158 BINOMIAL COEFFICIENTS property
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160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
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164 BINOMIAL COEFFICIENTS Table 164
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166 BINOMIAL COEFFICIENTS not rows.
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168 BINOMIAL COEFFICIENTS Equation
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170 BINOMIAL COEFFICIENTS that m =
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172 BINOMIAL COEFFICIENTS ifweuse (
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176 BINOMIAL COEFFICIENTS sum on th
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178 BINOMIAL COEFFICIENTS some data
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180 BINOMIAL COEFFICIENTS Problem 4
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182 BINOMIAL COEFFICIENTS We’re l
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184 BINOMIAL COEFFICIENTS We can’
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186 BINOMIAL COEFFICIENTS 5.3 TRICK
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188 BINOMIAL COEFFICIENTS if we app
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190 BINOMIAL COEFFICIENTS The Newto
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192 BINOMIAL COEFFICIENTS One examp
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194 BINOMIAL COEFFICIENTS We can de
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196 BINOMIAL COEFFICIENTS This obse
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198 BINOMIAL COEFFICIENTS Generatin
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200 BINOMIAL COEFFICIENTS can be pu
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204 BINOMIAL COEFFICIENTS This hold
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206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
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210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
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214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
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218 BINOMIAL COEFFICIENTS It’s al
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220 BINOMIAL COEFFICIENTS A similar
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222 BINOMIAL COEFFICIENTS One use o
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224 BINOMIAL COEFFICIENTS Therefore
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226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
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230 BINOMIAL COEFFICIENTS valid for
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232 BINOMIAL COEFFICIENTS 18 Find a
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234 BINOMIAL COEFFICIENTS 37 Show t
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236 BINOMIAL COEFFICIENTS 57 Use Go
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238 BINOMIAL COEFFICIENTS 72 Prove
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240 BINOMIAL COEFFICIENTS 86 Let al
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242 BINOMIAL COEFFICIENTS Research
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244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
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248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
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252 SPECIAL NUMBERS We can remember
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254 SPECIAL NUMBERS Table 254 Euler
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{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
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258 SPECIAL NUMBERS Table 258 Stirl
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260 SPECIAL NUMBERS ’ ’ ’ (Th
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262 SPECIAL NUMBERS we found that a
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264 SPECIAL NUMBERS Rearranging, we
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266 SPECIAL NUMBERS Thus we have th
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268 SPECIAL NUMBERS (We have used t
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270 SPECIAL NUMBERS Bernoulli numbe
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272 SPECIAL NUMBERS trigonometric f
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274 SPECIAL NUMBERS Recurrence (6.9
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276 SPECIAL NUMBERS Gnethingwecando
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278 SPECIAL NUMBERS cases are: a0 =
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280 SPECIAL NUMBERS The process of
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282 SPECIAL NUMBERS Fk < n < Fk+l;
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284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
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288 SPECIAL NUMBERS The continuant
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290 SPECIAL NUMBERS in four steps:
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292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
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296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
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300 SPECIAL NUMBERS 44 Prove the co
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302 SPECIAL NUMBERS 61 Prove the id
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304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
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308 GENERATING FUNCTIONS are added
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310 GENERATING FUNCTIONS (The last
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312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
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316 GENERATING FUNCTIONS determine
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318 GENERATING FUNCTIONS operation
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320 GENERATING FUNCTIONS Table 320
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322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
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326 GENERATING FUNCTIONS Once we’
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328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
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332 GENERATING FUNCTIONS Finally, s
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334 GENERATING FUNCTIONS Step 3 was
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336 GENERATING FUNCTIONS This is a
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338 GENERATING FUNCTIONS Table 337
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340 GENERATING FUNCTIONS F(z)' = i
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342 GENERATING FUNCTIONS We can app
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344 GENERATING FUNCTIONS Step 3 is
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346 GENERATING FUNCTIONS Raney’s
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348 GENERATING FUNCTIONS Notice tha
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350 GENERATING FUNCTIONS Chapter 5
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352 GENERATING FUNCTIONS Now let’
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354 GENERATING FUNCTIONS function f
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356 GENERATING FUNCTIONS 7.7 DIRICH
Page 372 and 373: 358 GENERATING FUNCTIONS 5 Find a g
Page 374 and 375: 360 GENERATING FUNCTIONS 20 A power
Page 376 and 377: 362 GENERATING FUNCTIONS 29 What is
Page 378 and 379: 364 GENERATING FUNCTIONS 43 The New
Page 380 and 381: 366 GENERATING FUNCTIONS 53 The seq
Page 382 and 383: 368 DISCRETE PROBABILITY total of 6
Page 384 and 385: 370 DISCRETE PROBABILITY about R if
Page 386 and 387: 372 DISCRETE PROBABILITY The mean o
Page 388 and 389: 374 DISCRETE PROBABILITY more sprea
Page 390 and 391: 376 DISCRETE PROBABILITY hence VS =
Page 392 and 393: 378 DISCRETE PROBABILITY And we can
Page 394 and 395: 380 DISCRETE PROBABILITY variance b
Page 396 and 397: 382 DISCRETE PROBABILITY (1 +z+...
