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using System;
using System.Collections.Generic;
using System.Linq;
/// <summary>
/// The building block of a tree that tracks nesting depth over a range.
/// The null node is considered to be an empty tree.
/// WARNING: Assumptions about the use case (interval references) are implicitely assumed in some places.
/// For example, the tree may or may not work properly if you don't keep it partitioned around holes before continuing to use it.
/// </summary>
public sealed class NestingDepthTreeNode {
///<summary>The place where this node's adjustment takes effect.</summary>
private readonly int _offset;
/// <summary>How much the nesting depth changes as you go from _offset-1 to _offset.</summary>
private int _adjust;
/// <summary>
/// The number of ranged arrays that point into this node.
/// An array needs its referenced node to be in the tree, so that it can find the root when deallocating.
/// Thus nodes with RefCount > 0 can't be removed, even if their Adjust=0.
/// </summary>
private int _refCount;
/// <summary>The root of the subtree containing indexes that are less than this node's offset.</summary>
private NestingDepthTreeNode _less;
/// <summary>The root of the subtree containing indexes that are more than this node's offset.</summary>
private NestingDepthTreeNode _more;
/// <summary>The only node that has this node as a Less or More child.</summary>
private NestingDepthTreeNode _parent;
/// <summary>The total amount the nesting depth changes as you cross the sub tree rooted at this node.</summary>
private int _subTreeTotalAdjust;
/// <summary>The lowest that the nesting depth gets as you cross the sub tree rooted at this node, ASSUMING your starting depth is 0.</summary>
private int _subTreeRelativeMinimum;
/// <summary>This is a constructor. You knew that already. Comments can be helpful sometimes.</summary>
private NestingDepthTreeNode(int offset, int adjust, int refAdjust) {
this._offset = offset;
this._adjust = adjust;
this._refCount = refAdjust;
///<summary>Gets the child in the given direction (negative -> lesser child, positive -> larger child).</summary>
private NestingDepthTreeNode Child(int childSign) {
if (childSign < 0) return _less;
if (childSign > 0) return _more;
throw new ArgumentException("childSign == 0");
///<summary>Sets the child in the given direction (negative -> lesser child, positive -> larger child).</summary>
private void SetChild(int childSign, NestingDepthTreeNode value) {
if (childSign < 0) _less = value;
else if (childSign > 0) _more = value;
else throw new ArgumentException("childSign == 0");
///<summary>+1 if the parent has a larger value, -1 if the parent has a lesser value, and 0 if we're the root</summary>
private int DirToParent() {
if (_parent == null) return 0;
if (_parent._less == this) return +1;
return -1;
///<summary>Travels upwards from the given node until the root of the tree containing it is found.</summary>
public static NestingDepthTreeNode RootOf(NestingDepthTreeNode node) {
if (node == null) return null;
if (node._parent == null) return node;
return RootOf(node._parent);
///<summary>Determines how much the nesting depth changes as you cross the tree rooted at the given node.</summary>
public static int GetTotalAdjust(NestingDepthTreeNode root) {
return root == null ? 0 : root._subTreeTotalAdjust;
/// <summary>Determines the range of indexes spanned by the sub tree rooted at the given node.</summary>
public static Interval GetInterval(NestingDepthTreeNode root) {
if (root == null) return default(Interval);
var min = root;
var max = root;
while (min._less != null) min = min._less;
while (max._more != null) max = max._more;
return new Interval(min._offset, max._offset - min._offset);
///<summary>Returns a result that is larger when the given value is a multiple of a higher power of 2.</summary>
private static int PowerOf2Ness(int value) {
// only the lowest set bit affects the result
// xxxxxxxxx1000000 - 1 == xxxxxxxxx0111111
// xxxxxxxxx1000000 ^ xxxxxxxxx0111111 == 0000000001111111
// larger powers of 2 have their first set bit higher
return value ^ (value - 1);
/// <summary>
/// Numbers with more power-of-2-ness are superior.
/// Also, power-of-2-ness makes a nice balanced binary structure: 1317131F1317131...
/// This is not optimal, because nodes may be sparse w.r.t. the range, but it's good enough for an experiment.
