Analysis of Hoare's FIND algorithm with Median-of-three partition

Abstract

Hoare's FIND algorithm can be used to select the jth element out of a file of n elements. It bears a remarkable similarity to Quicksort; in each pass of the algorithm, a pivot element is used to split the file into two subfiles, and recursively the algorithm proceeds with the subfile that contains the sought element. As in Quicksort, different strategies for selecting the pivot are reasonable. In this paper, we consider the Median‐of‐three version, where the pivot element is chosen as the median of a random sample of three elements. Establishing some hypergeometric differential equations, we find explicit formulae for both the average number of passes and comparisons. We compare these results with the corresponding ones for the basic partition strategy. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 143–156 (1997)

On Floyd and Rivest's SELECT algorithm

“…On the random sequences, the expected value of C avg is 2.75n + o(n) [15]. Twenty random instances of each size yield fairly accurate estimates, since the values of C avg in Table 5 are within 7% of 2.75n; Table 7 shows what happens when 1000 instances are used for each size.…”

On Floyd and Rivest's SELECT algorithm

“…On the random sequences, the expected value of C avg is 2.75n + o(n) [15]. Twenty random instances of each size yield fairly accurate estimates, since the values of C avg in Table 5 are within 7% of 2.75n; Table 7 shows what happens when 1000 instances are used for each size.…”

On the Variance of Quickselect

“…This reduces the probability of uneven partitions and there is a correspondDownloaded 05/10/18 to 54.149.104.195. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ing reduction in the average performance (see [4,7] and references therein).…”

Sesquickselect: One and a half pivots for cache-efficient selection

“…Like for Quicksort, better pivot selection methods are important in Quickselect. The widely used median-of-three version was analyzed in [2] and [20], and the generalization of using the median of any fixed size sample ("median-of-k") in [30] (expectation and variance for random ranks) and in [15] (for fixed ranks); refined limit laws for the case where k grows polynomially with n were recently derived in [37]. Choosing the order statistic of the sample depending on m/n was studied in [29] (w.r.t.…”

On Floyd and Rivest's SELECT algorithm

“…On the random sequences, the expected value of C avg is 2.75n + o(n) [15]. Twenty random instances of each size yield fairly accurate estimates, since the values of C avg in Table 5 are within 7% of 2.75n; Table 7 shows what happens when 1000 instances are used for each size.…”

On the Variance of Quickselect

“…This reduces the probability of uneven partitions and there is a correspondDownloaded 05/10/18 to 54.149.104.195. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ing reduction in the average performance (see [4,7] and references therein).…”

Sesquickselect: One and a half pivots for cache-efficient selection

“…Like for Quicksort, better pivot selection methods are important in Quickselect. The widely used median-of-three version was analyzed in [2] and [20], and the generalization of using the median of any fixed size sample ("median-of-k") in [30] (expectation and variance for random ranks) and in [15] (for fixed ranks); refined limit laws for the case where k grows polynomially with n were recently derived in [37]. Choosing the order statistic of the sample depending on m/n was studied in [29] (w.r.t.…”

On Floyd and Rivest's SELECT algorithm

“…On the random sequences, the expected value of C avg is 2.75n + o(n) [15]. Twenty random instances of each size yield fairly accurate estimates, since the values of C avg in Table 5 are within 7% of 2.75n; Table 7 shows what happens when 1000 instances are used for each size.…”

On the Variance of Quickselect

“…This reduces the probability of uneven partitions and there is a correspondDownloaded 05/10/18 to 54.149.104.195. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ing reduction in the average performance (see [4,7] and references therein).…”

Sesquickselect: One and a half pivots for cache-efficient selection

“…Like for Quicksort, better pivot selection methods are important in Quickselect. The widely used median-of-three version was analyzed in [2] and [20], and the generalization of using the median of any fixed size sample ("median-of-k") in [30] (expectation and variance for random ranks) and in [15] (for fixed ranks); refined limit laws for the case where k grows polynomially with n were recently derived in [37]. Choosing the order statistic of the sample depending on m/n was studied in [29] (w.r.t.…”