TAOCP 7.2.1.1 Exercise 12
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Section 7.2.1.1: Generating All n-Tuples
Exercise 12. ▶ [25] [25] The compositions of n are the sequences of positive integers that sum to n. For example, the compositions of 4 are 1111, 112, 121, 13, 211, 22, 31, and 4. An integer n has exactly 2n−1 compositions, corresponding to all subsets of the points {1, . . . , n−1} that might be used to break the interval (0 . . n) into integer-sized subintervals. a) Design a loopless algorithm to generate all compositions of n, representing each composition as a sequential array of integers s1s2 . . . sj. b) Similarly, design a loopless algorithm that represents the compositions implicitly in an array of pointers q0q1 . . . qt, where the elements of the composition are (q0 −q1)(q1 −q2) . . . (qt−1 −qt) and we have q0 = n, qt = 0. For example, the composition 211 would be represented under this scheme by the pointers q0 = 4, q1 = 2, q2 = 1, q3 = 0, and with t = 3.
7.2.1.1 GENERATING ALL n-TUPLES 309
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Solve time: 6m39s
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