/// </summary>
private bool ShouldBeParentOf(NestingDepthTreeNode other) {
if (other == null) return false;
return PowerOf2Ness(other._offset) < PowerOf2Ness(this._offset);
///<summary>Returns the nesting depth at a particular index, as determined by the tree rooted at the given node.</summary>
public static int QueryNestingDepthAt(NestingDepthTreeNode root, int index) {
if (root == null) return 0;
var pre = QueryNestingDepthAt(root._less, index);
if (root._offset > index) return pre;
var on = pre + root._adjust;
if (root._offset == index) return on;
var post = on + QueryNestingDepthAt(root._more, index);
return post;
/// <summary>Scans the tree for transitions into and out of holes, running a callback for each one.</summary>
private static void FindAndCallbackHoleTransitionsInOrder(NestingDepthTreeNode node, int initialNestingDepth, Action<NestingDepthTreeNode, bool> callback) {
if (node == null) return;
// we can skip this whole subtree if it's guaranteed to stay above water the whole time
var lowestNestingDepth = node._subTreeRelativeMinimum + initialNestingDepth;
if (initialNestingDepth > 0 && lowestNestingDepth > 0) return;
// scan left subtree
FindAndCallbackHoleTransitionsInOrder(node._less, initialNestingDepth, callback);
// check for transition here
var nestingDepthJustBeforeNode = initialNestingDepth + GetTotalAdjust(node._less);
var nestingDepthJustAfterNode = nestingDepthJustBeforeNode + node._adjust;
var wasInHole = nestingDepthJustBeforeNode <= 0;
var nowInHole = nestingDepthJustAfterNode <= 0;
if (wasInHole != nowInHole) {
callback(node, nowInHole);
// scan right subtree
FindAndCallbackHoleTransitionsInOrder(node._more, nestingDepthJustAfterNode, callback);
///<summary>Returns all of the areas in the given interval where the tree would return zero if you queried the nesting depth there.</summary>
public static IReadOnlyList<Interval> FindHolesIn(Interval interval, NestingDepthTreeNode root) {
var relevantSegments =
.SkipWhile(e => !e.Overlaps(interval))
.TakeWhile(e => e.Overlaps(interval))
var holeStarts =
new[] {interval.Offset}
.Select(e => e.Offset + e.Length));
var holeEnds =
.Select(e => e.Offset)
.Concat(new[] {interval.Offset + interval.Length});
return holeStarts
.Zip(holeEnds, (start, end) => new Interval(start, end - start))
.Where(e => e.Length > 0)
///<summary>Returns the areas that the tree has covered; that aren't holes in the nesting depth.</summary>
public static IReadOnlyList<Interval> FindCoveredIntervals(NestingDepthTreeNode root) {
var coveredStart = (int?)null;
var results = new List<Interval>();
(node, isIntoHole) => {
if (coveredStart.HasValue != isIntoHole) {
throw new InvalidOperationException("Invariant violated: in-out-repeat");
if (coveredStart.HasValue) {
results.Add(new Interval(coveredStart.Value, node._offset - coveredStart.Value));
coveredStart = null;
} else {
coveredStart = node._offset;
return results;
/// <summary>
/// When the nesting depth hits zero, we can actually split the tree in two around that hole and make future queries more efficient.
/// This only works because arrays only care about the segment they're in, as opposed to the global state.
/// </summary>
public static void PartitionAroundHoles(NestingDepthTreeNode root) {
var partitionsToPerform = new List<Action>();
// find the nodes that have a hole to their left or right, so we can cut there
(node, isIntoHole) => partitionsToPerform.Add(() => PartitionToSideOfNode(node, isIntoHole ? +1 : -1)));
// only do the partitioning after the search is done, so we don't interfere with it
foreach (var e in partitionsToPerform) {
/// <summary>Cuts the tree in two, to the left of the given node if d is negative and to the right if d is positive.</summary>
private static void PartitionToSideOfNode(NestingDepthTreeNode n, int d) {
var orphan = n.Child(d);
if (orphan != null) orphan._parent = null;
n.SetChild(d, null);
PartitionUpwardToParentSideOf(n, d, orphan);
/// <summary>Continues cutting the tree in two, assuming that child nodes have been handled.</summary>
private static void PartitionUpwardToParentSideOf(NestingDepthTreeNode n, int d, NestingDepthTreeNode orphan) {
if (n == null) return;
var p = n._parent;
if (n.DirToParent() == d) {
// going from n to p crosses the cut line, so we must disconnect them
n._parent = null;
// the orphan node takes n's place
p.SetChild(-d, orphan);
if (orphan != null) orphan._parent = p;
// switch directions back towards the cut, and onward with our new orphan n!
PartitionUpwardToParentSideOf(p, -d, n);
} else {
// didn't pass over the cut line, keep going upwards
PartitionUpwardToParentSideOf(p, d, orphan);
///<summary>Gets the child in the given direction (negative -> lesser child, positive -> larger child).</summary>
public struct IncludeResult {
/// <summary>
/// The node that was created or modified by the include operation.
/// Null when the node was imploded due to having no effect anymore.
/// </summary>
public readonly NestingDepthTreeNode AdjustedNode;
/// <summary>
/// The new root of the tree (or sub-tree).
/// The node is in there somewhere...
