Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Mass., 2nd edition, 1981. 18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which is a count on the number of available processors p. Recall from section 2.1 that the our parallel model has fixed topology and also the implementation of MPI used here does not permit variable number of processors. In normal hashing the size of the table is chosen to be a prime number [78], but this would be too restrictive on the limited number of processors available. Therefore the table size p can be statically chosen for any number of available processing elements. As the number of processors is small compared to the number of terms in a system of polynomial equations, ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, second edition, 1969.

....suitable for distributed memory implemenation. Each digit is signed therefore a number can be distributed across several processors without loss of sign information. The signed digit representation is used in the CALYPSO system [24] CALYPSO implements three multiplication algorithms: Karatsuba [77], 3 primes FFT and floating point FFT. The selection of which algorithm to use is based based on performance thresholds in asymptotic analysis: on smaller inputs, Karatsuba is best but for larger problems the FFT methods are better. 2.5.2 Parallel polynomial arithmetic The operations on ....

....i in A by the vector b. Then Cramer s rule is given in equation (2.4) x i = det(A i ) 2.4) Cramer s rule is impractical for small workstations because of the large determinant calculations. Consider that the determinant of a matrix A(n n) is a polynomial with n variables and n terms [77], therefore determinant calculations require a large amount of memory. Parallel systems with distributed memory can provide the memory required for large determinant calculations. Let A be a square matrix whose entries a ij are multivariate polynomials. The determinant of A is shown in equation ....

[Article contains additional citation context not shown here]

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, third edition, 1969.

....of catenate. It will be convenient for us to treat them as separate operations, however. In accordance with convention, we call a list subject only to push and pop (or inject and eject) a stack and a list subject only to inject and pop (or push and eject) a queue. Adopting the terminology of Knuth [29], we call a list subject to all four operations push, pop, inject, and eject a double ended queue, abbreviated deque (pronounced deck ) In a departure from existing terminology, we call a list subject only to push, pop, and inject a stack ended queue, or steque (pronounced steck ) Knuth called ....

Donald E. Knuth. Fundamental Algorithms, Volume 1 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, second edition, 1973. 37

....4.4. Theoretical Work. Early theoretical analysis of hashing schemes was typically done under the assumption that hash function values were uniformly random and independent. Precise analyses of the average and expected worst case behaviors of the abovementioned schemes have been made, see e.g. [30, 47]. We mention just a few facts, disregarding asymptotically vanishing terms. For Linear Probing the expected number of memory probes for successful and unsuccessful lookups are ) and (1 #) 2 ) respectively, where # denotes the fraction of the table occupied by keys, 0 # 1. The ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley Publishing Co., Reading, Massachusetts, second edition, 1998.

....of N can be enumerated as the elements in level set of rank N of Y (N;N; N) 4. Permutations We rst show a small computation that generates all permutations of n symbols. The computation consists of n 1 processes. Process P i executes i 1 events. We use the notion of the inversion table[Knu98] to interpret the choices made by processes. The number of inversions of i in a permutation is the number of symbols less than i that appear to the right of i in . The way a permutation is generated from a global state is as follows. We begin the permutation by writing 1. P 1 decides where to ....

....(n 1; n 2; 1) Further by computing the slice, we can also calculate the number of permutations satisfying B. The level set at rank k of the permutation lattice consists of all permutations with total number of inversions equal to k and therefore such permutations can be eciently enumerated [Knu98,ER02]. ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, second edition, 1998.

....all child nodes marked. Proof follows by induction. For the number of nodes, n, in a minimum tree of rank k: n(k) n(k 1) n(k 2) 9) The sequence of Fibonacci numbers is defined as: F (k 2) F (k 1) F (k) 10) For a detailed description of the Fibonacci sequence, refer to Knuth [4]. This recurrence relation can be solved to give: F (k) # 5 # # # 5 (11) The first term in this equation is more dominant as n tends toward infinity. The golden ratio is defined as: # = 1 # 5 2 (12) This approximates the ratio between two consecutive large Fibonacci numbers. ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, third edition, 1997.

....for the CRC, and we should realize the function in picotode, which is too expensive in terms of CPU cycles. Figure 3.13: Jenkins function with INPUT IP Figure 3.14: Wang s function with IN PUT IP Knuth s standard reference The standard rerencefor the integer hashfunctions is Ref. [24]. Knuth recommends the hash for (hash=len; len ; hash = hash 5) hash 27) key ; hash = hash table size; Unfortunately, that hash is only mediocre. The problem is the per character mixing: it only rotates bits, it does not really mix them. It was demonstrated that every input bit ....

