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Mergeable Trees 1
Mergeable TreesMergeable Trees
Robert E. TarjanPrinceton University and HP Labs
Joint work with Loukas Georgiadis, Haim Kaplan, Nira Shafrir and Renato Werneck
Partial results: Georgiadis, Tarjan, and Werneck, Design of data structures for mergeable trees, SODA 2006
Mergeable Trees 2
Outline
1. The Problem
2. The Motivating Application
3. Dynamic Trees
4. Mergeable Trees via Dynamic Trees
5. Implicit Mergeable Trees
6. Mergeable Trees via Partition by Rank
7. Open Problems
Mergeable Trees 3
Mergeable Trees
• Goal: Maintain a forest of rooted trees.
• Queries: parent(v)
nca(v,w)
root(v)
Mergeable Trees 4
Mergeable Trees
• Goal: Maintain a forest of rooted trees.
• Queries: parent(v)
nca(v,w)
root(v)
• Trees are heap-ordered (root=smallest)
• Updates:
insert(v)merge(v,w) : merge the v-to-root and w-to-root paths,
preserving the heap order.
Mergeable Trees 5
Mergeable Trees
• Goal: Maintain a forest of rooted trees.
• Queries: parent(v)
nca(v,w)
root(v)
• Trees are heap-ordered (root=smallest)
• Updates:
insert(v)merge(v,w) : merge the v-to-root and w-to-root paths,
preserving the heap order.
• Other updates:
link(v,w) : make root v a child of node w in another tree = merge(v,w). cut(v) : make v a root by disconnecting from parent. delete(v) : delete leaf v = cut(v), discard v.
Mergeable Trees 6
Example
3 2
6 7 4
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Mergeable Trees 7
Example
merge(5,2)
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6 7 4
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6 7 4
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8 11
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Mergeable Trees 8
Example
merge(6,11)
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Mergeable Trees 9
Example
merge(6,11)
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merge(7,8)
Mergeable Trees 10
Mergeable Trees: Motivation
Used in [Agarwal, Edelsbrunner, Harer and Wang ’04];
sub-problem in their algorithm for computing the structure of 2-manifolds embedded in RR3.
nodes = critical points (minima, maxima, saddle points), heap-ordered by height.
merging is used for pairing critical points;
no cut operation;
link only attaches leaves to the tree;
the arguments of merge are always leaves.
Mergeable Trees 11
Turbo Exhaust Manifold
Mergeable Trees 12
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Reeb Graph encodes skeleton of manifold
sink (0 out-degree)
source (0 in-degree)
up-fork (1 in, 2 out)
down-fork (2 in, 1 out)
Nodes in Reeb Graph
Mergeable Trees 13
Pairing Algorithm on Reeb Graph
x
x
x
v
v w
v
merge(x,v)
root(v) ≠ root(w): pair x with max{root(v),root(w)}root(v) = root(w): pair x with nca(v, w)
merge(x,v); merge(x,w)
merge(x,v) while v is paired, replace by parent(v)
pair x with v
sink
Mergeable Trees 14
12
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Reeb Graph Mergeable Trees
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Example
Mergeable Trees 15
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Reeb Graph Mergeable Trees
12 (3)
3 (2)
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pair(3,2)merge(3,1)merge(3,2)
Example
Mergeable Trees 16
12
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Reeb Graph Mergeable Trees
12 (3)
3 (2)
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merge(4,3)merge(5,4)merge(6,4)
Example
Mergeable Trees 17
12
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Reeb Graph Mergeable Trees
12 (3)
3 (2)
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5 (7)
7 (5)
10
pair(7,5)merge(7,5)
Example
Mergeable Trees 18
12
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Reeb Graph Mergeable Trees
12 (3)
3 (2)
4 (8)
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8 (4)
9
5 (7)
7 (5)
10
pair(8,4)merge(8,7)merge(8,6)
Example
Mergeable Trees 19
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Reeb Graph Mergeable Trees
12 (3)
3 (2)
