INTRODUCTION TO ALGORITHMS 3rd phần 1 docx

... summations 11 49 B Sets, Etc. 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 ... proofs 10 78 34.5 NP-complete problems 10 86 35 Approximation Algorithms 11 06 35 .1 The vertex-cover problem 11 08 35.2 The traveling-salesman problem 11 11...

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... model. 10 72 Chapter 34 NP-Completeness x 3 x 2 x 1 (a) 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 x 3 x 2 x 1 (b) Figure 34.8 Two instances of the circuit-satisfiability problem. (a) The assignment hx 1 D 1; x 2 D ... up of symbols from †. For example, if † D f 0; 1 g , the set L D f 10 ; 11 ; 10 1; 11 1; 10 11; 11 01; 10 0 01; : : : g is the language of binary represen...

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... algorithm 16 1 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) 12 3478 910 1 416 10 2 13 4789 16 14 1 23 4789 16 1 410 3 21 9874 10 14 16 4 23 98 71 10 14 16 8 37 4 219 16 1 410 7 43 98 21 10 14 16 9 83 217 4 16 1 410 10 89 317 4 16 142 14 810 3974 16 12 16 14 ... wish to hire. 15 8 Chapter 6 Heapsort 1 23 4567 8 910 1 23 4567 8 910 1 23 4567 8 910...

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... the pair of decimal integers .11 2; 11 6/,sincep D 11 2 and t D 11 6 in the ASCII character set; then, expressed as a radix -12 8 integer, pt becomes .11 2  12 8/ C 11 6 D 14 452. In the context of a given ... probes in a successful search: 1 n n 1 X iD0 m m i D m n n 1 X iD0 1 m i D 1 ˛ m X kDmnC1 1 k Ä 1 ˛ Z m mn .1= x/ dx (by inequality (A .12 )) D 1 ˛ ln m m n...

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... Algorithms 012 345678 910 111 213 14 time 235 306 457 539 659 7 610 8 811 9 812 10 214 11 12 16 11 4 k s k f k a 1 a 2 a 1 a 3 a 1 a 4 a 1 a 4 a 5 a 1 a 4 a 6 a 1 a 4 a 7 a 1 a 4 a 8 a 1 a 4 a 8 a 9 a 1 a 4 a 8 a 10 a 1 a 4 a 8 a 11 a 1 a 4 a 8 a 11 0–0 a 1 a 0 a 0 RECURSIVE-ACTIVITY-SELECTOR(s, ... root. O PTIMAL-BST.p;q;n/ 1lete 1 : : n C 1; 0::n, w...

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... Fibonacci 20 .1 Preliminary approaches 533 0 0 0 1 1 2 1 3 1 4 1 5 0 6 1 7 0 8 0 9 0 10 0 11 0 12 0 13 1 14 1 15 0 1 1 1 0 0 0 1 1 1 0 1 1 1 1 A Figure 20 .1 A binary tree of bits superimposed on top of ... Section 19 .4, we will show that the maxi- 20 .1 Preliminary approaches 535 0 0 0 1 1 2 1 3 1 4 1 5 0 6 1 7 0 8 0 9 0 10 0 11 0 12 0 1...

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... Paths 2 ∞∞0 5 16 34 ∞ ∞ ∞ 7 1 2 2 (a) xtsryz 25 16 34 7 1 2 2 (c) xtsryz 25 16 34 7 1 2 2 (e) xtsryz 25 16 34 7 1 2 2 (g) xtsryz 2 5 16 34 7 1 2 2 (b) xtsryz 25 16 34 7 1 2 2 (d) xtsryz 25 16 34 7 1 2 2 (f) xtsryz ∞ ... number a ¤ 1, wehavea C1D1Ca D1. Also, to make our proofs hold in the presence of negative-weight cycles, we shall assume that for any real number a 1, wehavea...

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... w T =a 11 D L 0 U 0 ; where L 0 is unit lower-triangular and U 0 is upper-triangular. Then, using matrix algebra, we have A D Â 10 =a 11 I n 1 ÃÂ a 11 w T 0A 0  w T =a 11 Ã D Â 10 =a 11 I n 1 ÃÂ a 11 w T 0L 0 U 0 Ã D Â 10 =a 11 L 0 ÃÂ a 11 w T 0U 0 Ã D ... input. P-M ERGE.T; p 1 ;r 1 ;p 2 ;r 2 ;A;p 3 / 1 n 1 D r 1  p 1 C 1 2 n 2 D r 2  p 2 C 1 3...

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... 2. 31. 3 Modular arithmetic 9 41 012 345 0 1 2 3 4 5 012 345 012 345 012 345 012 345 012 345 012 345 (a) 1 2 4 7 8 11 13 14 1 2 4 7 8 11 13 14 1 2 4 7 8 11 13 14 2 4 8 14 1 7 11 13 4 811 3 214 711 714 13 411 2 1 ... 11 13 14 2 4 8 14 1 7 11 13 4 811 3 214 711 714 13 411 2 1 8 8 1 211 413 147 11 714 213 1 8 4 13 11 7 1 14 8 4 2 14 1 311...

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... hands. 12 02 Appendix C Counting and Probability 0.05 0 .10 0 .15 0.20 0.25 12 345678 910 111 213 1 415 0.30 0.35 k  2 3 à k 1  1 3 à Figure C .1 A geometric distribution with probability p D 1= 3 of ... We have Pr f A 1 g D 1= 2 ; Pr f A 2 g D 1= 2 ; Pr f A 3 g D 1= 2 ; Pr f A 1 \ A 2 g D 1= 4 ; Pr f A 1 \ A 3 g D 1= 4 ; Pr f A 2 \ A 3 g D 1= 4 ; Pr f A 1 \ A 2 \ A...

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