double RangeSearchRules<MetricType, TreeType>::Score(const size_t queryIndex,
TreeType& referenceNode)
{
// We must get the minimum and maximum distances and store them in this
// object.
math::Range distances;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// In this situation, we calculate the base case. So we should check to be
// sure we haven't already done that.
double baseCase;
if (tree::TreeTraits<TreeType>::HasSelfChildren &&
(referenceNode.Parent() != NULL) &&
(referenceNode.Point(0) == referenceNode.Parent()->Point(0)))
{
// If the tree has self-children and this is a self-child, the base case
// was already calculated.
baseCase = referenceNode.Parent()->Stat().LastDistance();
lastQueryIndex = queryIndex;
lastReferenceIndex = referenceNode.Point(0);
}
else
{
// We must calculate the base case by hand.
baseCase = BaseCase(queryIndex, referenceNode.Point(0));
}
// This may be possibly loose for non-ball bound trees.
distances.Lo() = baseCase - referenceNode.FurthestDescendantDistance();
distances.Hi() = baseCase + referenceNode.FurthestDescendantDistance();
// Update last distance calculation.
referenceNode.Stat().LastDistance() = baseCase;
}
else
{
distances = referenceNode.RangeDistance(querySet.unsafe_col(queryIndex));
}
// If the ranges do not overlap, prune this node.
if (!distances.Contains(range))
return DBL_MAX;
// In this case, all of the points in the reference node will be part of the
// results.
if ((distances.Lo() >= range.Lo()) && (distances.Hi() <= range.Hi()))
{
AddResult(queryIndex, referenceNode);
return DBL_MAX; // We don't need to go any deeper.
}
// Otherwise the score doesn't matter. Recursion order is irrelevant in
// range search.
return 0.0;
}
long RecComputeDegree(long u, const ZZ_pEX& h, const ZZ_pEXModulus& F,
FacVec& fvec)
{
if (IsX(h)) return 1;
if (fvec[u].link == -1) return BaseCase(h, fvec[u].q, fvec[u].a, F);
ZZ_pEX h1, h2;
long q1, q2, r1, r2;
q1 = fvec[fvec[u].link].val;
q2 = fvec[fvec[u].link+1].val;
TandemPowerCompose(h1, h2, h, q1, q2, F);
r1 = RecComputeDegree(fvec[u].link, h2, F, fvec);
r2 = RecComputeDegree(fvec[u].link+1, h1, F, fvec);
return r1*r2;
}
void LSHSearch<SortPolicy>::
Search(const size_t k,
arma::Mat<size_t>& resultingNeighbors,
arma::mat& distances,
const size_t numTablesToSearch)
{
// Set the size of the neighbor and distance matrices.
resultingNeighbors.set_size(k, querySet.n_cols);
distances.set_size(k, querySet.n_cols);
distances.fill(SortPolicy::WorstDistance());
resultingNeighbors.fill(referenceSet.n_cols);
size_t avgIndicesReturned = 0;
Timer::Start("computing_neighbors");
// Go through every query point sequentially.
for (size_t i = 0; i < querySet.n_cols; i++)
{
// Hash every query into every hash table and eventually into the
// 'secondHashTable' to obtain the neighbor candidates.
arma::uvec refIndices;
ReturnIndicesFromTable(i, refIndices, numTablesToSearch);
// An informative book-keeping for the number of neighbor candidates
// returned on average.
avgIndicesReturned += refIndices.n_elem;
// Sequentially go through all the candidates and save the best 'k'
// candidates.
for (size_t j = 0; j < refIndices.n_elem; j++)
BaseCase(distances, resultingNeighbors, i, (size_t) refIndices[j]);
}
Timer::Stop("computing_neighbors");
distanceEvaluations += avgIndicesReturned;
avgIndicesReturned /= querySet.n_cols;
Log::Info << avgIndicesReturned << " distinct indices returned on average." <<
std::endl;
}
inline double NeighborSearchRules<SortPolicy, MetricType, TreeType>::Score(
const size_t queryIndex,
TreeType& referenceNode)
{
++scores; // Count number of Score() calls.
