tesseract.js

(function(exports){
tesseract.version = "1.0.1";
function tesseract_identity(d) {
  return d;
}
tesseract.permute = permute;

function permute(array, index) {
  for (var i = 0, n = index.length, copy = new Array(n); i < n; ++i) {
    copy[i] = array[index[i]];
  }
  return copy;
}
var bisect = tesseract.bisect = bisect_by(tesseract_identity);

bisect.by = bisect_by;

function bisect_by(f) {

  // Locate the insertion point for x in a to maintain sorted order. The
  // arguments lo and hi may be used to specify a subset of the array which
  // should be considered; by default the entire array is used. If x is already
  // present in a, the insertion point will be before (to the left of) any
  // existing entries. The return value is suitable for use as the first
  // argument to `array.splice` assuming that a is already sorted.
  //
  // The returned insertion point i partitions the array a into two halves so
  // that all v < x for v in a[lo:i] for the left side and all v >= x for v in
  // a[i:hi] for the right side.
  function bisectLeft(a, x, lo, hi) {
    while (lo < hi) {
      var mid = lo + hi >> 1;
      if (f(a[mid]) < x) lo = mid + 1;
      else hi = mid;
    }
    return lo;
  }

  // Similar to bisectLeft, but returns an insertion point which comes after (to
  // the right of) any existing entries of x in a.
  //
  // The returned insertion point i partitions the array into two halves so that
  // all v <= x for v in a[lo:i] for the left side and all v > x for v in
  // a[i:hi] for the right side.
  function bisectRight(a, x, lo, hi) {
    while (lo < hi) {
      var mid = lo + hi >> 1;
      if (x < f(a[mid])) hi = mid;
      else lo = mid + 1;
    }
    return lo;
  }

  bisectRight.right = bisectRight;
  bisectRight.left = bisectLeft;
  return bisectRight;
}
var heap = tesseract.heap = heap_by(tesseract_identity);

heap.by = heap_by;

function heap_by(f) {

  // Builds a binary heap within the specified array a[lo:hi]. The heap has the
  // property such that the parent a[lo+i] is always less than or equal to its
  // two children: a[lo+2*i+1] and a[lo+2*i+2].
  function heap(a, lo, hi) {
    var n = hi - lo,
        i = (n >>> 1) + 1;
    while (--i > 0) sift(a, i, n, lo);
    return a;
  }

  // Sorts the specified array a[lo:hi] in descending order, assuming it is
  // already a heap.
  function sort(a, lo, hi) {
    var n = hi - lo,
        t;
    while (--n > 0) t = a[lo], a[lo] = a[lo + n], a[lo + n] = t, sift(a, 1, n, lo);
    return a;
  }

  // Sifts the element a[lo+i-1] down the heap, where the heap is the contiguous
  // slice of array a[lo:lo+n]. This method can also be used to update the heap
  // incrementally, without incurring the full cost of reconstructing the heap.
  function sift(a, i, n, lo) {
    var d = a[--lo + i],
        x = f(d),
        child;
    while ((child = i << 1) <= n) {
      if (child < n && f(a[lo + child]) > f(a[lo + child + 1])) child++;
      if (x <= f(a[lo + child])) break;
      a[lo + i] = a[lo + child];
      i = child;
    }
    a[lo + i] = d;
  }

  heap.sort = sort;
  return heap;
}
var heapselect = tesseract.heapselect = heapselect_by(tesseract_identity);

heapselect.by = heapselect_by;

function heapselect_by(f) {
  var heap = heap_by(f);

  // Returns a new array containing the top k elements in the array a[lo:hi].
  // The returned array is not sorted, but maintains the heap property. If k is
  // greater than hi - lo, then fewer than k elements will be returned. The
  // order of elements in a is unchanged by this operation.
  function heapselect(a, lo, hi, k) {
    var queue = new Array(k = Math.min(hi - lo, k)),
        min,
        i,
        x,
        d;

    for (i = 0; i < k; ++i) queue[i] = a[lo++];
    heap(queue, 0, k);

    if (lo < hi) {
      min = f(queue[0]);
      do {
        if (x = f(d = a[lo]) > min) {
          queue[0] = d;
          min = f(heap(queue, 0, k)[0]);
        }
      } while (++lo < hi);
    }

    return queue;
  }

  return heapselect;
}
var insertionsort = tesseract.insertionsort = insertionsort_by(tesseract_identity);

insertionsort.by = insertionsort_by;

function insertionsort_by(f) {

  function insertionsort(a, lo, hi) {
    for (var i = lo + 1; i < hi; ++i) {
      for (var j = i, t = a[i], x = f(t); j > lo && f(a[j - 1]) > x; --j) {
        a[j] = a[j - 1];
      }
      a[j] = t;
    }
    return a;
  }

  return insertionsort;
}
// Algorithm designed by Vladimir Yaroslavskiy.
// Implementation based on the Dart project; see lib/dart/LICENSE for details.

var quicksort = tesseract.quicksort = quicksort_by(tesseract_identity);

quicksort.by = quicksort_by;

function quicksort_by(f) {
  var insertionsort = insertionsort_by(f);

  function sort(a, lo, hi) {
    return (hi - lo < quicksort_sizeThreshold
        ? insertionsort
        : quicksort)(a, lo, hi);
  }

  function quicksort(a, lo, hi) {

    // Compute the two pivots by looking at 5 elements.
    var sixth = (hi - lo) / 6 | 0,
        i1 = lo + sixth,
        i5 = hi - 1 - sixth,
        i3 = lo + hi - 1 >> 1,  // The midpoint.
        i2 = i3 - sixth,
        i4 = i3 + sixth;

    var e1 = a[i1], x1 = f(e1),
        e2 = a[i2], x2 = f(e2),
        e3 = a[i3], x3 = f(e3),
        e4 = a[i4], x4 = f(e4),
        e5 = a[i5], x5 = f(e5);

    // Sort the selected 5 elements using a sorting network.
    if (x1 > x2) t = e1, e1 = e2, e2 = t, t = x1, x1 = x2, x2 = t;
    if (x4 > x5) t = e4, e4 = e5, e5 = t, t = x4, x4 = x5, x5 = t;
    if (x1 > x3) t = e1, e1 = e3, e3 = t, t = x1, x1 = x3, x3 = t;
    if (x2 > x3) t = e2, e2 = e3, e3 = t, t = x2, x2 = x3, x3 = t;
    if (x1 > x4) t = e1, e1 = e4, e4 = t, t = x1, x1 = x4, x4 = t;
    if (x3 > x4) t = e3, e3 = e4, e4 = t, t = x3, x3 = x4, x4 = t;
    if (x2 > x5) t = e2, e2 = e5, e5 = t, t = x2, x2 = x5, x5 = t;
    if (x2 > x3) t = e2, e2 = e3, e3 = t, t = x2, x2 = x3, x3 = t;
    if (x4 > x5) t = e4, e4 = e5, e5 = t, t = x4, x4 = x5, x5 = t;

    var pivot1 = e2, pivotValue1 = x2,
        pivot2 = e4, pivotValue2 = x4;

    // e2 and e4 have been saved in the pivot variables. They will be written
    // back, once the partitioning is finished.
    a[i1] = e1;
    a[i2] = a[lo];
    a[i3] = e3;
    a[i4] = a[hi - 1];
    a[i5] = e5;

    var less = lo + 1,   // First element in the middle partition.
        great = hi - 2;  // Last element in the middle partition.

    // Note that for value comparison, <, <=, >= and > coerce to a primitive via
    // Object.prototype.valueOf; == and === do not, so in order to be consistent
    // with natural order (such as for Date objects), we must do two compares.
    var pivotsEqual = pivotValue1 <= pivotValue2 && pivotValue1 >= pivotValue2;
    if (pivotsEqual) {

      // Degenerated case where the partitioning becomes a dutch national flag
      // problem.
      //
      // [ |  < pivot  | == pivot | unpartitioned | > pivot  | ]
      //  ^             ^          ^             ^            ^
      // left         less         k           great         right
      //
      // a[left] and a[right] are undefined and are filled after the
      // partitioning.
      //
      // Invariants:
      //   1) for x in ]left, less[ : x < pivot.
      //   2) for x in [less, k[ : x == pivot.
      //   3) for x in ]great, right[ : x > pivot.
      for (var k = less; k <= great; ++k) {
        var ek = a[k], xk = f(ek);
        if (xk < pivotValue1) {
          if (k !== less) {
            a[k] = a[less];
            a[less] = ek;
          }
          ++less;
        } else if (xk > pivotValue1) {

          // Find the first element <= pivot in the range [k - 1, great] and
          // put [:ek:] there. We know that such an element must exist:
          // When k == less, then el3 (which is equal to pivot) lies in the
          // interval. Otherwise a[k - 1] == pivot and the search stops at k-1.
          // Note that in the latter case invariant 2 will be violated for a
          // short amount of time. The invariant will be restored when the
          // pivots are put into their final positions.
          while (true) {
            var greatValue = f(a[great]);
            if (greatValue > pivotValue1) {
              great--;
              // This is the only location in the while-loop where a new
              // iteration is started.
              continue;
            } else if (greatValue < pivotValue1) {
              // Triple exchange.
              a[k] = a[less];
              a[less++] = a[great];
              a[great--] = ek;
              break;
            } else {
              a[k] = a[great];
              a[great--] = ek;
              // Note: if great < k then we will exit the outer loop and fix
              // invariant 2 (which we just violated).
              break;
            }
          }
        }
      }
    } else {

      // We partition the list into three parts:
      //  1. < pivot1
      //  2. >= pivot1 && <= pivot2
      //  3. > pivot2
      //
      // During the loop we have:
      // [ | < pivot1 | >= pivot1 && <= pivot2 | unpartitioned  | > pivot2  | ]
      //  ^            ^                        ^              ^             ^
      // left         less                     k              great        right
      //
      // a[left] and a[right] are undefined and are filled after the
      // partitioning.
      //
      // Invariants:
      //   1. for x in ]left, less[ : x < pivot1
      //   2. for x in [less, k[ : pivot1 <= x && x <= pivot2
      //   3. for x in ]great, right[ : x > pivot2
      for (var k = less; k <= great; k++) {
        var ek = a[k], xk = f(ek);
        if (xk < pivotValue1) {
          if (k !== less) {
            a[k] = a[less];
            a[less] = ek;
          }
          ++less;
        } else {
          if (xk > pivotValue2) {
            while (true) {
              var greatValue = f(a[great]);
              if (greatValue > pivotValue2) {
                great--;
                if (great < k) break;
                // This is the only location inside the loop where a new
                // iteration is started.
                continue;
              } else {
                // a[great] <= pivot2.
                if (greatValue < pivotValue1) {
                  // Triple exchange.
                  a[k] = a[less];
                  a[less++] = a[great];
                  a[great--] = ek;
                } else {
                  // a[great] >= pivot1.
                  a[k] = a[great];
                  a[great--] = ek;
                }
                break;
              }
            }
          }
        }
      }
    }

    // Move pivots into their final positions.
    // We shrunk the list from both sides (a[left] and a[right] have
    // meaningless values in them) and now we move elements from the first
    // and third partition into these locations so that we can store the
    // pivots.
    a[lo] = a[less - 1];
    a[less - 1] = pivot1;
    a[hi - 1] = a[great + 1];
    a[great + 1] = pivot2;

    // The list is now partitioned into three partitions:
    // [ < pivot1   | >= pivot1 && <= pivot2   |  > pivot2   ]
    //  ^            ^                        ^             ^
    // left         less                     great        right

    // Recursive descent. (Don't include the pivot values.)
    sort(a, lo, less - 1);
    sort(a, great + 2, hi);

    if (pivotsEqual) {
      // All elements in the second partition are equal to the pivot. No
      // need to sort them.
      return a;
    }

    // In theory it should be enough to call _doSort recursively on the second
    // partition.
    // The Android source however removes the pivot elements from the recursive
    // call if the second partition is too large (more than 2/3 of the list).
    if (less < i1 && great > i5) {
      var lessValue, greatValue;
      while ((lessValue = f(a[less])) <= pivotValue1 && lessValue >= pivotValue1) ++less;
      while ((greatValue = f(a[great])) <= pivotValue2 && greatValue >= pivotValue2) --great;

      // Copy paste of the previous 3-way partitioning with adaptions.
      //
      // We partition the list into three parts:
      //  1. == pivot1
      //  2. > pivot1 && < pivot2
      //  3. == pivot2
      //
      // During the loop we have:
      // [ == pivot1 | > pivot1 && < pivot2 | unpartitioned  | == pivot2 ]
      //              ^                      ^              ^
      //            less                     k              great
      //
      // Invariants:
      //   1. for x in [ *, less[ : x == pivot1
      //   2. for x in [less, k[ : pivot1 < x && x < pivot2
      //   3. for x in ]great, * ] : x == pivot2
      for (var k = less; k <= great; k++) {
        var ek = a[k], xk = f(ek);
        if (xk <= pivotValue1 && xk >= pivotValue1) {
          if (k !== less) {
            a[k] = a[less];
            a[less] = ek;
          }
          less++;
        } else {
          if (xk <= pivotValue2 && xk >= pivotValue2) {
            while (true) {
              var greatValue = f(a[great]);
              if (greatValue <= pivotValue2 && greatValue >= pivotValue2) {
                great--;
                if (great < k) break;
                // This is the only location inside the loop where a new
                // iteration is started.
                continue;
              } else {
                // a[great] < pivot2.
                if (greatValue < pivotValue1) {
                  // Triple exchange.
                  a[k] = a[less];
                  a[less++] = a[great];
                  a[great--] = ek;
                } else {
                  // a[great] == pivot1.
                  a[k] = a[great];
                  a[great--] = ek;
                }
                break;
              }
            }
          }
        }
      }
    }

    // The second partition has now been cleared of pivot elements and looks
    // as follows:
    // [  *  |  > pivot1 && < pivot2  | * ]
    //        ^                      ^
    //       less                  great
    // Sort the second partition using recursive descent.

    // The second partition looks as follows:
    // [  *  |  >= pivot1 && <= pivot2  | * ]
    //        ^                        ^
    //       less                    great
    // Simply sort it by recursive descent.

    return sort(a, less, great + 1);
  }

  return sort;
}

var quicksort_sizeThreshold = 32;
var tesseract_array8 = tesseract_arrayUntyped,
    tesseract_array16 = tesseract_arrayUntyped,
    tesseract_array32 = tesseract_arrayUntyped,
    tesseract_arrayLengthen = tesseract_identity,
    tesseract_arrayWiden = tesseract_identity;

if (typeof Uint8Array !== "undefined") {
  tesseract_array8 = function(n) { return new Uint8Array(n); };
  tesseract_array16 = function(n) { return new Uint16Array(n); };
  tesseract_array32 = function(n) { return new Uint32Array(n); };

  tesseract_arrayLengthen = function(array, length) {
    var copy = new array.constructor(length);
    copy.set(array);
    return copy;
  };

  tesseract_arrayWiden = function(array, width) {
    var copy;
    switch (width) {
      case 16: copy = tesseract_array16(array.length); break;
      case 32: copy = tesseract_array32(array.length); break;
      default: throw new Error("invalid array width!");
    }
    copy.set(array);
    return copy;
  };
}

function tesseract_arrayUntyped(n) {
  return new Array(n);
}
function tesseract_filterExact(bisect, value) {
  return function(values) {
    var n = values.length;
    return [bisect.left(values, value, 0, n), bisect.right(values, value, 0, n)];
  };
}

function tesseract_filterRange(bisect, range) {
  var min = range[0],
      max = range[1];
  return function(values) {
    var n = values.length;
    return [bisect.left(values, min, 0, n), bisect.left(values, max, 0, n)];
  };
}

function tesseract_filterAll(values) {
  return [0, values.length];
}
function tesseract_null() {
  return null;
}
function tesseract_zero() {
  return 0;
}
function tesseract_reduceIncrement(p) {
  return p + 1;
}

function tesseract_reduceDecrement(p) {
  return p - 1;
}

function tesseract_reduceAdd(f) {
  return function(p, v) {
    return p + +f(v);
  };
}

function tesseract_reduceSubtract(f) {
  return function(p, v) {
    return p - f(v);
  };
}
exports.tesseract = tesseract;

function tesseract() {
  var tesseract = {
    add: add,
    dimension: dimension,
    groupAll: groupAll,
    size: size
  };

  var data = [], // the records
      n = 0, // the number of records; data.length
      m = 0, // number of dimensions in use
      M = 8, // number of dimensions that can fit in `filters`
      filters = tesseract_array8(0), // M bits per record; 1 is filtered out
      filterListeners = [], // when the filters change
      dataListeners = []; // when data is added

  // Adds the specified new records to this tesseract.
  function add(newData) {
    var n0 = n,
        n1 = newData.length;

    // If there's actually new data to add…
    // Merge the new data into the existing data.
    // Lengthen the filter bitset to handle the new records.
    // Notify listeners (dimensions and groups) that new data is available.
    if (n1) {
      data = data.concat(newData);
      filters = tesseract_arrayLengthen(filters, n += n1);
      dataListeners.forEach(function(l) { l(newData, n0, n1); });
    }

    return tesseract;
  }

  // Adds a new dimension with the specified value accessor function.
  function dimension(value) {
    var dimension = {
      filter: filter,
      filterExact: filterExact,
      filterRange: filterRange,
      filterAll: filterAll,
      top: top,
      group: group,
      groupAll: groupAll
    };

    var one = 1 << m++, // bit mask, e.g., 00001000
        zero = ~one, // inverted one, e.g., 11110111
        values, // sorted, cached array
        index, // value rank ↦ object id
        newValues, // temporary array storing newly-added values
        newIndex, // temporary array storing newly-added index
        sort = quicksort_by(function(i) { return newValues[i]; }),
        refilter = tesseract_filterAll, // for recomputing filter
        indexListeners = [], // when data is added
        lo0 = 0,
        hi0 = 0;

    // Updating a dimension is a two-stage process. First, we must update the
    // associated filters for the newly-added records. Once all dimensions have
    // updated their filters, the groups are notified to update.
    dataListeners.unshift(preAdd);
    dataListeners.push(postAdd);

    // Incorporate any existing data into this dimension, and make sure that the
    // filter bitset is wide enough to handle the new dimension.
    if (m > M) filters = tesseract_arrayWiden(filters, M <<= 1);
    preAdd(data, 0, n);
    postAdd(data, 0, n);

    // Incorporates the specified new records into this dimension.
    // This function is responsible for updating filters, values, and index.
    function preAdd(newData, n0, n1) {

      // Permute new values into natural order using a sorted index.
      newValues = newData.map(value);
      newIndex = sort(tesseract_range(n1), 0, n1);
      newValues = permute(newValues, newIndex);

      // Bisect newValues to determine which new records are selected.
      var bounds = refilter(newValues), lo1 = bounds[0], hi1 = bounds[1], i;
      for (i = 0; i < lo1; ++i) filters[newIndex[i] + n0] |= one;
      for (i = hi1; i < n1; ++i) filters[newIndex[i] + n0] |= one;

      // If this dimension previously had no data, then we don't need to do the
      // more expensive merge operation; use the new values and index as-is.
      if (!n0) {
        values = newValues;
        index = newIndex;
        lo0 = lo1;
        hi0 = hi1;
        return;
      }

      var oldValues = values,
          oldIndex = index,
          i0 = 0,
          i1 = 0;

      // Otherwise, create new arrays into which to merge new and old.
      values = new Array(n);
      index = tesseract_index(n, n);

      // Merge the old and new sorted values, and old and new index.
      for (i = 0; i0 < n0 && i1 < n1; ++i) {
        if (oldValues[i0] < newValues[i1]) {
          values[i] = oldValues[i0];
          index[i] = oldIndex[i0++];
        } else {
          values[i] = newValues[i1];
          index[i] = newIndex[i1++] + n0;
        }
      }

      // Add any remaining old values.
      for (; i0 < n0; ++i0, ++i) {
        values[i] = oldValues[i0];
        index[i] = oldIndex[i0];
      }

      // Add any remaining new values.
      for (; i1 < n1; ++i1, ++i) {
        values[i] = newValues[i1];
        index[i] = newIndex[i1] + n0;
      }

      // Bisect again to recompute lo0 and hi0.
      bounds = refilter(values), lo0 = bounds[0], hi0 = bounds[1];
    }

    // When all filters have updated, notify index listeners of the new values.
    function postAdd(newData, n0, n1) {
      indexListeners.forEach(function(l) { l(newValues, newIndex, n0, n1); });
      newValues = newIndex = null;
    }

    // Updates the selected values based on the specified bounds [lo, hi].
    // This implementation is used by all the public filter methods.
    function filterIndex(bounds) {
      var i,
          j,
          k,
          lo1 = bounds[0],
          hi1 = bounds[1],
          added = [],
          removed = [];

      // Fast incremental update based on previous lo index.
      if (lo1 < lo0) {
        for (i = lo1, j = Math.min(lo0, hi1); i < j; ++i) {
          filters[k = index[i]] ^= one;
          added.push(k);
        }
      } else if (lo1 > lo0) {
        for (i = lo0, j = Math.min(lo1, hi0); i < j; ++i) {
          filters[k = index[i]] ^= one;
          removed.push(k);
        }
      }

      // Fast incremental update based on previous hi index.
      if (hi1 > hi0) {
        for (i = Math.max(lo1, hi0), j = hi1; i < j; ++i) {
          filters[k = index[i]] ^= one;
          added.push(k);
        }
      } else if (hi1 < hi0) {
        for (i = Math.max(lo0, hi1), j = hi0; i < j; ++i) {
          filters[k = index[i]] ^= one;
          removed.push(k);
        }
      }

      lo0 = lo1;
      hi0 = hi1;
      filterListeners.forEach(function(l) { l(one, added, removed); });
      return dimension;
    }

    // Filters this dimension using the specified range, value, or null.
    // If the range is null, this is equivalent to filterAll.
    // If the range is an array, this is equivalent to filterRange.
    // Otherwise, this is equivalent to filterExact.
    function filter(range) {
      return range == null
          ? filterAll() : Array.isArray(range)
          ? filterRange(range)
          : filterExact(range);
    }

    // Filters this dimension to select the exact value.
    function filterExact(value) {
      return filterIndex((refilter = tesseract_filterExact(bisect, value))(values));
    }

    // Filters this dimension to select the specified range [lo, hi].
    // The lower bound is inclusive, and the upper bound is exclusive.
    function filterRange(range) {
      return filterIndex((refilter = tesseract_filterRange(bisect, range))(values));
    }

    // Clears any filters on this dimension.
    function filterAll() {
      return filterIndex((refilter = tesseract_filterAll)(values));
    }

    // Returns the top K selected records, based on this dimension's order.
    // Note: observes this dimension's filter, unlike group and groupAll.
    function top(k) {
      var array = [],
          i = hi0,
          j;

      while (--i >= lo0 && k > 0) {
        if (!filters[j = index[i]]) {
          array.push(data[j]);
          --k;
        }
      }

      return array;
    }

    // Adds a new group to this dimension, using the specified key function.
    function group(key) {
      var group = {
        top: top,
        all: all,
        reduce: reduce,
        reduceCount: reduceCount,
        reduceSum: reduceSum,
        order: order,
        orderNatural: orderNatural,
        size: size
      };

      var groups, // array of {key, value}
          groupIndex, // object id ↦ group id
          groupWidth = 8,
          groupCapacity = tesseract_capacity(groupWidth),
          k = 0, // cardinality
          select,
          heap,
          reduceAdd,
          reduceRemove,
          reduceInitial,
          update = tesseract_null,
          reset = tesseract_null,
          resetNeeded = true;

      if (arguments.length < 1) key = tesseract_identity;

      // The group listens to the tesseract for when any dimension changes, so
      // that it can update the associated reduce values. It must also listen to
      // the parent dimension for when data is added, and compute new keys.
      filterListeners.push(update);
      indexListeners.push(add);

      // Incorporate any existing data into the grouping.
      add(values, index, 0, n);

      // Incorporates the specified new values into this group.
      // This function is responsible for updating groups and groupIndex.
      function add(newValues, newIndex, n0, n1) {
        var oldGroups = groups,
            reIndex = tesseract_index(k, groupCapacity),
            add = reduceAdd,
            initial = reduceInitial,
            k0 = k, // old cardinality
            i0 = 0, // index of old group
            i1 = 0, // index of new record
            j, // object id
            g0, // old group
            x0, // old key
            x1, // new key
            g, // group to add
            x; // key of group to add

        // If a reset is needed, we don't need to update the reduce values.
        if (resetNeeded) add = initial = tesseract_null;

        // Reset the new groups (k is a lower bound).
        // Also, make sure that groupIndex exists and is long enough.
        groups = new Array(k), k = 0;
        groupIndex = k0 > 1 ? tesseract_arrayLengthen(groupIndex, n) : tesseract_index(n, groupCapacity);

        // Get the first old key (x0 of g0), if it exists.
        if (k0) x0 = (g0 = oldGroups[0]).key;

        // Find the first new key (x1), skipping NaN keys.
        while (i1 < n1 && !((x1 = key(newValues[i1])) >= x1)) ++i1;

        // While new keys remain…
        while (i1 < n1) {

          // Determine the lesser of the two current keys; new and old.
          // If there are no old keys remaining, then always add the new key.
          if (g0 && x0 <= x1) {
            g = g0, x = x0;

            // Record the new index of the old group.
            reIndex[i0] = k;

            // Retrieve the next old key.
            if (g0 = oldGroups[++i0]) x0 = g0.key;
          } else {
            g = {key: x1, value: initial()}, x = x1;
          }

          // Add the lesser group.
          groups[k] = g;

          // Add any selected records belonging to the added group, while
          // advancing the new key and populating the associated group index.
          while (!(x1 > x)) {
            groupIndex[j = newIndex[i1] + n0] = k;
            if (!(filters[j] & zero)) g.value = add(g.value, data[j]);
            if (++i1 >= n1) break;
            x1 = key(newValues[i1]);
          }

          groupIncrement();
        }

        // Add any remaining old groups that were greater than all new keys.
        // No incremental reduce is needed; these groups have no new records.
        // Also record the new index of the old group.
        while (i0 < k0) {
          groups[reIndex[i0] = k] = oldGroups[i0++];
          groupIncrement();
        }

        // If we added any new groups before any old groups,
        // update the group index of all the old records.
        if (k > i0) for (i0 = 0; i0 < n0; ++i0) {
          groupIndex[i0] = reIndex[groupIndex[i0]];
        }

        // Modify the update and reset behavior based on the cardinality.
        // If the cardinality is less than or equal to one, then the groupIndex
        // is not needed. If the cardinality is zero, then there are no records
        // and therefore no groups to update or reset. Note that we also must
        // change the registered listener to point to the new method.
        j = filterListeners.indexOf(update);
        if (k > 1) {
          update = updateMany;
          reset = resetMany;
        } else {
          if (k === 1) {
            update = updateOne;
            reset = resetOne;
          } else {
            update = tesseract_null;
            reset = tesseract_null;
          }
          groupIndex = null;
        }
        filterListeners[j] = update;

        // Count the number of added groups,
        // and widen the group index as needed.
        function groupIncrement() {
          if (++k === groupCapacity) {
            reIndex = tesseract_arrayWiden(reIndex, groupWidth <<= 1);
            groupIndex = tesseract_arrayWiden(groupIndex, groupWidth);
            groupCapacity = tesseract_capacity(groupWidth);
          }
        }
      }

      // Reduces the specified selected or deselected records.
      // This function is only used when the cardinality is greater than 1.
      function updateMany(filterOne, added, removed) {
        if (filterOne === one || resetNeeded) return;

        var i,
            k,
            n;

        // Add the added values.
        for (i = 0, n = added.length; i < n; ++i) {
          if (!(filters[k = added[i]] & zero)) {
            g = groups[groupIndex[k]];
            g.value = reduceAdd(g.value, data[k]);
          }
        }

        // Remove the removed values.
        for (i = 0, n = removed.length; i < n; ++i) {
          if ((filters[k = removed[i]] & zero) === filterOne) {
            g = groups[groupIndex[k]];
            g.value = reduceRemove(g.value, data[k]);
          }
        }
      }

      // Reduces the specified selected or deselected records.
      // This function is only used when the cardinality is 1.
      function updateOne(filterOne, added, removed) {
        if (filterOne === one || resetNeeded) return;

        var i,
            k,
            n,
            g = groups[0];

        // Add the added values.
        for (i = 0, n = added.length; i < n; ++i) {
          if (!(filters[k = added[i]] & zero)) {
            g.value = reduceAdd(g.value, data[k]);
          }
        }

        // Remove the removed values.
        for (i = 0, n = removed.length; i < n; ++i) {
          if ((filters[k = removed[i]] & zero) === filterOne) {
            g.value = reduceRemove(g.value, data[k]);
          }
        }
      }

      // Recomputes the group reduce values from scratch.
      // This function is only used when the cardinality is greater than 1.
      function resetMany() {
        var i,
            g;

        // Reset all group values.
        for (i = 0; i < k; ++i) {
          groups[i].value = reduceInitial();
        }

        // Add any selected records.
        for (i = 0; i < n; ++i) {
          if (!(filters[i] & zero)) {
            g = groups[groupIndex[i]];
            g.value = reduceAdd(g.value, data[i]);
          }
        }
      }

      // Recomputes the group reduce values from scratch.
      // This function is only used when the cardinality is 1.
      function resetOne() {
        var i,
            g = groups[0];

        // Reset the singleton group values.
        g.value = reduceInitial();

        // Add any selected records.
        for (i = 0; i < n; ++i) {
          if (!(filters[i] & zero)) {
            g.value = reduceAdd(g.value, data[i]);
          }
        }
      }

      // Returns the array of group values, in the dimension's natural order.
      function all() {
        if (resetNeeded) reset(), resetNeeded = false;
        return groups;
      }

      // Returns a new array containing the top K group values, in reduce order.
      function top(k) {
        var top = select(all(), 0, groups.length, k);
        return heap.sort(top, 0, top.length);
      }

      // Sets the reduce behavior for this group to use the specified functions.
      // This method lazily recomputes the reduce values, waiting until needed.
      function reduce(add, remove, initial) {
        reduceAdd = add;
        reduceRemove = remove;
        reduceInitial = initial;
        resetNeeded = true;
        return group;
      }

      // A convenience method for reducing by count.
      function reduceCount() {
        return reduce(tesseract_reduceIncrement, tesseract_reduceDecrement, tesseract_zero);
      }

