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By R Cooke, K L. Lockett, J A Bellman

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54) above, when multiplied by n, is a m e m b e r of this broad class. D e f i n i t i o n . A real valued function g(n) is convex if A2g(n) > 0 for all n. This means that g(n + 2) - g(n + 1) >_ g(n + 1) - g(n), n > 0. 56) Let a(n) and b(n) be convex functions. Then the "minvolution" Lemma. 57) is also convex. Moreover if c(n) = a(k) + b ( n - k) then c(n + 1) = min(a(k) + b(n + 1 - k), a(k + 1) + b ( n - k)). (In other words, the location of the m i n i m u m does not shift drastically as n increases.

We assume without loss of generality that each key hashing to the leftmost cell has probe sequence 1, 2, 3, . . Then in rule S1 with probability p we hash to the leftmost cell, reuse the same hash sequence 1, 2, 3, . . , and occupy the leftmost empty cell. The score is the distance to the first free cell, and we have two scoregenerating functions for the two rules: Hmn(x) for SO and Gmn(x) for S1. Let's look first at Gmn(x): Gmn(X) -- (px + q(~m)Gm-l,n-l(x). " With probability p the key lands at location one and increases the monster by one.

In the special cases r -- 2 and r = 4, the constant k is known to be equal to vf2 and the golden ratio respectively. In other cases the constant can 30 RECURRENCE RELATIONS be estimated by iterating the recurrence and solving for k. The doubly exponential growth of the sequence makes such estimates converge rapidly; but it also makes the estimates inexact for further terms in the sequence. 2) Balanced Trees. 3], counts the number of balanced binary trees of height n. Y n + l - Y2n + 2 y n Y n - - X .

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