double distance;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// The first point in the tree is the centroid. So we can then calculate
// the base case between that and the query point.
double baseCase = -1.0;
if (tree::TreeTraits<TreeType>::HasSelfChildren)
{
// If the parent node is the same, then we have already calculated the
// base case.
if ((referenceNode.Parent() != NULL) &&
(referenceNode.Point(0) == referenceNode.Parent()->Point(0)))
baseCase = referenceNode.Parent()->Stat().LastDistance();
else
baseCase = BaseCase(queryIndex, referenceNode.Point(0));
// Save this evaluation.
referenceNode.Stat().LastDistance() = baseCase;
}
distance = SortPolicy::CombineBest(baseCase,
referenceNode.FurthestDescendantDistance());
}
else
{
distance = SortPolicy::BestPointToNodeDistance(querySet.col(queryIndex),
&referenceNode);
}
// Compare against the best k'th distance for this query point so far.
const double bestDistance = distances(distances.n_rows - 1, queryIndex);
return (SortPolicy::IsBetter(distance, bestDistance)) ? distance : DBL_MAX;
}
inline double NeighborSearchRules<SortPolicy, MetricType, TreeType>::Score(
TreeType& queryNode,
TreeType& referenceNode)
{
++scores; // Count number of Score() calls.
// Update our bound.
const double bestDistance = CalculateBound(queryNode);
// Use the traversal info to see if a parent-child or parent-parent prune is
// possible. This is a looser bound than we could make, but it might be
// sufficient.
const double queryParentDist = queryNode.ParentDistance();
const double queryDescDist = queryNode.FurthestDescendantDistance();
const double refParentDist = referenceNode.ParentDistance();
const double refDescDist = referenceNode.FurthestDescendantDistance();
const double score = traversalInfo.LastScore();
double adjustedScore;
// We want to set adjustedScore to be the distance between the centroid of the
// last query node and last reference node. We will do this by adjusting the
// last score. In some cases, we can just use the last base case.
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
adjustedScore = traversalInfo.LastBaseCase();
}
else if (score == 0.0) // Nothing we can do here.
{
adjustedScore = 0.0;
}
else
{
// The last score is equal to the distance between the centroids minus the
// radii of the query and reference bounds along the axis of the line
// between the two centroids. In the best case, these radii are the
// furthest descendant distances, but that is not always true. It would
// take too long to calculate the exact radii, so we are forced to use
// MinimumBoundDistance() as a lower-bound approximation.
const double lastQueryDescDist =
traversalInfo.LastQueryNode()->MinimumBoundDistance();
const double lastRefDescDist =
traversalInfo.LastReferenceNode()->MinimumBoundDistance();
adjustedScore = SortPolicy::CombineWorst(score, lastQueryDescDist);
adjustedScore = SortPolicy::CombineWorst(score, lastRefDescDist);
}
// Assemble an adjusted score. For nearest neighbor search, this adjusted
// score is a lower bound on MinDistance(queryNode, referenceNode) that is
// assembled without actually calculating MinDistance(). For furthest
// neighbor search, it is an upper bound on
// MaxDistance(queryNode, referenceNode). If the traversalInfo isn't usable
// then the node should not be pruned by this.
if (traversalInfo.LastQueryNode() == queryNode.Parent())
{
const double queryAdjust = queryParentDist + queryDescDist;
adjustedScore = SortPolicy::CombineBest(adjustedScore, queryAdjust);
}
else if (traversalInfo.LastQueryNode() == &queryNode)
{
adjustedScore = SortPolicy::CombineBest(adjustedScore, queryDescDist);
}
else
{
// The last query node wasn't this query node or its parent. So we force
// the adjustedScore to be such that this combination can't be pruned here,
// because we don't really know anything about it.
