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March 1975
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21 Reads
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356 Citations
Communications of the ACM
A new selection algorithm is presented which is shown to be very efficient on the average, both theoretically and practically. The number of comparisons used to select the ith smallest of n numbers is n + min(i,n-i) + o(n). A lower bound within 9 percent of the above formula is also derived.
March 1975
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35 Reads
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57 Citations
Communications of the ACM
SELECT will rearrange the values of array segment X[L: R] so that X[K] (for some given K; L ≤ K ≤ R) will contain the (K-L+1)-th smallest value, L ≤ I ≤ K will imply X[I] ≤ X[K], and K ≤ I ≤ R will imply X[I] ≥ X[K. While SELECT is thus functionally equivalent to Hoare's algorithm FIND [1], it is significantly faster on the average due to the effective use of sampling to determine the element T about which to partition X. The average time over 25 trials required by SELECT and FIND to determine the median of n elements was found experimentally to be: n 500 1000 5000 10000 SELECT 89 ms. 141 ms. 493 ms. 877 ms. FIND 104 ms. 197 ms. 1029 ms. 1964 ms. The arbitrary constants 600, .5, and .5 appearing in the algorithm minimize execution time on the particular machine used. SELECT has been shown to run in time asymptotically proportional to N + min (I, N-I), where N = L - R + 1 and I = K - L + 1. A lower bound on the running time within 9 percent of this value has also been proved [2]. Sites [3] has proved SELECT terminates.
January 1975
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16 Reads
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2 Citations
January 1975
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45 Reads
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28 Citations
Communications of the ACM
December 1973
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14 Reads
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13 Citations
This chapter discusses the Bose–Nelson sorting problem. This chapter illustrates a typical sorting network for four numbers; the network involves five comparators, shown as directed wires connecting two lines. Four numbers are input at the left, and as they move toward the right, each comparator causes an interchange of two numbers if necessary so that the larger number appears at the point of the arrow. At the right of the network, the numbers have been sorted into nondecreasing order from top to bottom; it is easy to verify that this would be the case no matter what numbers are input, because the first four comparators select the smallest and the largest elements and the final comparator ranks the middle two. Bose and Nelson gave an upper bound for S(n), and conjectured that their method actually gave S(n) exactly; but subsequent constructions have shown that their upper bound can be improved for all n > 8. However, the methods of proof used to establish the lower bounds on S(n) are quite unsatisfactory for larger values of n.
August 1973
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60 Reads
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1,269 Citations
Journal of Computer and System Sciences
The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm—PICK. Specifically, no more than 5.4305 n comparisons are ever required. This bound is improved for extreme values of i, and a new lower bound on the requisite number of comparisons is also proved.
January 1973
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1 Read
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2 Citations
Information Processing Letters
January 1973
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45 Reads
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27 Citations
August 1972
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8 Reads
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11 Citations
Journal of Computer and System Sciences
A special purpose theorem prover for establishing the validity of expressions over integer variables was developed as part of a program verifier. It is built around a powerful system for manipulating and simplifying integer expressions.
June 1972
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19 Reads
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60 Citations
Information Processing Letters
... As a solution to this problem, other faster sorting algorithms such as SELECT [54] or SORT [55] can be used as a solution to this problem. ...
March 1975
Communications of the ACM
... Then the resulting flow diagram cannot be expected to be more transparent than the original one" [2]. Indeed, in their original paper, Böhm and Jacopini conjectured, and Floyd and Knuth later proved [23], that the auxiliary variables form a critical part of this proof-there are some programs which cannot be 'structured' without additional variables. Lemma 4.3, originally proved in [8] and reproved in [10], [24], [1], has similarly been used as a theoretical justification for structuring control architectures as BTs. ...
June 1972
Information Processing Letters
... Formally, Theorem 2.4 follows from a result in [Schmid et al. 2021, Appendix D]; we will explain more in Section 4. A different program inexpressible in GKAT forms the main result of [Kozen and Tseng 2008]; part of the reason why this program is not expressible can already be found in [Ashcroft and Manna 1972;Knuth and Floyd 1971], although this is not formalized in KAT. ...
February 1971
Information Processing Letters
... But finding the real median is expensive; at it basics, it requires the array to be sorted in the first place. There exists better algorithms like the Floyd-Rivest algorithm [8] which has 3 a running time of O(n), specifically it requires about 1.5n comparisons. Unfortunately even those fast methods are not applicable for sorting since Quicksort needs to determine a median at every recursive stage in its call tree which would result in more than doubling the number of comparisons at each stage. ...
January 1975
... The first is to use textbook sorting networks [20]. For N = 4 (5 terms to sort) a bitonic sort has a depth of 5 which is optimal [21]. For 9 terms, the network from [22] has a depth of 7 which is proved optimal in [23]. ...
Reference:
Exact Fused Dot Product Add Operators
January 1973
... It follows that we can perform the steps 1. and 2. in O(|F (r T )|) time. To find S in the case of additive utilities, we only need to find ℓ i many largest valued agents in F (r T ), which can be done in linear time [13]. The distribution of agents in F (r T ) to parts in step 5. can also be straightforwardly done in time O(|F (r T )|). ...
August 1973
Journal of Computer and System Sciences
... Automated theorem provers and interactive proof checkers have been associated with verification from the beginning [King 1969;King and Floyd 1970]. Jones [1992] covers the history of verification research. ...
Reference:
Automated Deduction for Verification
January 1970
... the following corollary by setting B D 6k, R D d16 ln.1=ı/e. The query time follows since the median of T numbers can be found in linear time O.T /[8]. ...
Reference:
Forty years of frequent items
January 1972
... We give a conceptual review of transposing a matrix with vector instructions. The idea was introduced by [Flo72] under the context of permuting with pages and more recently by [War12, Section 7.3] for permuting bit matrices. It was also used in [NG21, BBCT22, BHK + 22b, CCHY24, Hwa24, HLY24] for vectorized polynomial multiplications. ...
January 1972
... That component of the EFFIGY system was, in fact, borrowed from a program verifier previously built by the author [8]. The capabilities and limitations of EFFIGY'S formula manipulation are inherited directly from that earlier system which is discussed in detail in [9]. ...
Reference:
Symbolic Execution and Program Testing
August 1972
Journal of Computer and System Sciences
