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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 131347, 3344]*) (*NotebookOutlinePosition[ 132400, 3378]*) (* CellTagsIndexPosition[ 132356, 3374]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[ "Nearest Neighbours Using k-d Trees"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This program calculates the set of nearest neighbours for each point in the \ input data set using the algorithm descibed in[1]. A k-d tree is built from \ the data set in O(n log n) time which allows the computation of the nearest \ neighbours for a particular query point in O(log n) time."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Reading In The Data"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "First the preliminaries - reading the data file. The following assumes a \ file format of\n \" ... , ... ,\", which allows multiple inputs and outputs:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["filename = \"Math/Gamma/hen100.asc\";"], "Input", AspectRatioFixed->True], Cell[TextData[ "filedata = Partition[ReadList[filename, Word, \n RecordLists -> \ True,\n WordSeparators -> {\" \"},\n RecordSeparators \ -> {\",\", \"\\n\"}], 2];\n(* WARNING USING THIS SYNTAX IS CONVENIENT BUT THE \ RESULTS\n * ARE STRINGS NOT NUMBERS. TO FIX THIS WE DO THE FOLLOWING\n *)\n\ data = Map[ToExpression, filedata];"], "Input", AspectRatioFixed->True], Cell[TextData[ "This package usefully provides all the statistical tools needed"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Needs[\"Statistics`DescriptiveStatistics`\"]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now the data is ready to use. It can be seen that the data consists of a \ list of input-output pairs. The nearest neighbours calculation is only to be \ performed on the input set, so the output set can be removed to simplify \ future processing. The following does this:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["data = Table[data[[i]][[1]], {i, 1, Length[data]}];"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Constructing the kd-Tree"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The next step is to build the k-d tree. This is done recursively using two \ basic steps at each stage. First the dimension (or key) with the largest \ spread of values over the records in this branch is found, and the median of \ the record values for this dimension is found. Second, the data is \ partitioned into two sets about the median previously found; the tree \ building then continues with each of the smaller sets. The recursion halts \ when the number of records to be processed falls below some preset value; the \ records become a terminal or leaf node in the tree."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In order to do this, the representation of the tree structure in \ Mathematica must be decided. At each non-terminal node, the dimension split \ on and the associated median value are recorded, together with the left and \ right branch of the recursively created subtrees. This leads to the \ following:\n\n Tree = Terminal Node |\n {Dimension \ Number, Partition Value, Left Tree, Right Tree}"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This is complicted somewhat by the need to return a list of {record index, \ distance} pairs as the nearest neighbours for each node. As the tree-building \ mechanism destroys the ordering of the nodes, the simplest way to retain each \ node's index in the original list is to tag it with it. Each data point then \ becomes a pair {value vector, index number}. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In order to manipulate these without cluttering the code to much, a few data \ access functions have been defined. Given a list of index-tagged data points, \ \"getValueFor\" returns the value for dimension (d) at position (i) in the \ list, \"getIndex\" returns the index for the record at position (i) and \ \"getPoint\" returns the data vector (without index) for the record at \ position (i)."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "getValueFor[in_, i_, d_] := in[[i]][[1]][[d]];\ngetIndex[in_, i_] := \ in[[i]][[2]];\ngetPoint[in_, i_] := in[[i]][[1]];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The following function is used to tag each point in the data set with its \ index number. This is called before the tree is to be built:"], "Text", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[ "addIndex[data_] := Table[{data[[i]], i}, {i,1,Length[data]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The following module is used to create the k-d tree. It needs to be passed \ the data set and the dimensionality of the data (the variable \"k\"). The \ local variable \"b\" controls the number of records in each terminal node - \ the default is 1, but the paper by Bentley[1] mentions that: \"... increasing \ the bucket size from one record per bucket considerably improves the \ performance of the search...\". The paper by White and Jain[2] mentions that \ spliting the data on the dimension with the highest variance gives better \ query performance, and this should also be faster to compute. It might be \ interesting to compares these approaches."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ As the data set is approximately halved at each stage, the time required \ to build the k-d tree is O(n log n).\ \>"], "Text", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True, ImageCacheValid->False], Cell[TextData[ "buildTree[k_, inData_] := \n Module[\n (* \"b\" is the number \ of records per bucket *)\n {bucketsize = 1, maxspread = 0, i, \ dimension,\n hi, low, thisHi, thisLow, medianval, pair, temp},\n \ \n If[Length[inData] <= bucketsize,\n (* This is a \ terminal node, so simply return the\n * list as it was passed \ *)\n Return[inData],\n (* Find the dimension \ with the largest spread *)\n For[i = 1, i <= k, i++,\n \ hi = -Infinity; low = Infinity;\n For \ [j = 1, j <= Length[inData], j++,\n temp = \ getValueFor[inData, j, i];\n If[hi < temp, hi \ = temp, ];\n If[low > temp, low = temp, ]\n \ ];\n If[hi - low >= maxspread,\n \ maxspread = hi - low;\n \ thisHi = hi; thisLow = low;\n dimension \ = i, ]\n ];\n \n (* Extract \ values at [dimension] and take median *)\n medianval = \ Median[Map[Part[Part[#,1],dimension]&,\n \ inData]];\n \ \n (* If the median is the same as the hi or low values\ \n * for the dimension, then repeated values must exist\n \ * in the data set. Return the set without splitting\n \ *)\n If[thisLow == medianval || thisHi == medianval,\n \ Return[inData], \n (* Else return the node \ in the tree with the correct\n * subtrees *)\n \ pair = split[dimension, medianval, inData];\n \ Return[{dimension, medianval,\n buildTree[k, \ First[pair]],\n buildTree[k, Last[pair]]}]\n \ ]\n ]\n];"], "Input", PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[ "The final conditional is necessary to deal correctly with data sets where \ one or vectors are equal. If a particular vecotr is repeated enough, it is \ possible that the median can be\nskewed to the maximum or minium of the range \ for the dimension. If this occurs, then the entire