// It would be possible to modify this section to try and make a prune based
// on the query descendant distance and the distance between the query node
// and last traversal query node, but this case doesn't actually happen for
// kd-trees or cover trees.
adjustedScore = SortPolicy::BestDistance();
}
if (traversalInfo.LastReferenceNode() == referenceNode.Parent())
{
const double refAdjust = refParentDist + refDescDist;
adjustedScore = SortPolicy::CombineBest(adjustedScore, refAdjust);
}
else if (traversalInfo.LastReferenceNode() == &referenceNode)
{
adjustedScore = SortPolicy::CombineBest(adjustedScore, refDescDist);
}
else
{
// The last reference node wasn't this reference node or its parent. So we
// force the adjustedScore to be such that this combination can't be pruned
// here, because we don't really know anything about it.
// It would be possible to modify this section to try and make a prune based
// on the reference descendant distance and the distance between the
// reference node and last traversal reference node, but this case doesn't
// actually happen for kd-trees or cover trees.
adjustedScore = SortPolicy::BestDistance();
}
// Can we prune?
if (SortPolicy::IsBetter(bestDistance, adjustedScore))
{
if (!(tree::TreeTraits<TreeType>::FirstPointIsCentroid && score == 0.0))
{
// There isn't any need to set the traversal information because no
// descendant combinations will be visited, and those are the only
// combinations that would depend on the traversal information.
return DBL_MAX;
}
}
double distance;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// The first point in the node is the centroid, so we can calculate the
// distance between the two points using BaseCase() and then find the
// bounds. This is potentially loose for non-ball bounds.
double baseCase = -1.0;
if (tree::TreeTraits<TreeType>::HasSelfChildren &&
(traversalInfo.LastQueryNode()->Point(0) == queryNode.Point(0)) &&
(traversalInfo.LastReferenceNode()->Point(0) == referenceNode.Point(0)))
{
// We already calculated it.
baseCase = traversalInfo.LastBaseCase();
}
else
{
baseCase = BaseCase(queryNode.Point(0), referenceNode.Point(0));
}
distance = SortPolicy::CombineBest(baseCase,
queryNode.FurthestDescendantDistance() +
referenceNode.FurthestDescendantDistance());
lastQueryIndex = queryNode.Point(0);
lastReferenceIndex = referenceNode.Point(0);
lastBaseCase = baseCase;
traversalInfo.LastBaseCase() = baseCase;
}
else
{
distance = SortPolicy::BestNodeToNodeDistance(&queryNode, &referenceNode);
}
if (SortPolicy::IsBetter(distance, bestDistance))
{
// Set traversal information.
traversalInfo.LastQueryNode() = &queryNode;
traversalInfo.LastReferenceNode() = &referenceNode;
traversalInfo.LastScore() = distance;
return distance;
}
else
{
// There isn't any need to set the traversal information because no
// descendant combinations will be visited, and those are the only
// combinations that would depend on the traversal information.
return DBL_MAX;
}
}
double RangeSearchRules<MetricType, TreeType>::Score(TreeType& queryNode,
TreeType& referenceNode)
{
math::Range distances;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// It is possible that the base case has already been calculated.
double baseCase = 0.0;
bool alreadyDone = false;
if (tree::TreeTraits<TreeType>::HasSelfChildren)
{
TreeType* lastQuery = (TreeType*) referenceNode.Stat().LastDistanceNode();
TreeType* lastRef = (TreeType*) queryNode.Stat().LastDistanceNode();
// Did the query node's last combination do the base case?
if ((lastRef != NULL) && (referenceNode.Point(0) == lastRef->Point(0)))
{
baseCase = queryNode.Stat().LastDistance();
alreadyDone = true;
}
// Did the reference node's last combination do the base case?
if ((lastQuery != NULL) && (queryNode.Point(0) == lastQuery->Point(0)))
{
baseCase = referenceNode.Stat().LastDistance();
alreadyDone = true;
}
// If the query node is a self-child, did the query parent's last
// combination do the base case?