....Also regarding to the NASA input, Figs. 3.15 and 3.16 show that the spreading is highly uniform. For both algorithm the greatest frequency is 3. Figure 3.17: Knuth s function with INPUT IP Figure 3.18: Knuth s function with INPUT IP (NASA input) Knuth s integer hash function In Section 6. 4 of [24], another multiplicative hashing scheme is introduced as a way to write the hash function. The key is multiplied by the golden ratio 4) computed over 32 bits (2654435769) to produce a hash result. return (key 2654435769) OxFFFFFFFF; Because 2654435769 and 232 have no common factors, the ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998.

....tremendous source of inspiration a majority of published papers in the field are about Counting Networks. Their close relatives, the Sorting Networks, are even more popular in theoretical computer science. They re often treated in introductory algorithm courses and in distinguished text books [Knu73, Man89, CLR92, OW93] Due to the amount of published material, we re not in the position to discuss every aspect of Counting Networks. If one is interested in a more detailed treatment, we d recommend the journal version of the seminal paper [AHS94] or the dissertation of Michael Klugerman ....

....known; let us restrict on the most prominent first: The Bitonic Counting Network. This structure is isomorphic to Batcher s Bitonic Sorting Network [Bat68] a very popular structure within the field of communication networks; It is presented in many comprehensive computer science text books [Knu73, Man89, CLR92, OW93] The only parameter of a Bitonic Counting Network is its width w, which is always a power of two. A Counting Network with width w has w input wires and w output wires. In the following, let us describe the structure of the Bitonic Counting Network in an inductive manner. To ....

[Article contains additional citation context not shown here]

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, 1973.

....Figure 4: Sorter class hierarchy As shown in Figure 4, the sorting algorithms normally discussed in the second course can be organized into a class hierarchy. Indeed the abstract sorters of this hierarchy correspond to way in which sorting algorithms have been traditionally categorized[3]. For example, consider the quicksort algorithm: 1. Select a pivot; and then 2. Using the pivot split the data to be sorted into two sublists and then recursively quicksort the sublists. As shown in Figure 4 we can define the abstract class AbstractQuickSorter and the concrete class ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, 1973.

....16 row boundary, rows after the 11th are free to hold a smaller configuration that will fit, such as a 4 row configuration beginning at the 12th row. The power of two hierarchy suggests a version of the buddy system, a technique for allocating storage described by numerous sources including Knuth [43]. To reduce the work associated with allocating array space, the smallest unit of allocation will be four rows. The maximum number of one row configurations a cache plane can store is therefore 92 32 4 = 8, even though it would be possible to fit 32 one row configurations into a single cache ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley Publishing Company, second edition, 1973.

....of early hashing schemes Early theoretical analysis of hashing schemes was done under the assumption that hash function values are uniformly random and independent. Precise analyses of the average and expected worst case behaviors of the abovementioned schemes have been made, see for example [14, 18]. We mention just a few facts, disregarding asymptotically vanishing terms. Note that some figures depend on implementation details the below hold for the implementations described in Section 5. We first consider the expected number of memory probes needed by the two open addressing schemes ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley Publishing Co., Reading, Mass., second edition, 1998.

....the performance of trajectory sampling. Our overall goal is to obtain as many pseudo random trajectory samples from the network as possible, without using too many resources (network bandwidth, collection system memory) In this paper we demonstrate hashes based on modular arithmetic (see e.g. [17]) and show that the parameters of this scheme can be chosen such that the hashes appear statistically independent from the original packet content, thus enabling unbiased sampling. 3 We then compute the optimal choice of the total number of samples to be collected from the network and the ....

....IP addresses are interchanged, arising from the two way flow of traffic in TCP sessions. The hash function, and hence the modulus, must be chosen to avoid collisions in which a pair of packets that differ little by such an interchange are mapped onto the same remainder. Knuth (see f 6. 4 in [17]) formulates a condition for avoidance of such collisions, namely that g Uh ikj l,mZn ] for small i o where g is a radix of the alphabet used to describe the header. Including g Dp0q r in this criterion suppresses collisions of the type described above. Moduli obeying these ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1998.

....last, and will generate duplicates, or it can do so because a relation may contain, apart from its own special I , other prime ideals that are used as special prime ideals as well. The simultaneous use of line by line and lattice siever also causes overlap. Duplicates are tracked down by hashing [12]. Since it is easier and cheaper to use a number instead of a relation as a hash table entry, we identify a relation with a number. The user specifies how many relations he expects to be in the input file(s) maxrelsinp) This figure is used to choose the size of the in memory tables needed ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, second edition, 1998.

