Seriation methods order a set of descriptions given some criterion (e.g.,
unimodality or minimum distance between similarity scores). Seriation is thus
inherently a problem of finding the optimal solution among a set of
permutations of objects. In this short technical note, we review the
combinatorial structure of the classical seriation problem, which seeks a
single solution out of a set of objects. We then extend those results to the
iterative frequency seriation approach introduced by Lipo (1997), which finds
optimal subsets of objects which each satisfy the unimodality criterion within
each subset. The number of possible solutions across multiple solution subsets
is larger than $n!$, which underscores the need to find new algorithms and
heuristics to assist in the deterministic frequency seriation problem.Comment: 8 pages, 2 figure

Lipo, Carl P.

Madsen, Mark E.

English

arXiv

Seriation methods order a set of descriptions given some criterion (e.g., unimodality or minimum distance between similarity scores). Seriation is thus inherently a problem of finding the optimal solution among a set of permutations of objects. In this short technical note, we review the combinatorial structure of the classical seriation problem, which seeks a single solution out of a set of objects. We then extend those results to the iterative frequency seriation approach introduced by Lipo et al. (1997), which finds optimal subsets of objects which each satisfy the unimodality criterion within each subset. The number of possible solutions across multiple solution subsets is larger than n!, which underscores the need to find new algorithms and heuristics to assist in the deterministic frequency seriation problem

