Ebook Introduction to algorithms (3rd edition) Part 2

... 11 58 B .1 Sets 11 58 B.2 Relations 11 63 B.3 Functions 11 66 B.4 Graphs 11 68 B.5 Trees 11 73 C Counting and Probability 11 83 C .1 Counting 11 83 C.2 Probability 11 89 C.3 Discrete random variables 11 96 ... problem 11 28 11 23 10 21 Contents xi VIII Appendix: Mathematical Background Introduction A 11 43 Summations 11 45 A .1 Summation formulas and properties A.2 Boundi...

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... 6; 22 ; 9i 6.3 -2 Why we want the loop index i in line of B UILD -M AX -H EAP to decrease from bA:length=2c to rather than increase from to bA:length=2c? 6.3-3 ˙ Show that there are at most n=2hC1 ... blg nc l n m X hD0 2hC1 159 X h O.h/ D O n 2h blg nc ! : hD0 We evalaute the last summation by substituting x D 1 =2 in the formula (A.8), yielding X h 2h D hD0 1 =2 1 =2/ 2 D 2: Thus...

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... 401, 39 8, 33 0, 34 4, 39 7, 36 3 b 924, 220, 911, 244, 898, 258, 36 2, 36 3 c 925, 202, 911, 240, 912, 245, 36 3 d 2, 39 9, 38 7, 219, 266, 38 2, 38 1, 278, 36 3 e 935 , 278, 34 7, 621, 299, 39 2, 35 8, 36 3 12.2-2 ... be the fraction of the p form s= 232 that is closest to 1/=2, so that A D 2654 435 769= 232 Then k s D 32 7706022297664 D 7 630 0 232 / C 17612864, and so...

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... -ACTIVITY-SELECTOR(s, f, 1, 11) a1 a3 a1 a4 a1 m =4 RECURSIVE -ACTIVITY-SELECTOR(s, f, 4, 11) 9 10 8 11 12 10 14 11 12 a5 16 a1 a4 a1 a4 a1 a4 a1 a4 a6 a7 a8 m=8 a9 RECURSIVE -ACTIVITY -SELECTOR (s, ... 11) a1 a4 a8 a10 a1 a4 a8 a1 a4 a8 m = 11 a8 a11 a11 RECURSIVE -ACTIVITY -SELECTOR (s, f, 11, 11) a1 a4 time 10 11 12 13 14 15 16 Figure 16.1 The operation of R ECURSIVE -ACTIVITY-S ELECTO...

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... 23 38 21 30 39 41 52 35 H:min (c) 15 18 H:min 24 17 26 30 26 18 23 24 21 38 39 (d) 15 41 26 24 52 17 18 23 30 21 38 39 41 52 H:min (e) 15 17 30 18 23 21 38 39 41 52 Figure 19 .5 Two calls of F ... 4 ,5 proto-vEB(4) u summary u A 1 u A 0 cluster proto-vEB(2) A 0 proto-vEB(2) u u A 1 elements 6,7 cluster proto-vEB(2) clusters 2,3 proto-vEB(2) A 1 proto-vEB(2) u cluster proto-vEB(2) A...

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... Paths 24 -6 Bitonic shortest paths A sequence is bitonic if it monotonically increases and then monotonically decreases, or if by a circular shift it monotonically increases and then monotonically ... are given an m n matrix A, an m-vector b, and an n-vector c.P wish to find a vector x of n elements that We n maximizes the objective function i D1 ci xi subject to the m constraints given b...

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... ( 27. 2), the span law ( 27. 3), and inequality ( 27. 5) from Exercise 27. 1-3.) 79 2 Chapter 27 Multithreaded Algorithms 27. 1-6 Give a multithreaded algorithm to multiply an n n matrix by an n-vector ... use to analyze multithreaded algorithms Section 27. 2 investigates how to multiply matrices with multithreading, and Section 27. 3 tackles the tougher problem of multithreading...

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... The DFT and FFT i 907 !8 !8 !8 !8 !8 D !8 1 !8 i !8 !8 Figure 30.2 The values of !8 ; !8 ; : : : ; !8 in the complex plane, where !8 D e cipal 8th root of unity !n D e i=n i =8 is the prin- (30.6) ... have to combine DFTs on each level of the tree, we introduce a variable s to count the levels, ranging from (at the bottom, when we are combining pairs to form 2-element DFTs...

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... a tautology if it evaluates to for every assignment of and to the input variables Define TAUTOLOGY as the language of boolean formulas that are tautologies Show that TAUTOLOGY co-NP 34.2 -9 Prove ... you to show that all such graphs are nonhamiltonian We can define the hamiltonian-cycle problem, “Does a graph G have a hamiltonian cycle?” as a formal language: HAM-CYCLE D fhGi W G is a ha...

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... AT D : A vector is a one-dimensional array of numbers For example, xD is a vector of size We sometimes call a vector of length n an n-vector We use lowercase letters to denote vectors, and we ... element of a size-n vector x by xi , for i D 1; 2; : : : ; n We take the standard form of a vector to be as a column vector equivalent to an n matrix; the corresponding row vector is obtained b...

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... j k 5 ∞ R j ∞ 10 11 12 13 14 15 16 17 2 k … 4 i ∞ R j ∞ 10 11 12 13 14 15 16 17 2 k … 4 i ∞ R ∞ j 10 11 12 13 14 15 16 17 A … 4 ∞ i R ∞ j 2 2 6 … k 10 11 12 13 14 15 16 17 A … 1 L 4 ∞ i R ∞ j ... 14 15 16 17 A … 1 L 4 ∞ i R ∞ j … k L L … L … 10 11 12 13 14 15 16 17 i 2 A … A … 1 L … L 10 11 12 13 14 15 16 17 i k i A … A … 10 11 12...

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... lg(n − 2) − 2c lg(n − 2) + dn − 2d + lg n > cn lg(n − 2) − 2c lg n + dn − 2d + lg n (since − lg n < − lg(n − 2) ) = cn lg(n − 2) − 2( c − 1) lg n + dn − 2d ≥ cn lg(n /2) − 2( c − 1) lg n + dn − 2d (by ... 1, 2, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2, probability 4 /27 5 /27 5 /27 5 /27 4 /27 4 /27 Solutions for Chapter 5: Probabilistic Analysis and Randomized Algorithms 5-13 Although these...

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... array A on digit i Example: sorted 32 6 4 53 608 835 751 435 704 690 690 751 4 53 704 835 435 32 6 608 704 608 32 6 835 435 751 4 53 690 32 6 435 4 53 608 690 704 751 835 Lecture Notes for Chapter 8: Sorting ... A[1] to A[2] 1:2 A[1] ≤ A[2] ≤ 2 :3 ≤ A[1] ≤ A[2] > A[2] > A [3] 〈1,2 ,3 〈1 ,3, 2〉 1 :3 ≤ A[1] > A[2] > A[1] > A [3] 〈2,1 ,3 1 :3 2 :3 > ≤ A[1] ≤ A[2] ≤ A [3] > A[1]...

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... we increment top[S], set S[top[S]] ← k, set S [top[S]] ← x, and set T [k] ← top[S] D ELETE: To delete object x with key k, assuming that this object is in the dictionary, we need to break the ... pointer in the slot that pointed to j to point to the new slot Then insert the new element in the now-empty slot as usual To update the pointer to j , it is necessary to Þnd it by sea...

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