....destination IP addresses are interchanged, arising from the two way ow of trac in TCP sessions. The hash function, and hence the modulus, must be chosen to avoid collisions in which a pair of packets that di er little by such an interchange are mapped onto the same remainder. Knuth (see x6.4 in [17]) formulates a condition for avoidance of such collisions, namely that q k a 6= 0 mod A for small a; k where q is a radix of the alphabet used to describe the header. Including q k = 2 32 in this criterion suppresses collisions of the type described above. Moduli obeying these conditions ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1998.

....the performance of trajectory sampling. Our overall goal is to obtain as many pseudorandom trajectory samples from the network as possible, without using too many resources (network bandwidth, collection system memory) In this paper we demonstrate hashes based on modular arithmetic (see e.g. [16]) and show that the parameters of this scheme can be chosen such that the hashes appear statistically independent from the original packet content, thus enabling unbiased sampling. 3 We then compute the optimal choice of the total number of samples to be collected from the network and the number ....

....IP addresses are interchanged, arising from the two way flow of traffic in TCP sessions. The hash function, and hence the modulus, must be chosen to avoid collisions in which a pair of packets that differ little by such an interchange are mapped onto the same remainder. Knuth (see x6.4 in [16]) formulates a condition for avoidance of such collisions, namely that q k Sigma a 6= 0 mod A for small a; k where q is a radix of the alphabet used to describe the header. Including q k = 2 32 in this criterion suppresses collisions of the type described above. Moduli obeying these ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1998.

....a smaller fi 0 only. Similarily, setting right( R q ) further right could only decrease i 0 , which would enlarge 0 x in the next round and hence provide a smaller fi 0 then, see below. Evaluating fi 0 for the above situation is simple: By using (59) and Gammadxe = b Gammaxc (see [Knu73, Sec. 1.2.4, Ex. 4] we find fi 0 = fi H Gamma Gamma j H Gamma H x k GS = fi l H H H Gamma Gamma H Gamma H x m GS ; 28) which easily yields expression (26) for ( The required monotonicity is immediately apparent, given that H = H ....

.... we obtain i 0 = I j H Gamma H x k GS Gamma j H Gamma H ( I x ) k GS 30 = j H Gamma H x k GS l H H I Gamma H Gamma H x m GS l H H I m GS ; 29) where we used the triangle inequality dx ye dxe dye (see [Knu73, Sec. 1.2.4, Ex. 7] This eventually confirms the expression for = in (27) The required monotonicity of = is again immediately apparent. The analogous expressions for Gamma ( resp. Gamma ( can be obtained by considering Figure 7 mirrored at the (dashed) vertical ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, 2nd edition, 1973.

....be [1, 1, 1, we have K(#) 1 and thus the upper bound of Theorem 6, using Theorem 8, has the form # 1 1 # # 2 x 1 for appropriate constants # 1 and # 2 . The fact that # is so well suited to measurement can be immediately related to its usefulness in multiplicative hashing [Knu97] Every point of the sequence t# = t # #t## on the unit interval bisects (following the golden ratio) one of the longest intervals not containing previous points [ Swi58] i.e. the sequence is most uniformly distributed (more precisely, this is true of # 1 , but t# 1 = t# ) ....

.... 0 # t F n 1 has length greater than 1 2 # 2n . Proof. We use the fact that the point k# bisects following the golden ratio one of the longest intervals determined by t# 0 # t k on the unit interval, and that new lengths appear only when k is a Fibonacci number (see [Knu97] for a full discussion) Since each bisection possibly reduces an interval by a factor of 1 # 1 = # 2 , all intervals determined by t# 0 # t F n 1 have at least length # 2n (the same holds for the other set of points) Hence, the statement can be easily obtained observing ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, second edition, 1997.

....[1, 1, 1, we have K(#) 1 and thus the upper bound of Theorem 11, using Theorem 15, has the form # 1 1 # # 2 x 1 for appropriate constants # 1 and # 2 . The fact that # is so well suited to measurement can be immediately related to its usefulness in multiplicative hashing [3]. Every point of the sequence t# = t # #t## on the unit interval bisects (following the 11 golden ratio) one of the longest intervals not containing previous points [7] that is, the sequence is most uniformly distributed (more precisely, this is true of # 1 , but t# 1 = ....

