Today I want to talk about the paradigms of programming, how they affect our success as designers of computer programs, how they should be taught, and how they should be embodied in our programming languages.
Robert W. Floyd
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Algorithm 489: the algorithm SELECT—for finding the ith smallest of n elements [M1]
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.
Expected time bounds for selection
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.
Bounded context syntactic analysis
Certain phase structure grammars define languages in which the phrasehood and structure of a substring of a sentence may be determined by consideration of only a bounded context of the substring. It is possible to determine, for any specified bound on the number of contextual characters considered, whether a given grammar is such a bounded context grammar. Such grammars are free from syntactic ambiguity. Syntactic analysis of sentences in a bounded context language may be performed by a standard process and requires a number of operations proportional to the length of sentence analyzed.
Bounded context grammars form models for most languages used in computer programming, and many methods of syntactic analysis, including analysis by operator precedence, are special cases of bounded context analysis.
On the nonexistence of a phrase structure grammar for ALGOL 60
ALGOL 60 is defined partly by formal mechanisms of phrase structure grammar, partly by informally stated restrictions. It is shown that no formal mechanisms of the type used are sufficient to define ALGOL 60.
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