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Binghamton UniversityThe Open Repository @ Binghamton (The ORB)Anthropology Faculty Scholarship Anthropology12-2014Combinatorial Structure of the DeterministicSeriation Method with Multiple Subset SolutionsMark E. MadsenUniversity of Washington - Seattle CampusCarl P. LipoBinghamton University--SUNY, clipo@binghamton.eduFollow this and additional works at: https://orb.binghamton.edu/anthropology_facPart of the Anthropology CommonsThis Data Set is brought to you for free and open access by the Anthropology at The Open Repository @ Binghamton (The ORB). It has been acceptedfor inclusion in Anthropology Faculty Scholarship by an authorized administrator of The Open Repository @ Binghamton (The ORB). For moreinformation, please contact ORB@binghamton.edu.Recommended CitationMadsen, Mark E. and Lipo, Carl P., "Combinatorial Structure of the Deterministic Seriation Method with Multiple Subset Solutions"(2014). Anthropology Faculty Scholarship. 17.https://orb.binghamton.edu/anthropology_fac/17Combinatorial Structure of the Deterministic SeriationMethod with Multiple Subset SolutionsMark E. MadsenDepartment of Anthropology, Box 353100, University of Washington, Seattle WA, 98195 USACarl P. LipoDepartment of Anthropology and IIRMES, 1250 Bellflower Blvd, California State University at LongBeach, Long Beach CA, 90840 USAAbstractSeriation methods order a set of descriptions given some criterion (e.g., unimodal-ity or minimum distance between similarity scores). Seriation is thus inherently aproblem of finding the optimal solution among a set of permutations of objects. Inthis short technical note, we review the combinatorial structure of the classical seri-ation problem, which seeks a single solution out of a set of objects. We then extendthose results to the iterative frequency seriation approach introduced by Lipo et al.(1997), which finds optimal subsets of objects which each satisfy the unimodalitycriterion within each subset. The number of possible solutions across multiple so-lution subsets is larger than n!, which underscores the need to find new algorithmsand heuristics to assist in the deterministic frequency seriation problem.Keywords: seriation, combinatorics1. Single Seriation CombinatoricsSeriation, whether employing class frequencies or simple occurrence to orderassemblages, yields solutions which are permutations of the set of assemblages. Be-cause we cannot determine the “polarity” of a seriation solution—which ends rep-resent early and late—from the class data alone, a unique seriation solution is thusEmail addresses: mark@madsenlab.org (Mark E. Madsen), Carl.Lipo@csulb.edu (Carl P. Lipo)URL: http://notebook.madsenlab.org (Mark E. Madsen), http://lipolab.org (Carl P.Lipo)Working Paper No. 1 January 3, 2013formally a pair of mirror-image permutations:fa; d; b; c; eg  fe; c; b; d; ag (1)This means that a set of n assemblages can yield n!=2 distinct solutions, regard-less of whether solutions are composed of ordered similarity matrices or“Fordian”frequency curves. With small numbers of assemblages, enumeration and testing ofall possible solutions is easy, even without parallel testing across many processors.The ability to test solutions by enumeration quickly breaks downwith only a modestnumber of assemblages. Table 1 gives the number of unique solutions for selectedproblem sizes between 4 and 100 assemblages, and estimates of processing time toenumerate and test all solutions, assuming a cluster of 64 cores, and 0.005 secondsper solution test.1 With 10 assemblages, we can test all solutions quickly enoughthat even a serial algorithm on a single core will be adequate to find the global bestsolution in a matter of hours, with parallelism improving this to real time responses.A typical characteristic of many combinatorial algorithms is that small changesin problem size can have massive changes in processing time. 13 assemblages willturn out to be the practical limit for direct enumeration, even given parallel process-ing with circa-2012 technology, with total processing time of nearly 3 days running64 cores at full capacity.2 Problems involving 14 and 15 assemblages reach the pointwhere large clusters requiremore than amonth and 19months respectively, to solve.Beyond 15 assemblages, a “combinatorial explosion” sets in, with 20 assemblages re-quiring more than 3 million years, before solution times quickly exceed the lifetimeof the universe.In short, top-down enumerative methods are feasible for small sets of assem-blages, and given widespread availability of multiple core computers, seriation pack-ages should employ enumeration for small problems, or to build and test smallerparts of larger seriation solutions.1These assumptions concerning per-trial processing time and parallelism are arbitrary but withinreach of social scientists given Amazon’s EC2 cloud computing infrastructure, without requiring for-mal “supercomputer” access. Modification by a factor of 10 has little effect on the results, perhapsshifting feasibility upward slightly before combinatorial explosion occurs.2Realistically, almost nobody would contemplate doing this, given the expense of the computingtime relative to the value of guaranteeing the optimal solution, but the hypothetical example demon-strates that such solutions are feasible.2N Seriation Solutions Seconds Years4 12 0.00094 3e-116 3.6e+02 0.028 8.9e-108 2e+04 1.6 5e-0810 1.8e+06 1.4e+02 4.5e-0612 2.4e+08 1.9e+04 0.0005913 3.1e+09 2.4e+05 0.007714 4.4e+10 3.4e+06 0.1115 6.5e+11 5.1e+07 1.616 1e+13 8.2e+08 2620 1.2e+18 9.5e+13 3e+0640 4.1e+47 3.2e+43 1e+3660 4.2e+81 3.3e+77 1e+7080 3.6e+118 2.8e+114 8.9e+106100 4.7e+157 3.6e+153 1.2e+146Table 1: Number of unique seriation solutions and parallel processing time for sets of assemblages4 < n < 100, testing solutions across 64 cores, assuming 5ms per trial2. Deterministic Seriation with Multiple Solution GroupsIn an earlier paper (Lipo et al., 1997), we introduced an iterative method forfinding deterministic solutions to the frequency seriation problem by partitioningassemblages into subsets, each of which meets the unimodal ordering principle,within tolerance limits governed by sample size. Lipo (2001) extended and refinedthe method in his dissertation research. Our initial work on the method employeda combination of automated calculations (e.g., bootstrap significance tests for pair-wise orderings), and manual sorting of assemblages into groups and specific posi-tions (using an Excel macro package available at http://lipolab.org/seriation.html). Figure 1 is an example of seriation with multiple solution groups, from Lipo’sdissertation research in the Lower Mississippi Valley.Our initial work suggests assemblages seriate together into groups reflectingvariation in the intensity of cultural transmission among assemblages, over their du-ration of accumulation. In most cases, solution groups tend to be spatiotemporallycompact, and form clusters when mapped on the landscape, although