....# t F n 1 # t# 0 # t F n 1 has length greater than 1 2 # 2n . PROOF. The point k# bisects following the golden ratio one of the longest intervals determined by t# 0 # t k on the unit interval, and new lengths appear only when k is a Fibonacci number (see [3] for a full discussion) Since each bisection possibly reduces an interval by a factor of 1 # 1 = # 2 , all intervals determined by t# 0 # t F n 1 have at least length # 2n (the same holds for the other set of points) Hence, the statement can be easily obtained observing ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, second edition, 1997.

.... to the in memory sort phase External merge phase: if there are multiple runs in tmp table [check adjust memory] merge them to produce the final results for multiple pass merge, check adjust memory] before each merge The data in the buffers can be sorted using any internal sorting algorithm [3]. Therefore, many sort algorithms can be modified to produce memory adaptive versions. 3.2 Memory adjustment mechanism 3.2.1 In memory sort phase During this phase, the sort process collects data into buffers and sorts each buffer using some inmemory sort algorithm. When it runs out of free ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, second edition, 1973.

....(with their outputs) must be less than the square of the size of memory. This is a well known constraint on hashing; a similar constraint applies to sorting. These constraints can be overcome by recursive application of either partitioning (for hashing) or merging (for sorting) DKO 84, Knu73] Our implementation of Hybrid Cache in Illustra includes recursive partitioning to handle this situation. The number of input values v can be estimated by using stored statistics [SAC 79] or via sampling [HOT88, HNSS95] Unfortunately this estimation is subject to error, so we must consider ....

.... (Note that in the cases of grouping and duplicate elimination, these two analyses are identical, since the number of tuples in the output is equal to the number of values in the input) Graefe s analysis of the situation leads him to a modification of sorting via replacement selection [Knu73] which allows sort based grouping and duplicate elimination to utilize an amount of memory closer to the number of tuples in its output rather than that in its input. However, note that the number of tuples in the input and output are identical for caching: one output tuple is produced for ....

Donald Ervin Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley Publishing Co., 1973.

....space can be used to store keys. In the rest of this paper we shall consider only enhanced B trees as described above; however, we may sometimes use B trees for brevity. Hash: Hash indices can also benefit from cache optimization. The most common hashing method is chained bucket hashing [Knu73] In [GBC98] the authors use the cache line size as the bucket size and squeeze in as many key, RID pairs as possible. This can reduce the number of cache misses when scanning through the buckets. Hash indices are fast for searching only if the length of each bucket chain is small. This ....

Donald Ervin Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, USA, 1973.

....necessary, and of marginal value. I agree with this assessment, and believe that the CFG is the main cause of the first two afflictions. 2 Kicking the Control Flow Graph Habit Why are CFGs treated as a sine qua non of code optimization One reason is probably historical in nature. According to [Knu73], Goldstine and von Neumann began using flow charts to describe programs in 1946. This apparently set a precedent that subsequent workers followed with little question. Another more important reason for the prevalence of CFGs is that they describe control flow in a way that is easy to express with ....

Donald E. Knuth. Fundamental Algorithms. Volume 1 of The Art of Computer Programming, Addison-Wesley, 1973.

....IP addresses are interchanged, arising from the two way flow of traffic in TCP sessions. The hash function, and hence the modulus, must be chosen to avoid collisions in which a pair of packets that differ little by such an interchange are mapped onto the same remainder. Knuth (see x6.4 in [17]) formulates a condition for avoidance of such collisions, namely that q k Sigma a 6= 0 mod A for small a; k where q is a radix of the alphabet used to describe the header. Including q k = 2 32 in this criterion suppresses collisions of the type described above. Moduli obeying these ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1998.

....that is suitable for all collection type constructors. However, most modules implementing collections will de ne a type constructor named either Bag, Set, or Heap. Edison currently supports one implementation of sets (unbalanced binary search trees) four implementations of heaps (leftist heaps [5], skew heaps [13] splay heaps [9] and lazy pairing heaps [9] and one heap adaptor that maintains the minimum element of a heap separate from the rest of the heap. This heap adaptor is particularly useful in conjunction with splay heaps. 4.1 Observability Note that the equality relation de ....

Donald E. Knuth. Searching and Sorting, volume 3 of The Art of Computer Programming. Addison-Wesley, 1973.