Page 398 and 399: 384 DISCRETE PROBABILITY ~1 and ~2
Page 400 and 401: 386 DISCRETE PROBABILITY where Ug i
Page 402 and 403: 388 DISCRETE PROBABILITY Repeating
Page 404 and 405: 390 DISCRETE PROBABILITY There’s
Page 406 and 407: 392 DISCRETE PROBABILITY But the co
Page 408 and 409: 394 DISCRETE PROBABILITY In the spe
Page 410 and 411: 396 DISCRETE PROBABILITY Here 1 is
Page 412 and 413: 398 DISCRETE PROBABILITY For exampl
Page 414 and 415: 400 DISCRETE PROBABILITY Thus the m
Page 416 and 417: 402 DISCRETE PROBABILITY Let sj be
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Page 420 and 421: 406 DISCRETE PROBABILITY The mean v
Page 424 and 425: 410 DISCRETE PROBABILITY all k and
Page 426 and 427: 412 DISCRETE PROBABILITY to be a pg
Page 428 and 429: 414 DISCRETE PROBABILITY 10 What’
Page 430 and 431: 416 DISCRETE PROBABILITY 28 What is
Page 432 and 433: 418 DISCRETE PROBABILITY 38 What is
Page 434 and 435: 420 DISCRETE PROBABIL1T.Y a Once he
Page 436 and 437: 422 DISCRETE PROBABILITY 50 Conside
Page 438 and 439: 424 DISCRETE PROBABILITY 58 Are the
Page 440 and 441: 426 ASYMPTOTICS The word asymptotic
Page 442 and 443: 428 ASYMPTOTIC3 How about the funct
Page 444 and 445: 430 ASYMPTOTICS N. G. de Bruijn beg
Page 446 and 447: 432 ASYMPTOTICS subtle consideratio
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Page 450 and 451: 436 ASYMPTOTICS 9.3 0 MANIPULATION
Page 452 and 453: 138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455: 440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457: 442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459: 444 ASYMPTOTICS Problem 4: A sum fr
Page 460 and 461: 446 ASYMPTOTICS Problem 5: An infin
Page 462 and 463: 448 ASYMPTOTICS We might as well do
Page 464 and 465: 450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467: 452 ASYMPTOTICS This last estimate
Page 468 and 469: 454 ASYMPTOTICS Here’s another ex
Page 470 and 471: 456 ASYMPTOTICS On the left is a ty
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458 ASYMPTOTICS (The Bernoulli poly
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460 ASYMPTOTICS The graph of B,(x)
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462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
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464 ASYMPTOTICS The integral JT B4(
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466 ASYMPTOTIC3 where 0 < a,,,, < 1
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468 ASYMPTOTICS In Chapter 5, equat
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470 ASYMPTOTICS For example, we hav
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472 ASYMPTOTICS The final task seem
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474 ASYMPTOTICS This is our approxi
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476 ASYMPTOTIC23 9 Prove (9.22) rig
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478 ASYMPTOTICS 39 Evaluate xOsk
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480 ASYMPTOTICS a What is the asymp
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482 ASYMPTOTIC3 Research problems 6
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484 ANSWERS TO EXERCISES Venn [294]
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486 ANSWERS TO EXERCISES move the b
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488 ANSWERS TO EXERCISES 2.2 This i
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490 ANSWERS TO EXERCISES 2.24 Summi
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492 ANSWERS TO EXERCISES 2.36 (a) B
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494 ANSWERS TO EXERCISES 3.21 If 10
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496 ANSWERS TO EXERCISES the circle
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498 ANSWERS TO EXERCISES 3.40 Let L
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500 ANSWERS TO EXERCISES 3.50 H. S.