long-distanceconnections between past communities can also yield patterns which aremore com-plex and less cohensive when mapped. Madsen’s dissertation research is aimed attying the properties seriation solution groups to their causes in regional patterns of3Parkin PunctateBarton IncisedOld Town Red PolychromeAvenue KentIncisedRanch IncisedPaul AppliqueFortune NodedManly Punctated RhodesIncisedWalls EngravedHullEngravedPlace IncisedRed-on-BuffOwensPunctated 10-P-110-P-410-Q-111-N-411-N-1311-O-1011-O-811-O-411-N-1511-N-911-N-112-N-3/A&B11-N-1213-P-1012-N-213-N-2113-N-712-N-412-O-212-O-513-N-15/C,D,E12-P-112-P-213-P-1113-P-113-P-813-P-413-P-713-O-513-O-713-O-1113-O-1014-O-413-O-613-N-1413-N-513-N-4/B,C,D,E13-N-1613-N-1116-N-2/B-I14-N-114-O-216-N-615-N-614-O-1014-O-8Group 1Group 2AGroup 2bGroup 3Group 4Group 5Group 6Group 7Group 8Vernon Mound CarsonFigure 1: Example of a deterministic frequency seriation with assemblages partitioned into multiplesubsets or solution groups. From Lipo (2001), Figure 4.4.4interaction and the dynamics of specific cultural transmission models.# of Solution Groups (m) 20 40 603 5.8e+08 2e+18 7.1e+274 4.5e+10 5e+22 5.5e+346 4.3e+12 1.8e+28 6.8e+438 1.5e+13 3.2e+31 3.8e+4910 5.9e+12 2.4e+33 2.7e+5315 2.9e+34 2.2e+5820 1.6e+32 1.7e+5925 3.7e+5730 9.6e+53Table 2: Number of ways to form m subsets (seriation solutions) from 20, 40, and 60 assemblagesIn this section, the goal is to understand the complexity of the multiple seriationgroups problem, constructing reasonable upper bounds for a given problem size,even if some problems encountered in real analyses do not approach the worst case.From a combinatorial standpoint, seriation with multiple solution groups has thefollowing structure. We begin with n assemblages in total, and seek a solution orsolutions whereby we end up withm solution groups, wherem < n. Each solutionmust have at least one assemblage, and in practice will often have 3 or more (single-tons may indicate assemblages which simply do not “fit” with anything else in thedata set). The number of ways that n objects can be partitioned intom non-emptysubsets (or solution groups) is given by the Stirling numbers of the second kind,which are given by the recursion equation:nm= mn   1m+n   1m   1(2)Table 2 gives the number of ways to form a specific number of subsets (or seriationsolution groups) from sets of assemblages ranging from 20 to 60. Each column runsfrom 3 solution groups to half of the number of assemblages, since the number ofpossible subsets is maximized just before n=2 and declines thereafter (Figure 2).We can immediately see that there are an enormous number of possible subsetsfor any assemblage size. There are fewer subsets, of course, than complete permuta-tions of the set of assemblages since subsets are unordered (i.e., fnmg < n! for allm).However, in the multiple seriation group problem, the problem size is larger thanthe corresponding Stirling number because we do not know in advance how many51e+061e+141e+221e+300 10 20 30 40Number of SubsetsNumber of Unique SubsetsFigure 2: Number of Unique Solution Sets for 40 Assemblages When Partitioned Into m SolutionGroupsgroups (subsets) a set of assemblages will seriate into. Thus, the total number ofunique subsets which might contain the optimal solution is the total of the numberof subsets, across all subset sizes:nXi=1ni(3)This result is still smaller than the total permutations for a set of n assemblages. Forexample, given 40 assemblages, n! = 8:159  1047, whereas the total from Equation3 for 40 assemblages is 1:575  1035.Another factor to consider is that each of these unique subsets resulting from apartition of n assemblages into seriation groups is still unordered. For example, ifwe partition 10 assemblages into 3 solution groups, there are 9330 unique ways ofassigning the 10 assemblages to the 3 solution groups. Each group within a partitionwill have ni members, wherePni = n. The number of unique seriations for each ofthe 3 solution groups is ni!=2, but we cannot assume that solution groups will have6a balanced or equal number of assemblages (as Figure 1 does). Partitions such as:f1; 2; 3; 4; 5; 6gf7; 8gf9; 10gare common in seriating real assemblages (Lipo, 2001).Since the factorial function grows so quickly, the computational cost of deter-mining the correct permutation within a given seriation solution group is controlledby the size of the largest subset, especially if the other subsets are relatively small, asin the previous example. At worst, for a solution set withm solution groups,m   1solution groups will contain 1 assemblage each, and the last solution group will con-sist of the remaining n  m   1 assemblages. This means, of course, that the worstcase would involve consideration of on the order of (n m 1)! permutations withineach solution group, for each of the subsets given by Equation 3. This yields:nXm=1nm(n  m   1)! (4)Table 3 gives the total number of possible solutions for assemblages ranging from 4to 100, where solutions may fall into multiple seriation groups of any size.N Total Solutions Seconds Years4 15 0.0012 3.7e-116 4.7e+02 0.037 1.2e-098 5.2e+04 4 1.3e-0710 1.5e+07 1.1e+03 3.6e-0512 8.5e+09 6.6e+05 0.02113 2.6e+11 2e+07 0.6414 8.9e+12 7e+08 2215 3.5e+14 2.8e+10 8.7e+0216 1.6e+16 1.2e+12 3.9e+0420 1.7e+23 1.3e+19 4.2e+1140 9e+65 7e+61 2.2e+5460 5.1e+116 4e+112 1.3e+10580 5.1e+172 4e+168 1.3e+161100 4.4e+232 3.4e+228 1.1e+221Table 3: Number of total solutions with multiple seriation groups and processing time for sets ofassemblages 4 < n < 100, testing solutions across 64 cores73. DiscussionClearly, in the worst case, the combinatorial complexity of the multiple seri-ation groups problem is much worse than even the straight factorial case involvedin single solution permutations. The feasibility of parallelized enumerative methodsstill explodes after 13 assemblages, but much more steeply. The goal of a new algo-rithm for deterministic multiple group seriations is, therefore, to employ heuristicsto drastically reduce the size of the solution space. Vast amounts of the solutionspace involve partial orders which violate unimodality, but of course we cannot eas-ily identify those regions of solution space a priori without testing possibilities. Butgiven small partial solutions which do meet the seriation model, we can easily testsolutions which are “adjacent” to the partial solutions, suggesting that agglomerativeheuristics may be the best approach to finding a computationally feasible method.References CitedLipo, C., Madsen, M., Dunnell, R., Hunt, T., 1997. Population structure, culturaltransmission, and frequency seriation. Journal of Anthropological Archaeology16, 33.Lipo, C.P., 2001. Science, Style and the Study of Community Structure: An Exam-ple from the Central Mississippi River Valley. British Archaeological Reports,International Series, no. 918, Oxford.8

Combinatorial Structure of the Deterministic Seriation Method with Multiple Subset Solutions

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