4 (8)
6 (9)
8 (4)
9 (6)
5 (7)
7 (5)
10
merge(9,8)pair(9,6)(6 is the closestunpaired ancestor of 9)
Example
Mergeable Trees 20
12
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10
Reeb Graph Mergeable Trees
2 (3)
3 (2)
4 (8)
6 (9)
8 (4)
9 (6)
5 (7)
7 (5)
10
merge(10,6)pair(10,1)(1 is the closestunpaired ancestor of 10)
Example
Mergeable Trees 21
Two-Pass Algorithm
To avoid parent operation
Pair max with min
Run algorithm forward and backward
Do no pairing in “sink” case
Mergeable Trees 22
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Reverse Reeb Graph Mergeable Trees
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10 (1)
1 (10)
10 (max) is paired with 1 (min)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 23
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Reverse Reeb Graph Mergeable Trees
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10 (1)
merge(8,9)merge(7,8)
1 (10)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 24
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Reverse Reeb Graph Mergeable Trees
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6 (9)
8
9 (6)
5
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10 (1)
pair(9,6)* merge(6,8)merge(6,10)
*pair 6 with root of min label;this pair is missing from forward pass.
1 (10)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 25
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Reverse Reeb Graph Mergeable Trees
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6 (9)
8
9 (6)
5 (7)
7 (5)
10 (1)
pair(5,7) merge(5,7)
1 (10)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 26
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Reverse Reeb Graph Mergeable Trees
2
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4 (8)
6 (9)
8 (4)
9 (6)
5 (7)
7 (5)
10 (1)
pair(4,8) merge(4,5)merge(4,6)
1 (10)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 27
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Reverse Reeb Graph Mergeable Trees
1 (10)2
3
4 (8)
6 (9)
8 (4)
9 (6)
5 (7)
7 (5)
10 (1)
merge(3,4)merge(1,3)merge(2,3)
Reverse Pass of Two-Pass Algorithm
Mergeable Trees 28
Mergeable Trees: Results
n = number of nodes in mergesm = number of merges
O(log2n) amortized time per operation
O(logn) amortized time without cuts
O(logn) worst-case without cuts, parent queries
Mergeable Trees 29
Dynamic Trees
Goal: maintain a forest of trees with values on vertices and/or edges.
Operations:
link(v,w) : add an edge between v and w. (no cycles allowed)
cut(v,w) : delete edge (v,w).
various operations (e.g. find a vertex of minimum value on a path or in a tree).
Trees can be free (unrooted), rooted, or ordered.
Lots of applications: network flows, static and dynamic graph algorithms, computational geometry, …
Several data structures with optimal O(logn) time per operation (worst case, amortized or randomized)
Mergeable Trees 30
Dynamic Trees
Optimal Data Structures:
• Sleator and Tarjan (‘83, ‘85): Link-Cut Trees – Worst Case and Amortized.
• Frederickson (‘85 , ‘97): Topology Trees – Worst Case.
• Alstrup, Holm, de Lichtenberg and Thorup (‘97, ’03): Top Trees – Worst Case.
• Acar, Blelloch, Harper, Vittes and Woo (’03): RC-Trees – Randomized.
• Tarjan and Werneck (‘05): Self-Adjusting Top Trees – Amortized.
Mergeable Trees 31
merge(w,p) :
Suppose we have computed nca(w,p) and isolated the two paths to be merged. It remains to do the actual merge by changing parents.
Mergeable Trees via Dynamic Trees
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Mergeable Trees 32
Mergeable Trees via Dynamic Trees
First idea: Iterated Insertions Insert the nodes of the shorter path into the longer. (Suggested in Agarwal et al.)
Cost is bounded below by the sum of the lengths of the shorter paths.
Unfortunately, this is
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£(n3=2)
merge(w,p) :
Suppose we have computed nca(w,p) and isolated the two paths to be merged. It remains to do the actual merge by changing parents.