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502 ANSWERS TO EXERCISES 4.22 (b,
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504 ANSWERS TO EXERCISES 4.40 Let f
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506 ANSWERS TO EXERCISES 4.50 (a) I
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508 ANSWERS TO EXERCISES and (k- 1)
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510 ANSWERS TO EXERCISES In both ca
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512 ANSWERS TO EXERCISES 5.7 Yes, b
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514 ANSWERS TO EXERCISES because al
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516 ANSWERS TO EXERCISES 5.44 Cance
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518 ANSWERS TO EXERCISES and this i
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520 ANSWERS TO EXERCISES The infini
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522 ANSWERS TO EXERCISES 5.82 Let ~
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524 ANSWERS TO EXERCISES Since ~06j
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526 ANSWERS TO EXERCISJES 5.94 This
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528 ANSWERS TO EXERCISES 6.21 The h
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530 ANSWERS TO EXERCISIES 6.40 If 6
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532 ANSWERS TO EXERCISES solutions
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534 ANSWERS TO EXERCISES 6.62 (a) A
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536 ANSWERS TO EXERCISES 6.74 If p(
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538 ANSWERS TO EXERCISIES One of th
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540 ANSWERS TO EXERCISES 6.85 The p
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542 ANSWERS TO EXERCISES Hence gzn+
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544 ANSWERS TO EXERCISES 7.24 ntk,+
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546 ANSWERS TO EXERCISES 7.40 The e
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548 ANSWERS TO EXERCISES 1 + a. Hen
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550 ANSWERS TO EXERCISES applied to
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552 ANSWERS TO EXERCISES 8.13 (Solu
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554 ANSWERS TO EXERCISES 8.24 (a) A
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556 ANSWERS TO EXERCISES 8.32 By sy
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558 ANSWERS TO EXERCISES 8.37 The n
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560 ANSWERS TO EXERCISES probabilit
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562 ANSWERS TO EXERCISES And since
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564 ANSWERS TO EXERCISES 8.56 If m
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566 ANSWERS TO EXERCISES We can now
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568 ANSWERS TO EXERCISES 9.19 Hlo =
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570 ANSWERS TO EXERCIS:ES 9.30 Let
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572 ANSWERS TO EXERCISES 9.42 The h
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574 ANSWERS TO EXERCISIES length of
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576 ANSWERS TO EXERCISES 9.58 Let 0
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B Bibliography HERE ARE THE WORKS c
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580 BIBLIOGRAPHY 24' 25 26 27 28 29
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582 BIBLIOGRAPHY 51 52 53 54 55 56
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584 BIBLIOGRAPHY 78 79 80 81 82 83
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586 BIBLIOGRAPHY 100 R. A. Fisher,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
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590 BIBLIOGRAPHY 160 K. Inkeri, “
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592 BIBLIOGRAPHY 188 E.E. Kummer,
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594 BIBLIOGRAPHY 214 Z.A. Melzak, C
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596 BIBLIOGRAPHY 243 George N. Rane
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598 BIBLIOGRAPHY 271 A. D. Solov’
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600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
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604 CREDITS FOR EXERCISES 6.46 6.47
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Index WHEN AN INDEX ENTRY refers to
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608 INDEX Bois-Reymond, Paul David
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610 INDEX Degenerate hypergeometric
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612 INDEX Feder, Tom&s, 604. Feigen
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614 INDEX Hall, Marshall, Jr., 588.
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616 INDEX Lincoln, Abraham, 387. Li
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618 INDEX Patashnik, Oren, iii, iv,
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620 INDEX Running time, 411-412. Ru
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622 INDEX Sylvester, James Joseph,
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