....objects in such systems. The issue of locality of reference of heap allocated objects has been investigated extensively, although more so for garbage collected languages than for languages with explicit storage allocation. The poor reference locality characteristics of rst t storage allocation [13] has prompted the improved better t methods [18] that are now often used. Grunwald et al. surveyed existing malloc implementations with the goal of understanding what techniques they provide to support cache locality [11] Their conclusion was that the existing methods, including eliminating ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming, chapter 2, pages 435-451. Addison Wesley, Reading, MA, 2nd edition, 1973.

....two smaller formulae: a half plane becomes a leaf node, while 5 the binary operator becomes the root of a tree whose two children are subtrees formed by the subformulae. The binary tree representation is consolidated into an n ary form by merging adjacent Boolean operator nodes of the same type [11]. It might be possible to alter the tree construction algorithm to directly create the n ary form; however, since we have not attempted to optimize the preprocessing stage of this algorithm, the existing algorithm of Dobkin et al. was not modified. 3.2 Building the Edge Sequence Graph For a ....

Donald E. Knuth. FundamentalAlgorithms, volume 1 of The Art of ComputerProgramming. AddisonWesley, second edition, 1973.

....intervals in the tree nodes. The solution to the second group is called a merge sort aggregation 0 7 8 12 18 20 21 31 S S S S E E E E 1 232 0 0 11 Figure 2. Example of count Aggregation by Sorting Timestamps and Tags algorithm, which is similar to the classical merge sort algorithm [10]. This algorithm will be presented in Section 3.2. In this section, we assume that the memory is large enough to store the entire data structures required by each aggregation algorithm. In the rest of this paper, we use the count and max as the representatives of the two groups of operators, ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley Publishing Company, Reading, Mass., 1973.

....or finite algorithms, i.e algorithms that terminate in a finite number of steps. See [FNO87] for a survey of iterative algorithms for inverse eigenvalue problems. 3.1 Euclid Sturm Algorithms The integer Euclid s algorithm is one of the oldest nontrivial algorithms. For a full description see [Knu81] and for a survey see [Bar74] Definition 3.1 Let f and g be two polynomials of degree n and m respectively. We call a generalised Euclid algorithm ( Knu81] any algorithm based on the following recursion: ae pn = f and pn Gamma1 = g; ffi k Delta p k = q k Delta p k Gamma1 sign( k ) ....

....3.1 Euclid Sturm Algorithms The integer Euclid s algorithm is one of the oldest nontrivial algorithms. For a full description see [Knu81] and for a survey see [Bar74] Definition 3.1 Let f and g be two polynomials of degree n and m respectively. We call a generalised Euclid algorithm ([Knu81]) any algorithm based on the following recursion: ae pn = f and pn Gamma1 = g; ffi k Delta p k = q k Delta p k Gamma1 sign( k ) Delta fl k Delta p k Gamma2 for k = n; n Gamma 1; Delta Delta Delta (7) where (sign( k ) Delta fl k Delta p k Gamma2 ) is the remainder of the polynomial ....

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Massachusetts, second edition, 10 January 1981.

....O( f (n) is the time re7 quired for sorting n objects. Since comparison sorts such as merge sort or heap sort take O(nlogn) time to sort n objects, sweeping can also be done in O(nlogn) time, which is independent of the size of the heap. Or a distribution sort such as radix sort could be used [19], which has a O(nlogH) worst case time bound, where H is the size of the heap, but is much faster than comparison sorts at large n and in fact can be done in O(n) time on average. These worst case time bounds are asymptotically worse compared to the O(H) time bound of traditional sweeping when n ....

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming, chapter 5, pages 168--179. Addison-Wesley, second edition, 1998.

....a fit is done by finding a fitting memory hole, which is unused, and using it to satisfy the memory allocation request. There are a variety of ways to find a fit, among them being first fit, best fit, and worst fit. Boundary tags are typically used to support fast coalescing of free memory chunks [2]. Most simply, right before and after each memory chunk (both used and free) there is a header and a footer containing the size of the memory chunk, and which also hold other information such as whether the chunk is in use. These can be used to find out whether a memory chunk which is being ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming, chapter 2, pages 438--442. Addison-Wesley, third edition, 1997.

....work. 2 Background The issue of locality of reference of heap allocated objects has been investigated extensively, although more so for garbage collected languages than for languages with explicit storage allocation. The poor reference locality characteristics of first fit storage allocation [11] has prompted the improved better fit methods [17] that are now often used. As mentioned, in our own previous work, we looked at existing malloc implementations with an eye to what impact they had on cache locality [10] This previous work did not consider the more speculative issue of ....

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming, chapter 2, pages 435--451. Addison Wesley, Reading, MA, 2nd edition, 1973.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Mass., 2nd edition, 1981. 18

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, Mass., 1969.