Mergeable Trees 33
Mergeable Trees via Dynamic Trees
merge(w,p):Suppose we have computed nca(w,p) and isolated the two paths;
it remains to do the actual merge by changing the parent pointers.
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Better idea: Interleaved Merges
Cut edges where parents change and link the pieces together.
Time is O(#cuts × logn).
Mergeable Trees 34
Mergeable Trees via Dynamic Trees
merge(w,p):Suppose we have computed nca(w,p) and isolated the two paths;
it remains to do the actual merge by changing the parent pointers.
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onqpTo enable fast merging:
Query:
topmost(v,w) : return the smallest (topmost) ancestor of v strictly greater than w. (assume v>w)
r
Better idea: Interleaved Merges
Cut edges where parents change and link the pieces together.
Time is O(#cuts × logn).
Mergeable Trees 35
Mergeable Trees via Dynamic Trees
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Mergeable Trees 36
Mergeable Trees via Dynamic Trees
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Mergeable Trees 37
Mergeable Trees via Dynamic Trees
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Mergeable Trees 38
Mergeable Trees via Dynamic Trees
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Mergeable Trees 39
Analysis of Merge
#parent changes = O(m logn).
w
v
Each node has parent potential (on its parent) has child potential (on its children).
Á(v;w) = 2log(v ¡ w) + 1
©=P
(v;w) Á(v;w)
1
log(v-w)
log(v-w)
Mergeable Trees 40
Analysis of Merge
#parent changes = O(m logn).
w
v
cut (or delete) : one parent change (to null) Φ 1.
Á(v;w) = 2log(v ¡ w) + 1
©=P
(v;w) Á(v;w)
1
log(v-w)
log(v-w)
Mergeable Trees 41
Analysis of Merge
#parent changes = O(m logn).
w
v
cut (or delete) : one parent change (to null) Φ 1.
Á(v;w) = 2log(v ¡ w) + 1
©=P
(v;w) Á(v;w)
1
log(v-w)
log(v-w)
merge(v,w) : Φ O(logn) if v,w in different trees (initial link).
Mergeable Trees 42
Analysis of Merge
#parent changes = O(m logn).
w
v
All other Φ changes are non-positive.
Á(v;w) = 2log(v ¡ w) + 1
©=P
(v;w) Á(v;w)
1
log(v-w)
log(v-w)
z
w
z
y
x
ww-x ≤ (w-z)/2 parent potential of w 1y-z ≤ (w-z)/2 child potent of z 1
Pays for parent change of w.
Mergeable Trees 43
Implicit Mergeable Trees
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Parents are computable but not efficiently.
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Mergeable Trees 44
Implicit Mergeable Trees
merge(v,w): Case (a) v,w in different trees: link(v,w)
3 2
8
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75
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6 10
9
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Mergeable Trees 45
Implicit Mergeable Trees
3 2
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6 10
9
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Real forestEquivalent forest
merge(v,w): Case (a) v,w in different trees: link(v,w)
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Mergeable Trees 46
Implicit Mergeable Trees
3 2
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75
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6 10
9
3 2
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9
Real forest
merge(10,11)
merge(v,w): Case (a) v,w in different trees: link(v,w)
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Equivalent forest
Mergeable Trees 47
Implicit Mergeable Trees
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6 10
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merge(10,11)
Real forest
merge(v,w): Case (a) v,w in different trees: link(v,w)
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Equivalent forest
Mergeable Trees 48
Implicit Mergeable Trees
merge(v,w): Case (b) v,w in same tree: cut (u,q); link (v,w) where
u = pathmin(v,w) , q = successor of u in path from u to v
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Real forest
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Equivalent forest
Mergeable Trees 49
Implicit Mergeable Trees
merge(6,12)
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Real forest
merge(v,w): Case (b) v,w in same tree: cut (u,q); link (v,w) where
u = pathmin(v,w) , q = successor of u in path from u to v
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Equivalent forest
Mergeable Trees 50
Implicit Mergeable Trees
3 2
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6 10
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3 2
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Real forest
merge(6,12)
merge(v,w): Case (b) v,w in same tree: cut (u,q); link (v,w) where
u = pathmin(v,w) , q = successor of u in path from u to v
Each mergeable tree represented by an equivalent unrooted dynamic tree:
root = treemin, nca = pathmin
Equivalent forest
Mergeable Trees 51
• Main idea: partition the vertices in a tree into disjoint solid paths connected by dashed edges, represent solid paths by search trees.
