Algorithms and Complexity by wilf

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The choice of the splitter element in the Quicksort algorithm is done very simply: at random. We just choose, using our favorite random number generator, one of the entries of the given subarray, let’s call it T , and declare it to be the splitter. To repeat the parenthetical comment above, T is the value of the array entry that was chosen, not its position in the array. Once the value is selected, the position will be what it has to be, namely to the right of all smaller entries, and to the left of all larger entries.

Attributed to Nico Lomuto by Jon Bentley, CACM 27 (April 1984). {split} We will now prove the correctness of split. 1. Procedure split correctly splits the array x around the chosen value T . Proof: We claim that as the loop in lines 7 and 8 is repeatedly executed for j := lef t + 1 to right, the following three assertions will always be true just after each execution of lines 7, 8: (a) x[lef t] = T and (b) x[r] < T for all lef t < r ≤ i and (c) x[r] ≥ T for all i < r ≤ j Fig. 1 illustrates the claim.

For a fast and easy way to do this see A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, 2nd ed. (New York: Academic Press, 1978), chap. 6. 3 Recursive graph algorithms 4. Consider the array x = {2, 4, 1, 10, 5, 3, 9, 7, 8, 6} with lef t = 1 and right = 10. Suppose that the procedure split is called, and suppose the random integer L in step 1 happens to be 5. Carry out the complete split algorithm (not on a computer; use pencil and paper). Particularly, record the condition of the array x after each value of j is processed in the f or j = .

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