if ((queryNode.Parent() != NULL) &&
(queryNode.Point(0) == queryNode.Parent()->Point(0)))
{
TreeType* lastParentRef = (TreeType*)
queryNode.Parent()->Stat().LastDistanceNode();
if ((lastParentRef != NULL) &&
(referenceNode.Point(0) == lastParentRef->Point(0)))
{
baseCase = queryNode.Parent()->Stat().LastDistance();
alreadyDone = true;
}
}
// If the reference node is a self-child, did the reference parent's last
// combination do the base case?
if ((referenceNode.Parent() != NULL) &&
(referenceNode.Point(0) == referenceNode.Parent()->Point(0)))
{
TreeType* lastQueryRef = (TreeType*)
referenceNode.Parent()->Stat().LastDistanceNode();
if ((lastQueryRef != NULL) &&
(queryNode.Point(0) == lastQueryRef->Point(0)))
{
baseCase = referenceNode.Parent()->Stat().LastDistance();
alreadyDone = true;
}
}
}
if (!alreadyDone)
{
// We must calculate the base case.
baseCase = BaseCase(queryNode.Point(0), referenceNode.Point(0));
}
else
{
// Make sure that if BaseCase() is called, we don't duplicate results.
lastQueryIndex = queryNode.Point(0);
lastReferenceIndex = referenceNode.Point(0);
}
distances.Lo() = baseCase - queryNode.FurthestDescendantDistance()
- referenceNode.FurthestDescendantDistance();
distances.Hi() = baseCase + queryNode.FurthestDescendantDistance()
+ referenceNode.FurthestDescendantDistance();
// Update the last distances performed for the query and reference node.
queryNode.Stat().LastDistanceNode() = (void*) &referenceNode;
queryNode.Stat().LastDistance() = baseCase;
referenceNode.Stat().LastDistanceNode() = (void*) &queryNode;
referenceNode.Stat().LastDistance() = baseCase;
}
else
{
// Just perform the calculation.
distances = referenceNode.RangeDistance(&queryNode);
}
// If the ranges do not overlap, prune this node.
if (!distances.Contains(range))
return DBL_MAX;
// In this case, all of the points in the reference node will be part of all
// the results for each point in the query node.
if ((distances.Lo() >= range.Lo()) && (distances.Hi() <= range.Hi()))
{
for (size_t i = 0; i < queryNode.NumDescendants(); ++i)
AddResult(queryNode.Descendant(i), referenceNode);
return DBL_MAX; // We don't need to go any deeper.
}
// Otherwise the score doesn't matter. Recursion order is irrelevant in range
// search.
return 0.0;
}
double FastMKSRules<KernelType, TreeType>::Score(TreeType& queryNode,
TreeType& referenceNode)
{
// Update and get the query node's bound.
queryNode.Stat().Bound() = CalculateBound(queryNode);
const double bestKernel = queryNode.Stat().Bound();
// First, see if we can make a parent-child or parent-parent prune. These
// four bounds on the maximum kernel value are looser than the bound normally
// used, but they can prevent a base case from needing to be calculated.
// Convenience caching so lines are shorter.
const double queryParentDist = queryNode.ParentDistance();
const double queryDescDist = queryNode.FurthestDescendantDistance();
const double refParentDist = referenceNode.ParentDistance();
const double refDescDist = referenceNode.FurthestDescendantDistance();
double adjustedScore = traversalInfo.LastBaseCase();
const double queryDistBound = (queryParentDist + queryDescDist);
const double refDistBound = (refParentDist + refDescDist);
double dualQueryTerm;
double dualRefTerm;
// The parent-child and parent-parent prunes work by applying the same pruning
// condition as when the parent node was used, except they are tighter because
// queryDistBound < queryNode.Parent()->FurthestDescendantDistance()
// and
// refDistBound < referenceNode.Parent()->FurthestDescendantDistance()
// so we construct the same bounds that were used when Score() was called with
// the parents, except with the tighter distance bounds. Sometimes this
// allows us to prune nodes without evaluating the base cases between them.
if (traversalInfo.LastQueryNode() == queryNode.Parent())
{
// We can assume that queryNode.Parent() != NULL, because at the root node
// combination, the traversalInfo.LastQueryNode() pointer will _not_ be
// NULL. We also should be guaranteed that
// traversalInfo.LastReferenceNode() is either the reference node or the
// parent of the reference node.
