INTRODUCTION TO ALGORITHMS 3rd phần 9 docx

...  is a tautology if it evaluates to 1 for every assignment of 1 and 0 to the input variables. Define TAUTOLOGY as the language of boolean formulas that are tautologies. Show that TAUTOLOGY 2 co-NP. 34.2 -9 Prove ... all three together do also, and so we have a way to decide on ˛ in polynomial time. In other words, by “reducing” solving problem A to solving problem B, we use the “easin...

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... an element 95 4 31.7 The RSA public-key cryptosystem 95 8 ? 31.8 Primality testing 96 5 ? 31 .9 Integer factorization 97 5 32 String Matching 98 5 32.1 The naive string-matching algorithm 98 8 32.2 The ... B-trees 491 18.3 Deleting a key from a B-tree 499 19 Fibonacci Heaps 505 19. 1 Structure of Fibonacci heaps 507 19. 2 Mergeable-heap operations 510 19. 3 Decreasing a key and...

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... (k) 12347 891 01416 10 2 13 47 89 1614 1 23 47 89 161410 3 21 98 74 10 14 16 4 23 98 71 10 14 16 8 37 42 19 161410 7 43 98 21 10 14 16 9 83 2174 161410 10 89 3174 16142 14 810 397 4 1612 16 14 10 397 8 142 A i i i ii ii i i Figure ... candidate we wish to hire. 158 Chapter 6 Heapsort 1 23 4567 891 0 1 23 4567 891 0 1 23 4567 891 0 1 23 4567 891 0 1 23 4567 891 0 1 23 4567...

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... key k not already stored in the table is equally likely to hash to any of the m slots. The expected time to search unsuccessfully for a key k is the expected time to search to the end of list ... interest are pointers to other nodes, and they vary according to the type of tree. Binary trees Figure 10 .9 shows how we use the attributes p, left,andright to store pointers to t...

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... Greedy Algorithms 01234567 891 011121314 time 235 306 457 5 39 6 59 7610 8811 98 12 10214 11 12 16 114 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, ... tables 2 for i D 1 to n C 1 3 eŒi;i  1 D q i1 4 wŒi;i  1 D q i1 5 for l D 1...

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... 14,15 u 2 0 1 A proto-vEB(2) 1 1 u 2 0 1 A proto-vEB(2) 0 0 u 2 0 1 A proto-vEB(2) 0 1 u 2 0 1 A proto-vEB(2) 0 0 u 2 0 1 A proto-vEB(2) 0 0 u 2 0 1 A proto-vEB(2) 0 0 u 2 0 1 A proto-vEB(2) 0 1 u 2 0 1 A proto-vEB(2) 1 1 u 2 0 1 A proto-vEB(2) 1 1 u 2 0 1 A proto-vEB(2) 1 1 u 2 0 1 A proto-vEB(2) 0 0 u 2 0 1 A proto-vEB(2) 0 1 u 2 0 1 A proto-vEB(2) 0 1 u 2 0 1 A proto-vEB(2) 1 1 u 2 Figure...

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... added to S,wehadx:d D ı.s; x/ when x was added 24.3 Dijkstra’s algorithm 6 59 0 ∞∞ ∞∞ 0 ∞ ∞ 1 2 10 5 (c) 10 5 0 8 5 14 7 0 8 5 13 7 0 8 5 9 7 0 5 9 7 8 6432 9 7 s tx yz 1 2 10 5 (f) 6432 9 7 s tx yz 1 2 10 5 (b) 6432 9 7 s tx yz 1 2 10 5 (e) 6432 9 7 s tx yz 1 2 10 5 (a) 6432 9 7 s tx yz 1 2 10 5 (d) 6432 9 7 s tx yz Figure ... there is at least one path, there is a shortes...

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... comes into the picture, executing instructions 4–6. Processor 2 reads x from memory into register r 2 ; increments it, producing the value 1 in r 2 ; and then stores this value into x, setting x to ... sorted recursively, and line 6 goes on to compute the number q 0 of elements in the first subarray AŒp : : q, which line 8 uses to determine the starting index in T of where to store th...

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... 2;3;:::;n. a. Is the sum of two Toeplitz matrices necessarily Toeplitz? What about the prod- uct? b. Describe how to represent a Toeplitz matrix so that you can add two n  n Toeplitz matrices in O.n/ ... this manner until the vector holds two .n=2/-element DFTs, which we combine using n=2 butterfly operations into the final n-element DFT. To turn this bottom-up approach into code, we use a...

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... created by G. Cantor during the period 1874–1 895 . Cantor focused primarily on sets of infinite cardinality. The term “function” is attributed to G. W. Leibniz, who used it to refer to several kinds ... vertices of the top graph are mapped to the vertices of the bottom graph by f.1/ D u; f .2/ D ; f .3/ D w; f.4/ D x;f .5/ D y;f .6/ D ´. (b) Two graphs that are not isomorphic, since...

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