....situations or multipass processes, analogous to the pipes of UNIX, when each coroutine transforms an input stream to an output stream; a sequence of such processes can be controlled in such a way that their intermediate data les need not be written in memory. See, for example, Section 1.4. 2 of [9]. The programming language SIMULA 67 [3] introduced support for coroutines in terms of fundamental operations named call, detach, and resume. Arne Wang and Ole Johan Dahl subsequently discovered [20] that an extremely simple computational model is able to accommodate these primitive operations. ....

....the dummy vertex 0 as a negative vertex that is parent to all the roots of the digraph. Thus we de ne prev [v] for all negative v as the largest element less than v in the set V k [f0g, where k = npro(parent(v) Notice that 9 is an element of both U 1 and U 8 in the example spider, so both gen [9](1) and gen [9] 8) will be invoked at various times. The former will invoke gen [6] 1) which will invoke gen [2] 1) the latter, however, will merely ip bit a 9 on and o , because prev [9] does not exceed 8. There is only one coroutine gen [9] its parameter l is reassigned each time gen [9] is ....

[Article contains additional citation context not shown here]

Donald E. Knuth, Fundamental Algorithms, Volume 1 of the Art of Computer Programming (Reading, Massachusetts: Addison{Wesley,

....transformation. Coroutines are analogous to subroutines, but they are symmetrical with respect to caller and callee: When coroutine A invokes coroutine B, the action of A is temporarily suspended and the action of B resumes where B had most recently left o . See, for example, Section 1.4. 2 of [3]. Programming languages such as SIMULA I [1] and its object oriented descendants have made it easy for programmers to specify families of parameterized coroutines that cooperate with each other in natural but nontrivial ways. In this paper we will study examples in which a compiler can transform ....

....a[k] 0; return true ; asleep0: while k n bump [k 1] do return true ; end. In this case, the driver program initiates action by involving bump [1] For example, the process plays out as follows when n = 3: 000 Initial state 123 001 01 1 bump [1] bump [2] true , bump [3] = false 12 1 11 bump [1] true , bump [2] false 1 111 bump [1] false 1 0 11 bump [1] true 1 2 0 0 1 bump [1] bump [2] true 1 2 3 0 0 0 2 3 000 bump [1] bump [2] bump [3] false 123 Each troll s action now depends on ....

[Article contains additional citation context not shown here]

Donald E. Knuth, Fundamental Algorithms, Volume 1 of the Art of Computer Programming (Reading, Massachusetts: Addison{Wesley, 1968.

....very rapidly, possibly in time O(1) i.e. the search time can be made independent of the amount of data to search . See section 2.1 for examples. The lower bound for general searching is, however, O(log 2 (N ) which can be approached by binary searching, see, e.g. CS202, 1994a ] or [ Knuth, 1973 ] Binary searching (as well as just about any kind of searching) can, however, only be performed on ordered sorted data and a significant part of a data organization to support searching is therefore the ordering sorting of the data before the actual searching. Section 2 describes some of the ....

....meaning that the array contains a matching piece of data, or an unused location is found, meaning that the array does not contain any matching data. If all matched should be found, then the linear search should always continue until an unused location is found. See, e.g. CS202, 1994c ] Knuth, 1973 ] Tucker, 1988 ] or [ Wiess, 1992 ] for a more thorough treatment of hashing into simple arrays . 2.1.1.1 Advantages ffl It can be extremely fast with a searching time complexity of O(1) if collisions are rare (best case) ffl It does not necessarily require much more memory than the ....

[Article contains additional citation context not shown here]

Donald Ervin Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley Publishing Company, 1973.

No context found.

Donald E. Knuth. Fundamental Algorithms. Volume 1 of The Art of Computer Programming, Addison-Wesley, Reading, Massachusetts, second edition, 1973.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, third edition, 1997.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, third edition, 1997.

No context found.

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, second edition, 1973.

No context found.

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, Reading, MA, USA, third edition, 1997.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley Longman, Reading, Massachusetts, U.S.A., second edition, 1998.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, 1969.

No context found.

Donald E. Knuth. Sorting and Searching, volume 2 of The Art of Computer Programming. Addison-Wesley, 1973.

No context found.

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, 1968.

No context found.

Donald E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, 1968.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, third edition, 1998.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley Publishing Company, second edition, 1981.

No context found.

Donald E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley Longman, 3rd edition, 1997.

No context found.

Donald E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, Mass., 1973.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC