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diff --git a/internal/ceres/parallel_utils.cc b/internal/ceres/parallel_utils.cc
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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2018 Google Inc. All rights reserved.
+// http://ceres-solver.org/
+//
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are met:
+//
+// * Redistributions of source code must retain the above copyright notice,
+// this list of conditions and the following disclaimer.
+// * Redistributions in binary form must reproduce the above copyright notice,
+// this list of conditions and the following disclaimer in the documentation
+// and/or other materials provided with the distribution.
+// * Neither the name of Google Inc. nor the names of its contributors may be
+// used to endorse or promote products derived from this software without
+// specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+// POSSIBILITY OF SUCH DAMAGE.
+//
+// Author: wjr@google.com (William Rucklidge)
+
+#include "ceres/parallel_utils.h"
+
+namespace ceres {
+namespace internal {
+
+void LinearIndexToUpperTriangularIndex(int k, int n, int* i, int* j) {
+ // This works by unfolding a rectangle into a triangle.
+ // Say n is even. 4 is a nice even number. The 10 i,j pairs that we
+ // want to produce are:
+ // 0,0 0,1 0,2 0,3
+ // 1,1 1,2 1,3
+ // 2,2 2,3
+ // 3,3
+ // This triangle can be folded into a 5x2 rectangle:
+ // 3,3 0,0 0,1 0,2 0,3
+ // 2,2 2,3 1,1 1,2 1,3
+
+ // If N is odd, say 5, then the 15 i,j pairs are:
+ // 0,0 0,1 0,2 0,3 0,4
+ // 1,1 1,2 1,3 1,4
+ // 2,2 2,3 2,3
+ // 3,3 3,4
+ // 4,4
+ // which folds to a 5x3 rectangle:
+ // 0,0 0,1 0,2 0,3 0,4
+ // 4,4 1,1 1,2 1,3 1,4
+ // 3,3 3,4 2,2 2,3 2,4
+
+ // All this function does is map the linear iteration position to a
+ // location in the rectangle and work out the appropriate (i, j) for that
+ // location.
+ if (n & 1) {
+ // Odd n. The tip of the triangle is on row 1.
+ int w = n; // Width of the rectangle to unfold
+ int i0 = k / w;
+ int j0 = k % w;
+ if (j0 >= i0) {
+ *i = i0;
+ *j = j0;
+ } else {
+ *i = n - i0;
+ *j = *i + j0;
+ }
+ } else {
+ // Even n. The tip of the triangle is on row 0, making it one wider.
+ int w = n + 1;
+ int i0 = k / w;
+ int j0 = k % w;
+ if (j0 > i0) {
+ *i = i0;
+ *j = j0 - 1;
+ } else {
+ *i = n - 1 - i0;
+ *j = *i + j0;
+ }
+ }
+}
+
+} // namespace internal
+} // namespace ceres