dataset is returned as a \ terminal node - no partitioning is possible as one branch will become empty, \ leading to an infinite loop."], "Text", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True], Cell[TextData[ "The following is used during the creation of the tree to split the records \ around a particular value (v) for a particular dimension (d). It may be \ quicker to do this \"in place\" in a programming language such as C."], "Text",\ Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[ "split[d_, v_, list_] := \n Module[\n {i, hi = {}, low = {}},\n \ For[i = 1, i <= Length[list], i++,\n If \ [getValueFor[list, i, d] >= v,\n hi = Append[hi, \ list[[i]]],\n low = Append[low, list[[i]]]\n \ ]\n ];\n Return[{low, hi}]\n];"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Finding the Nearest Neighbours"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Once the tree has been built, the task is then to search through it for the \ nearest neighbours of each point. The module takes parameters for the query \ vector to look for (query), the dimensionality of the data (k) and the \ previously constructed k-d tree (tree). Several \"global\" variables are \ used:\n nearest:This is best set of nearest neighbours found at any \ one point in the \n search. I\.05t is a pair of lists, one for \ the record indexes, the other for the\n distances to the query \ record - both lists are maintained in\n increasing distance \ from the query node.\n lowerBounds: This is the set of lower bounds \ currently in use to define the \n region being searched. There \ is one for each dimension.\n upperBounds: These are the equivilent \ set for the upper bounds on the\n search region"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The nearest neighbours are found by only searching those parts of the tree \ which contain data points \"close\" to the query point. The distance to the \ furthest nearest neighbour found so far and the size of the geometrical \ region defined by the tree branches traversed previously are compared to \ determine whether or not a branch should be examined. The comparison is \ carried out in the \"boundsOverlapBall\" procedure."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The search can terminate whenever the \"ballWithinBounds\" test returns \ True. Note that this test is not strictly required for the algorithm to work \ correctly - it is only used to indicate that the solution has been found."], "Text", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True], Cell[TextData[ "searchTree[query_, k_, tree_] :=\n Module[\n {medianval, \ dimension, temp, i, distance, maxloc, maxdist},\n \n \ Print[\"upper = \", upperBounds, \" lower = \", lowerBounds,\n \ \" tree = \", tree];\n If[Not[NumberQ[tree[[1]]]], \n \ (* Node is terminal - update nearest neighbours *)\n For[i = \ 1, i <= Length[tree], i++,\n (* Add points in bucket \ if closer than furthest\n * found so far, so long as \ this is not the query\n * record *)\n \ maxdist = Max[nearest[[2]]];\n distance = \ norm[query - tree[[i]][[1]]];\n If [(distance < \ maxdist) &&\n (query != getPoint[tree, i]),\n \ maxloc = Flatten[Position[nearest[[2]], \ maxdist]][[1]];\n nearest[[1, maxloc]] = \ getIndex[tree, i];\n nearest[[2, maxloc]] = \ distance, ]\n ];\n (* Sort nearest neighbours \ list *)\n nearest = Transpose[Sort[Transpose[nearest],\n \ (Part[#1, 2] < Part[#2, 2])&]];\n \ \n (* Check for termination and return *)\n \ If[ballWithinBounds[query, k],\n Print[\"**done**\"]; \ Return[ ],\n Return[ ]],\n ];\n \n \ (* Node not terminal - call searchTree recursively *) \n dimension \ = tree[[1]];\n medianval = tree[[2]];\n (* Recursive call on \ closer son *)\n If[query[[dimension]] <= medianval,\n \ temp = upperBounds[[dimension]];\n upperBounds[[dimension]] = \ medianval;\n searchTree[query, k, tree[[3]]];\n \ upperBounds[[dimension]] = temp, \n (* else *)\n temp = \ lowerBounds[[dimension]];\n lowerBounds[[dimension]] = \ medianval;\n searchTree[query, k, tree[[4]]];\n \ lowerBounds[[dimension]] = temp\n ];\n \n (* Recursive \ call on further son if necessary *)\n If[query[[dimension]] <= \ medianval,\n temp = lowerBounds[[dimension]];\n \ lowerBounds[[dimension]] = medianval;\n \ If[boundsOverlapBall[query, k],\n searchTree[query, k, \ tree[[4]]],\n ];\n lowerBounds[[dimension]] = \ temp,\n (* else *)\n temp = upperBounds[[dimension]];\n \ upperBounds[[dimension]] = medianval;\n \ If[boundsOverlapBall[query, k],\n searchTree[query, k, \ tree[[3]]],\n ];\n upperBounds[[dimension]] = \ temp\n ];\n \n (* See if we should terminate *)\n \ If[ballWithinBounds[query, k],\n Print[\"**done**\"]; Return[ \ ],\n Return[ ]] \n];\nnorm[x_] := N[Sqrt[x.x]];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The following two procedures are used to guide the search process. \ \"ballWithinBounds\" tests to see if the coordinate distance from the query \ record to the closer boundary along each dimension is less than the radius of \ the furthest nearest neighbour found so far. The test succeeds if all the \ coordinate distances are greater than the radius."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ballWithinBounds[query_, k_] := \n Module[\n {d},\n For \ [d = 1, d <= k, d++,\n (* \"Last[nearest[[2]]]\" is the \ furthest\n * nearest neighbour *)\n If \ [((query[[d]] - lowerBounds[[d]])^2 <=\n \ Last[nearest[[2]]]) ||\n ((query[[d]] - \ upperBounds[[d]])^2 <=\n Last[nearest[[2]]]),\n\ Return[False],\n ]\n ];\n \ Return[True]\n];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The \"boundsOverlapBall\" procedure is used on backtracking to decide if it \ is necessary to search down the opposite branch of the tree to the one \ previously searched. It determines whether the geometric boundaries of the \ subfile under consideration overlap a ball of radius equal to the distance of \ the furthest nearest neighbour found so far. The smallest distance between \ the query record and the bounded region is found - if this is less than the \ radius, then the branch of the subtree can be eliminated, as there is no \ overlap between the domain to be searched and the nearest neighbours."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "boundsOverlapBall[query_, k_] :=\n Module[\n {total = 0, d},\n \ (* \"Last[nearest[[2]]]\" is the furthest\n * nearest \ neighbour *)\n For[d = 1, d <= k, d++,\n If[query[[d]] \ < lowerBounds[[d]],\n total = total + (query[[d]] - \ lowerBounds[[d]])^2;\n If[total > \ (Last[nearest[[2]]])^2,\n Return[False],\n \ (* else nothing *)\n ],\n \ (* else *)\n If[query[[d]] > \ upperBounds[[d]],\n total = total + \ (query[[d]] - upperBounds[[d]])^2;\n If[total \ > (Last[nearest[[2]]])^2,\n \ Return[False],\n (* else nothing *)\n \ ],\n (* else nothing *)\n\ ]\n ]\n ];\n \ Return[True] \n];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The following module initialises the global variables to be used in the \ search. The number of nearest neighbours to be found is needed (numNearest), \ as is the query record (query), so that the dimensionality of the data cen be \ found. This initialisation must be repeated for each query point."