      // A convenience method for reducing by sum(value).
      function reduceSum(value) {
        return reduce(tesseract_reduceAdd(value), tesseract_reduceSubtract(value), tesseract_zero);
      }

      // Sets the reduce order, using the specified accessor.
      function order(value) {
        select = heapselect_by(valueOf);
        heap = heap_by(valueOf);
        function valueOf(d) { return value(d.value); }
        return group;
      }

      // A convenience method for natural ordering by reduce value.
      function orderNatural() {
        return order(tesseract_identity);
      }

      // Returns the cardinality of this group, irrespective of any filters.
      function size() {
        return k;
      }

      return reduceCount().orderNatural();
    }

    // A convenience function for generating a singleton group.
    function groupAll() {
      var g = group(tesseract_null), all = g.all;
      delete g.all;
      delete g.top;
      delete g.order;
      delete g.orderNatural;
      delete g.size;
      g.value = function() { return all()[0].value; };
      return g;
    }

    return dimension;
  }

  // A convenience method for groupAll on a dummy dimension.
  // This implementation can be optimized since it is always cardinality 1.
  function groupAll() {
    var group = {
      reduce: reduce,
      reduceCount: reduceCount,
      reduceSum: reduceSum,
      value: value
    };

    var reduceValue,
        reduceAdd,
        reduceRemove,
        reduceInitial,
        resetNeeded = true;

    // The group listens to the tesseract for when any dimension changes, so
    // that it can update the reduce value. It must also listen to the parent
    // dimension for when data is added.
    filterListeners.push(update);
    dataListeners.push(add);

    // For consistency; actually a no-op since resetNeeded is true.
    add(data, 0, n);

    // Incorporates the specified new values into this group.
    function add(newData, n0, n1) {
      var i;

      if (resetNeeded) return;

      // Add the added values.
      for (i = n0; i < n; ++i) {
        if (!filters[i]) {
          reduceValue = reduceAdd(reduceValue, data[i]);
        }
      }
    }

    // Reduces the specified selected or deselected records.
    function update(filterOne, added, removed) {
      var i,
          k,
          n;

      if (resetNeeded) return;

      // Add the added values.
      for (i = 0, n = added.length; i < n; ++i) {
        if (!filters[k = added[i]]) {
          reduceValue = reduceAdd(reduceValue, data[k]);
        }
      }

      // Remove the removed values.
      for (i = 0, n = removed.length; i < n; ++i) {
        if (filters[k = removed[i]] === filterOne) {
          reduceValue = reduceRemove(reduceValue, data[k]);
        }
      }
    }

    // Recomputes the group reduce value from scratch.
    function reset() {
      var i;

      reduceValue = reduceInitial();

      for (i = 0; i < n; ++i) {
        if (!filters[i]) {
          reduceValue = reduceAdd(reduceValue, data[i]);
        }
      }
    }

    // Sets the reduce behavior for this group to use the specified functions.
    // This method lazily recomputes the reduce value, waiting until needed.
    function reduce(add, remove, initial) {
      reduceAdd = add;
      reduceRemove = remove;
      reduceInitial = initial;
      resetNeeded = true;
      return group;
    }

    // A convenience method for reducing by count.
    function reduceCount() {
      return reduce(tesseract_reduceIncrement, tesseract_reduceDecrement, tesseract_zero);
    }

    // A convenience method for reducing by sum(value).
    function reduceSum(value) {
      return reduce(tesseract_reduceAdd(value), tesseract_reduceSubtract(value), tesseract_zero);
    }

    // Returns the computed reduce value.
    function value() {
      if (resetNeeded) reset(), resetNeeded = false;
      return reduceValue;
    }

    return reduceCount();
  }

  // Returns the number of records in this tesseract, irrespective of any filters.
  function size() {
    return n;
  }

  return arguments.length
      ? add(arguments[0])
      : tesseract;
}

// Returns an array of size n, big enough to store ids up to m.
function tesseract_index(n, m) {
  return (m < 0x101
      ? tesseract_array8 : m < 0x10001
      ? tesseract_array16
      : tesseract_array32)(n);
}

// Constructs a new array of size n, with sequential values from 0 to n - 1.
function tesseract_range(n) {
  var range = tesseract_index(n, n);
  for (var i = -1; ++i < n;) range[i] = i;
  return range;
}

function tesseract_capacity(w) {
  return w === 8
      ? 0x100 : w === 16
      ? 0x10000
      : 0x100000000;
}
})(this);