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Recently il Itas lee11 noticed I l l n l for semigroup conapulaItons nrld for s~lcrlintr, t*rrlrirrgrilnr ntesltes with rnttlfiple lroadcnstirig yicid Jnsler algoriibnas Lhan thew square co?~ntct*prrrls. T l r~ rorltribl~iion oj this paper is Lo y~rouide y c i n n o l l~r r eznrr11jlc 01 a fiinda7nentaI prolltrrl [or WILLCIA t h i s pbelior~ietlott occars. We show i l~n t the prabl~ul o j cotripuli~g lhe convex hull of n sel 4 11 ~o r t e d p o~l i t s in E6e plane 1 1 can be solved it1 O(t) log. 11) firire 011 a rectasgtalar OR d sq1111re mesh of stze fi x fi ruri in 0(7a i) time sn case the H P O~I I~S nre IHI-FIS 111 11 hin(lr*$ Y'IHIIJP, find Figure 1: A 4 x 5 mesh with multiple broadcnsting 1 = t n 0 1 7 1 6 I o g 3~t ) Itrtrr fill. c~l . f r r l pnitrlx rrr Ihc plarlc.

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Given a sequence of m items α1, α2,…, αm from a semigroup S with an associative operation ⊕, the semigroup computation problem involves computing α1 ⊕ α2 ⊕…⊕ αm. We consider the semigroup computation problem involving m (2≤m≤n) items on a mesh with multiple broadcasting of size [Formula: see text]. Our contribution is to present the first lower bound and the first time-optimal algorithm which apply to the entire range of m (2≤m≤n). Specifically, we show that any algorithm that solves the semigroup computation problem must take Ω( log m) time if [Formula: see text] and [Formula: see text] time for [Formula: see text]. We then show that our bound is tight by designing an algorithm whose running time matches the lower bound. These results unify and generalize all semigroup lower bounds and algorithms known to the authors.