adjustedScore += queryDistBound *
traversalInfo.LastReferenceNode()->Stat().SelfKernel();
dualQueryTerm = queryDistBound;
}
else
{
// The query parent could be NULL, which does weird things and we have to
// consider.
if (traversalInfo.LastReferenceNode() != NULL)
{
adjustedScore += queryDescDist *
traversalInfo.LastReferenceNode()->Stat().SelfKernel();
dualQueryTerm = queryDescDist;
}
else
{
// This makes it so a child-parent (or parent-parent) prune is not
// possible.
dualQueryTerm = 0.0;
adjustedScore = bestKernel;
}
}
if (traversalInfo.LastReferenceNode() == referenceNode.Parent())
{
// We can assume that referenceNode.Parent() != NULL, because at the root
// node combination, the traversalInfo.LastReferenceNode() pointer will
// _not_ be NULL.
adjustedScore += refDistBound *
traversalInfo.LastQueryNode()->Stat().SelfKernel();
dualRefTerm = refDistBound;
}
else
{
// The reference parent could be NULL, which does weird things and we have
// to consider.
if (traversalInfo.LastQueryNode() != NULL)
{
adjustedScore += refDescDist *
traversalInfo.LastQueryNode()->Stat().SelfKernel();
dualRefTerm = refDescDist;
}
else
{
// This makes it so a child-parent (or parent-parent) prune is not
// possible.
dualRefTerm = 0.0;
adjustedScore = bestKernel;
}
}
// Now add the dual term.
adjustedScore += (dualQueryTerm * dualRefTerm);
if (adjustedScore < bestKernel)
{
// It is not possible that this node combination can contain a point
// combination with kernel value better than the minimum kernel value to
// improve any of the results, so we can prune it.
return DBL_MAX;
}
// We were unable to perform a parent-child or parent-parent prune, so now we
// must calculate kernel evaluation, if necessary.
double kernelEval = 0.0;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// For this type of tree, we may have already calculated the base case in
// the parents.
if ((traversalInfo.LastQueryNode() != NULL) &&
(traversalInfo.LastReferenceNode() != NULL) &&
(traversalInfo.LastQueryNode()->Point(0) == queryNode.Point(0)) &&
(traversalInfo.LastReferenceNode()->Point(0) == referenceNode.Point(0)))
{
// Base case already done.
kernelEval = traversalInfo.LastBaseCase();
// When BaseCase() is called after Score(), these must be correct so that
// another kernel evaluation is not performed.
lastQueryIndex = queryNode.Point(0);
lastReferenceIndex = referenceNode.Point(0);
}
else
{
// The kernel must be evaluated, but it is between points in the dataset,
// so we can call BaseCase(). BaseCase() will set lastQueryIndex and
// lastReferenceIndex correctly.
kernelEval = BaseCase(queryNode.Point(0), referenceNode.Point(0));
}
traversalInfo.LastBaseCase() = kernelEval;
}
else
{
// Calculate the maximum possible kernel value.
arma::vec queryCentroid;
arma::vec refCentroid;
queryNode.Centroid(queryCentroid);
referenceNode.Centroid(refCentroid);
kernelEval = kernel.Evaluate(queryCentroid, refCentroid);
traversalInfo.LastBaseCase() = kernelEval;
}
++scores;
double maxKernel;
if (kernel::KernelTraits<KernelType>::IsNormalized)
{
// We have a tighter bound for normalized kernels.
const double querySqDist = std::pow(queryDescDist, 2.0);
const double refSqDist = std::pow(refDescDist, 2.0);
const double bothSqDist = std::pow((queryDescDist + refDescDist), 2.0);
if (kernelEval <= (1 - 0.5 * bothSqDist))
{
const double queryDelta = (1 - 0.5 * querySqDist);
const double queryGamma = queryDescDist * sqrt(1 - 0.25 * querySqDist);
const double refDelta = (1 - 0.5 * refSqDist);
const double refGamma = refDescDist * sqrt(1 - 0.25 * refSqDist);
maxKernel = kernelEval * (queryDelta * refDelta - queryGamma * refGamma) +
sqrt(1 - std::pow(kernelEval, 2.0)) *
(queryGamma * refDelta + queryDelta * refGamma);
}
else
{
maxKernel = 1.0;
}
}
else
{
// Use standard bound; kernel is not normalized.