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(* Initialisations for globals *)\ninitialise[query_, numNearest_] :=\n \ Module[\n {},\n nearest = {Table[-1, \ {numNearest}],Table[Infinity, {numNearest}]};\n upperBounds = \ Table[Infinity, {Length[query]}];\n lowerBounds = Table[-Infinity, \ {Length[query]}]\n];"], "Input", PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True], Cell[TextData[ "The following calculates the (numNearest) nearest neighbours of the input \ data entered (data). As an example, the 30 nearest neighbours of the points \ in \"data\" can be found by entering \"NNarray = nearestNeighbours[data, \ 30]\""], "Text", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True], Cell[TextData[ "nearestNeighbours[inData_, numNearest_] :=\n Module[\n {tree, \ query, indexedData, currentList = {}, i},\n indexedData = \ addIndex[inData];\n Print[\"Building Search Tree...\"];\n tree \ = buildTree[Length[inData[[1]]], indexedData];\n Print[\"Starting \ Nearest Neighbour Search...\"];\n For[i = 1, i <= Length[inData], i++,\ \n Print[\"Processing \", i];\n query = \ indexedData[[i]][[1]];\n initialise[query, numNearest];\n \ searchTree[query, Length[query], tree];\n \ currentList = Append[currentList, nearest]\n ];\n \ Return[currentList]\n];"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Experimental Results"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The algorithm above has been implemented in C, using long double arthimetic, \ togther with an naive O(n^2) algorithm to provide a comparison. Both were run \ initially on datafiles of sizes 100, 200, 400, 800, 1600 and 3200 \ 3-dimensional points. Further runs were carried out using bucket-sizes of 4 \ and 8 on the same set of points. The results are presented below (in a fixed \ height cell):"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ n2 = {{8,8,8}, {27,27,27}, {101,102,103}, {387,387,386}, {1490, 1490, 1490}, {5887, 5888, 5884}}; nlogn1 = {{8,8,8}, {16,16,16}, {34,34,34}, {67,67,67}, {145,145,146}, {290,289,290}}; nlogn4 = {{6,6,6}, {12,12,12}, {27,27,27}, {55,55,55}, {120,120,121}, {241,241,241}}; nlogn8 = {{6,6,6}, {12,12,12}, {26,26,26}, {53,53,53}, {116,116,117}, {234,233,233}}; xs = {100, 200, 400, 800, 1600, 3200}; n2bar = Transpose[{xs, Map[Mean, n2]}]; nlogn1bar = Transpose[{xs, Map[Mean, nlogn1]}]; nlogn4bar = Transpose[{xs, Map[Mean, nlogn4]}]; nlogn8bar = Transpose[{xs, Map[Mean, nlogn8]}]; p1 = ListPlot[n2bar, PlotJoined->True, DisplayFunction->Identity, AxesLabel->{\"(n)\", \"Time\"}]; p2 = ListPlot[nlogn1bar, PlotJoined->True, DisplayFunction->Identity, AxesLabel->{\"(n)\", \"Time\"}]; p3 = ListPlot[nlogn4bar, PlotJoined->True, DisplayFunction->Identity, AxesLabel->{\"(n)\", 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It can be seen that the kd-tree algorithm does indeed have \ O(n log n) performance. At 3200 points, the O(n^2) algorithm took on average \ 1 hour 38 minutes to complete, whereas the kd-tree took just under 5 \ minutes."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The second graph compares different setting for the bucket size on the same \ set of points. The top line is bucket size = 1, the middle is bucket size = \ 4, the lowest line if bucket size = 8. It can be seen that increasing the \ size does give a performance increase, particularly noticable in the jump \ from 1 to 4. The increase from 4 to 8 is less marked."