/// </summary>
public readonly NestingDepthTreeNode NewRoot;
public IncludeResult(NestingDepthTreeNode newRootAndAdjustedNode) {
AdjustedNode = NewRoot = newRootAndAdjustedNode;
public IncludeResult(NestingDepthTreeNode adjustedNode, NestingDepthTreeNode newRoot) {
AdjustedNode = adjustedNode;
NewRoot = newRoot;
/// <summary>
/// Adds an adjustment to the tree rooted at the given node.
/// The nesting depth will be perceived as the given adjust higher after the given index.
/// Can also adjust reference counts.
/// This can create, modify, or remove a node in the tree.
/// </summary>
public static IncludeResult Include(NestingDepthTreeNode root, int index, int adjust, int refAdjust) {
if (root != null && root._parent != null) throw new ArgumentException("root.Parent != null");
var preTotal = GetTotalAdjust(root);
var result = IncludeHelper(root, index, adjust, refAdjust);
var postTotal = GetTotalAdjust(result.NewRoot);
if (preTotal + adjust != postTotal) {
throw new InvalidOperationException("Invariant violated: total adjustment did not shift by the included adjustment.");
return result;
public static IncludeResult IncludeHelper(NestingDepthTreeNode root, int index, int adjust, int refAdjust) {
// Do we need to create a new node?
if (root == null) {
return new IncludeResult(new NestingDepthTreeNode(index, adjust, refAdjust));
// It is understood that the caller will fixup our parent afterwards; we must act independently of the tree above us
root._parent = null;
// Is this node the one we need to adjust?
if (index == root._offset) {
root._adjust += adjust;
root._refCount += refAdjust;
// nodes can be removed when they are not referenced and have no effect on the totals
if (root._adjust == 0 && root._refCount == 0) {
return new IncludeResult(null, root.Implode());
return new IncludeResult(root);
// Pick the subtree the node has to end up and recurse the inclusion that-a-way
var d = index.CompareTo(root._offset);
var subtree = root.Child(d);
var preTotal = GetTotalAdjust(subtree);
var subResult = IncludeHelper(subtree, index, adjust, refAdjust);
var postTotal = GetTotalAdjust(subResult.NewRoot);
if (preTotal + adjust != postTotal) {
throw new InvalidOperationException("Invariant violated: total adjustment did not shift by the included adjustment.");
// Great! Now we just need to fixup so the new subtree is linked in
var c = subResult.NewRoot;
root.SetChild(d, c);
if (c != null) c._parent = root;
// Oh, and do a token effort to keep things balanced using our hacky hierarchical ordering over the indexes
// Can we get away with not rotating the new child above ourselves to keep things sorta balanced?
if (c == null || !c.ShouldBeParentOf(root)) {
return new IncludeResult(subResult.AdjustedNode, root);
// darn, need to rotate
var s = c.Child(-d);
c.SetChild(-d, root);
root.SetChild(d, s);
// fixup
c._parent = null;
root._parent = c;
if (s != null) s._parent = root;
// finally
return new IncludeResult(subResult.AdjustedNode, c);
/// <summary>Removes this node from the tree, returning the child that moved into its place.</summary>
private NestingDepthTreeNode Implode() {
// I sure hope we're not in the inconvenient case...
if (_less != null && _more != null) {
// pick your favorite child to promote
var d = _less.ShouldBeParentOf(_more) ? -1 : +1;
var c = Child(d);
var s = Child(-d);
c._parent = null;
// find a place for orphaned s
var m = c;
while (m.Child(-d) != null) {
m = m.Child(-d);
m.SetChild(-d, s);
s._parent = m;
// a node can implode at most once, so we can afford to pay O(log n) for each
// not sure if this blows the per-operation worst case past logarithmic or not, though
return c;
// No extra child to put somewhere, so we just toss the one (or none) we have off on our parent so we can go have fun.
var p = _less ?? _more;
if (p != null) p._parent = null;
return p;
/// <summary>Fixes up the aggregate values stored by this node.</summary>
private void RecomputeAggregates() {
// total
_subTreeTotalAdjust = GetTotalAdjust(_less) + _adjust + GetTotalAdjust(_more);
// lowest dive
var totalAdjustJustBefore = GetTotalAdjust(_less) + _adjust;
var totalAdjustJustAfter = totalAdjustJustBefore + (_more == null ? 0 : _more._subTreeRelativeMinimum);
_subTreeRelativeMinimum = Math.Min(totalAdjustJustBefore, totalAdjustJustAfter);
if (_less != null) _subTreeRelativeMinimum = Math.Min(_less._subTreeRelativeMinimum, _subTreeRelativeMinimum);
/// <summary>Fixes up the aggregate values stored by this node and its ancestors.</summary>
private void RecomputeAggregatesToRoot() {
if (this._parent != null) this._parent.RecomputeAggregatesToRoot();
public override string ToString() {
return "" + _offset;