const double refKernelTerm = queryDescDist *
referenceNode.Stat().SelfKernel();
const double queryKernelTerm = refDescDist * queryNode.Stat().SelfKernel();
maxKernel = kernelEval + refKernelTerm + queryKernelTerm +
(queryDescDist * refDescDist);
}
// Store relevant information for parent-child pruning.
traversalInfo.LastQueryNode() = &queryNode;
traversalInfo.LastReferenceNode() = &referenceNode;
// We return the inverse of the maximum kernel so that larger kernels are
// recursed into first.
return (maxKernel > bestKernel) ? (1.0 / maxKernel) : DBL_MAX;
}
double FastMKSRules<KernelType, TreeType>::Score(const size_t queryIndex,
TreeType& referenceNode)
{
// Compare with the current best.
const double bestKernel = products(products.n_rows - 1, queryIndex);
// See if we can perform a parent-child prune.
const double furthestDist = referenceNode.FurthestDescendantDistance();
if (referenceNode.Parent() != NULL)
{
double maxKernelBound;
const double parentDist = referenceNode.ParentDistance();
const double combinedDistBound = parentDist + furthestDist;
const double lastKernel = referenceNode.Parent()->Stat().LastKernel();
if (kernel::KernelTraits<KernelType>::IsNormalized)
{
const double squaredDist = std::pow(combinedDistBound, 2.0);
const double delta = (1 - 0.5 * squaredDist);
if (lastKernel <= delta)
{
const double gamma = combinedDistBound * sqrt(1 - 0.25 * squaredDist);
maxKernelBound = lastKernel * delta +
gamma * sqrt(1 - std::pow(lastKernel, 2.0));
}
else
{
maxKernelBound = 1.0;
}
}
else
{
maxKernelBound = lastKernel +
combinedDistBound * queryKernels[queryIndex];
}
if (maxKernelBound < bestKernel)
return DBL_MAX;
}
// Calculate the maximum possible kernel value, either by calculating the
// centroid or, if the centroid is a point, use that.
++scores;
double kernelEval;
if (tree::TreeTraits<TreeType>::FirstPointIsCentroid)
{
// Could it be that this kernel evaluation has already been calculated?
if (tree::TreeTraits<TreeType>::HasSelfChildren &&
referenceNode.Parent() != NULL &&
referenceNode.Point(0) == referenceNode.Parent()->Point(0))
{
kernelEval = referenceNode.Parent()->Stat().LastKernel();
}
else
{
kernelEval = BaseCase(queryIndex, referenceNode.Point(0));
}
}
else
{
const arma::vec queryPoint = querySet.unsafe_col(queryIndex);
arma::vec refCentroid;
referenceNode.Centroid(refCentroid);
kernelEval = kernel.Evaluate(queryPoint, refCentroid);
}
referenceNode.Stat().LastKernel() = kernelEval;
double maxKernel;
if (kernel::KernelTraits<KernelType>::IsNormalized)
{
const double squaredDist = std::pow(furthestDist, 2.0);
const double delta = (1 - 0.5 * squaredDist);
if (kernelEval <= delta)
{
const double gamma = furthestDist * sqrt(1 - 0.25 * squaredDist);
maxKernel = kernelEval * delta +
gamma * sqrt(1 - std::pow(kernelEval, 2.0));
}
else
{
maxKernel = 1.0;
}
}
else
{
maxKernel = kernelEval + furthestDist * queryKernels[queryIndex];
}
// We return the inverse of the maximum kernel so that larger kernels are
// recursed into first.
return (maxKernel > bestKernel) ? (1.0 / maxKernel) : DBL_MAX;
}