Sleator and Tarjan’s Link-Cut Trees
Mergeable Trees 52
Sleator and Tarjan’s Link-Cut Trees
• Main idea: partition the vertices in a tree into disjoint solid paths connected by dashed edges, represent solid paths by search trees.
Mergeable Trees 53
Mergeable Trees via Partition by Rank
24(4)
16(4)
7(2)
6(2)
4(2)
1(0) 1(0
)
1(0)
1(0)
1(0)
1(0)
1(0)
1(0)
1(0)
2(1)
2(1)
3(1)
4(2)
5(2)
6(2)
7(2)
5(2)
6(2)
2(1)
We can do better when arbitrary cuts are disallowed.
(x,y) is solid if rank(x)=rank(y); dashed otherwise.
rank(x) = blgsize(x)c
Mergeable Trees 54
In every leaf to root path there are at most logn dashed arcs nca(·,·) can be implemented in O(logn)-time worst case.
We can do better when arbitrary cuts are disallowed.
(x,y) is solid if rank(x)=rank(y); dashed otherwise.
Mergeable Trees via Partition by Rank
24(4)
16(4)
7(2)
6(2)
4(2)
1(0) 1(0
)
1(0)
1(0)
rank(x) = blgsize(x)c
1(0)
1(0)
1(0)
1(0)
1(0)
2(1)
2(1)
3(1)
4(2)
5(2)
6(2)
7(2)
5(2)
6(2)
2(1)
Each node points to (a node that points to)The top node on its path.
Mergeable Trees 55
Mergeable Trees via Partition by Rank
We can do better when arbitrary cuts are disallowed.
(x,y) is solid if rank(x)=rank(y); dashed otherwise.
Merging idea: Insert nodes of lower rank into solid paths of higher rank.
rank(p) ≥ rank(q)
p
q
The rank of q increases by at least 1 O(nlogn) such insertions.
rank(x) = blgsize(x)c
Mergeable Trees 56
Mergeable Trees via Partition by Rank
We can do better when arbitrary cuts are disallowed.
(x,y) is solid if rank(x)=rank(y); dashed otherwise.
Merging idea: Insert nodes of lower rank into solid paths of higher rank.
rank(p) ≥ rank(q)
q
The rank of q increases by at least 1 O(nlogn) such insertions.
p
A prefix of the resulting path may change rank O(nlogn) such events.
rank(x) = blgsize(x)c
Mergeable Trees 57
Mergeable Trees via Partition by Rank
Operations: (1) insert a node in an arbitrary position on a solid path; (2) remove top node of a solid path.
Solid paths represented by finger search trees.
x
zy
P
x = most-recently-accessed node of Pz is inserted immediately below y
cost ~ log(d+2)d = length of path from y to x
d
Mergeable Trees 58
Mergeable Trees via Partition by Rank
x
zy
x = most-recently-accessed node of Pz is inserted immediately below y
cost ~ log(d+2)d = length of path from y to x
d
P
di = length of the (solid) insertion path for the i-th insertion (1≤i≤nlogn).
Claim:P
i logdi = O(n logn)
Mergeable Trees 59
Open Problems
Get an O(logn)-time algorithm when arbitrary cuts are allowed.
Conjecture:
The interleaved merging algorithm implemented with self-adjusting dynamic trees takes O(logn) amortized time per operation.
Other applications?