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In order to determine how the algorithm scales with increasing dimension, a \ series of tests were performed where the dimension ranged from 2 to 40 and \ the number of points ranged from 100 to 2000 (ie. 100 tests in all). All \ tests were perfomed on an\nAlpha workstation. The raw results are in the \ following fixed height cell:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ rawdata = {{2, 200, 0}, {2, 400, 0}, {2, 600, 0}, {2, 800, 0}, {2, \ 1000, 0}, {2, 1200, 0}, {2, 1400, 0}, {2, 1600, 0}, {2, 1800, 1}, {2, 2000, 1}, {4, 200, 0},{4, \ 400, 0}, {4, 600, 0}, {4, 800, 0}, {4, 1000, 1}, {4, 1200, 1}, {4, 1400, 2}, \ {4, 1600, 2}, {4, 1800, 3}, {4, 2000, 3}, {6, 200, 0}, {6, 400, 0}, {6, 600, \ 1}, {6, 800, 2}, {6, 1000, 3}, {6, 1200, 5}, {6, 1400, 6}, {6, 1600, 7}, {6, \ 1800, 9}, {6, 2000, 10}, {8, 200, 0}, {8, 400, 0}, {8, 600, 2}, {8, 800, 5}, \ {8, 1000, 7}, {8, 1200, 11}, {8, 1400, 14}, {8, 1600, 18}, {8, 1800, 21}, {8, \ 2000, 25}, {10, 200, 0}, {10, 400, 1}, {10, 600, 4}, {10, 800, 7}, {10, 1000, \ 11}, {10, 1200, 18}, {10, 1400, 23}, {10, 1600, 29}, {10, 1800, 36}, {10, 2000, \ 43}, {12, 200, 0}, {12, 400, 1}, {12, 600, 4}, {12, 800, 9}, {12, 1000, 13}, \ {12, 1200, 22}, {12, 1400, 29}, {12, 1600, 39}, {12, 1800, 44}, {12, 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Metetra .528 .629 .894 r .56364 .47964 .57885 .50173 .64149 .50105 .62662 .48076 Metetra .488 .633 .918 r .62662 .48076 .64149 .50105 .70637 .50036 .69216 .48527 Metetra .525 .725 .965 r .69216 .48527 .70637 .50036 .774 .50417 .75968 .48085 Metetra .544 .637 .893 r .75968 .48085 .774 .50417 .84518 .51627 .82994 .47974 Metetra .557 .551 .803 r .82994 .47974 .84518 .51627 .91879 .52177 .90318 .48095 Metetra .635 .721 .909 r .31326 .5086 .33008 .52535 .38628 .51048 .36975 .49443 Metetra .609 .726 .928 r .36975 .49443 .38628 .51048 .44377 .49944 .42758 .48305 Metetra .609 .726 .928 r .42758 .48305 .44377 .49944 .50285 .48915 .48704 .47136 Metetra .596 .707 .922 r .48704 .47136 .50285 .48915 .56364 .47964 .54821 .46149 Metetra .532 .681 .933 r .54821 .46149 .56364 .47964 .62662 .48076 .61144 .45899 Metetra .532 .642 .904 r .61144 .45899 .62662 .48076 .69216 .48527 .67706 .45862 Metetra .54 .603 .866 r .67706 .45862 .69216 .48527 .75968 .48085 .74526 .45939 Metetra .582 .679 .907 r .74526 .45939 .75968 .48085 .82994 .47974 .8153 .45091 Metetra .483 .562 .86 r .8153 .45091 .82994 .47974 .90318 .48095 .89007 .4622 Metetra .642 .736 .917 r .29605 .49258 .31326 .5086 .36975 .49443 .35294 .47701 Metetra .612 .714 .917 r .35294 .47701 .36975 .49443 .42758 .48305 .41111 .46528 Metetra .612 .714 .917 r .41111 .46528 .42758 .48305 .48704 .47136 .47092 .45322 Metetra .604 .711 .919 r .47092 .45322 .48704 .47136 .54821 .46149 .53246 .44189 Metetra .548 .674 .921 r .53246 .44189 .54821 .46149 .61144 .45899 .59601 .43791 Metetra .552 .664 .911 r .59601 .43791 .61144 .45899 .67706 .45862 .66183 .43379 Metetra .525 .616 .885 r .66183 .43379 .67706 .45862 .74526 .45939 .73041 .43521 Metetra .58 .655 .887 r .73041 .43521 .74526 .45939 .8153 .45091 .80095 .4274 Metetra .541 .646 .902 r .80095 .4274 .8153 .45091 .89007 .4622 .87464 .42521 Metetra .648 .751 .924 r .27842 .47728 .29605 .49258 .35294 .47701 .33574 .4603 Metetra .618 .729 .925 r .33574 .4603 .35294 .47701 .41111 .46528 .3943 .44715 Metetra .604 .711 .92 r .3943 .44715 .41111 .46528 .47092 .45322 .45446 .43578 Metetra .61 .726 .928 r .45446 .43578 .47092 .45322 .53246 .44189 .5164 .42299 Metetra .542 .685 .932 r .5164 .42299 .53246 .44189 .59601 .43791 .58033 .41864 Metetra .561 .693 .929 r .58033 .41864 .59601 .43791 .66183 .43379 .64647 .41188 Metetra .527 .652 .914 r .64647 .41188 .66183 .43379 .73041 .43521 .71528 .41059 Metetra .601 .664 .883 r .71528 .41059 .73041 .43521 .80095 .4274 .78592 .3988 Metetra .554 .61 .864 r .78592 .3988 .80095 .4274 .87464 .42521 .85994 .39602 Metetra .642 .723 .907 r .26059 .45956 .27842 .47728 .33574 .4603 .31819 .44325 Metetra .626 .731 .922 r .31819 .44325 .33574 .4603 .3943 .44715 .37717 .42866 Metetra .612 .714 .917 r .37717 .42866 .3943 .44715 .45446 .43578 .4377 .41583 Metetra .614 .702 .907 r .4377 .41583 .45446 .43578 .5164 .42299 .50002 .40262 Metetra .565 .682 .918 r .50002 .40262 .5164 .42299 .58033 .41864 .56427 .39564 Metetra .564 .657 .899 r .56427 .39564 .58033 .41864 .64647 .41188 .6308 .38954 Metetra .576 .675 .907 r .6308 .38954 .64647 .41188 .71528 .41059 .6996 .38094 Metetra .58 .615 .852 r .6996 .38094 .71528 .41059 .78592 .3988 .77092 .3732 Metetra .56 .644 .89 r .77092 .3732 .78592 .3988 .85994 .39602 .84511 .36752 Metetra .649 .738 .914 r .24232 .44255 .26059 .45956 .31819 .44325 .30035 .42479 Metetra .612 .714 .918 r .30035 .42479 .31819 .44325 .37717 .42866 .35957 .41195 Metetra .633 .746 .93 r .35957 .41195 .37717 .42866 .4377 .41583 .42062 .39547 Metetra .614 .702 .907 r .42062 .39547 .4377 .41583 .50002 .40262 .4833 .38183 Metetra .574 .686 .917 r .4833 .38183 .50002 .40262 .56427 .39564 .54792 .37331 Metetra .577 .675 .906 r .54792 .37331 .56427 .39564 .6308 .38954 .61475 .3645 Metetra .581 .654 .887 r .61475 .3645 .6308 .38954 .6996 .38094 .68392 .35538 Metetra .56 .644 .89 r .68392 .35538 .6996 .38094 .77092 .3732 .75582 .34942 Metetra .584 .679 .906 r .75582 .34942 .77092 .3732 .84511 .36752 .83001 .33851 Metetra .656 .752 .921 r .22357 .42625 .24232 .44255 .30035 .42479 .28208 .40702 Metetra .634 .733 .92 r .28208 .40702 .30035 .42479 .35957 .41195 .34184 .39055 Metetra .622 .693 .895 r .34184 .39055 .35957 .41195 .42062 .39547 .40315 .3758 Metetra .62 .716 .915 r .40315 .3758 .42062 .39547 .4833 .38183 .46625 .36062 Metetra .582 .689 .915 r .46625 .36062 .4833 .38183 .54792 .37331 .53124 .35054 Metetra .569 .671 .908 r .53124 .35054 .54792 .37331 .61475 .3645 .59848 .34237 Metetra .59 .693 .914 r .59848 .34237 .61475 .3645 .68392 .35538 .66799 .33048 Metetra .593 .671 .894 r .66799 .33048 .68392 .35538 .75582 .34942 .73989 .31818 Metetra .567 .609 .854 r .73989 .31818 .75582 .34942 .83001 .33851 .81487 .31132 Metetra .656 .74 .911 r .20453 .40853 .22357 .42625 .28208 .40702 .26341 .38888 Metetra .634 .733 .92 r .26341 .38888 .28208 .40702 .34184 .39055 .32356 .37199 Metetra .627 .731 .922 r .32356 .37199 .34184 .39055 .40315 .3758 .38532 .35572 Metetra .62 .716 .915 r .38532 .35572 .40315 .3758 .46625 .36062 .44883 .34008 Metetra .588 .704 .923 r .44883 .34008 .46625 .36062 .53124 .35054 .51422 .32843 Metetra .548 .674 .921 r .51422 .32843 .53124 .35054 .59848 .34237 .58192 .32205 Metetra .625 .732 .924 r .58192 .32205 .59848 .34237 .66799 .33048 .6517 .30394 Metetra .559 .644 .89 r .6517 .30394 .66799 .33048 .73989 .31818 .72441 .29686 Metetra .592 .72 .932 r .72441 .29686 .73989 .31818 .81487 .31132 .79933 .28242 Metetra .656 .728 .902 r .1852 .38937 .20453 .40853 .26341 .38888 .24445 .36928 Metetra .635 .721 .91 r .24445 .36928 .26341 .38888 .32356 .37199 .30497 .35194 Metetra .635 .721 .91 r .30497 .35194 .32356 .37199 .38532 .35572 .36717 .33411 Metetra .621 .705 .905 r .36717 .33411 .38532 .35572 .44883 .34008 .43106 .31799 Metetra .598 .696 .911 r .43106 .31799 .44883 .34008 .51422 .32843 .49686 .30473 Metetra .607 .688 .9 r .49686 .30473 .51422 .32843 .58192 .32205 .56481 .28991 Metetra .578 .607 .846 r .56481 .28991 .58192 .32205 .6517 .30394 .63522 .28031 Metetra 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