Amortized Rigidness in Dynamic Cartesian Trees Iwona Białynicka-Birula and Roberto Grossi...

Amortized Rigidness in Dynamic Cartesian TreesIwona Białynicka-Birula and Roberto GrossiUniversità di Pisa

STACS 2006

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Cartesian trees Vuillemin 1980 Nodes store points ⟨x, y⟩

y value can be viewed as priority Recursive definition

Root stores point with greatest y value x value partitions remaining points(left and right subtrees)

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Cartesian tree example ⟨2, 22⟩ ⟨21, 20⟩⟨18, 19⟩⟨6, 17⟩ ⟨20, 16⟩⟨7, 15⟩⟨8, 13⟩⟨5, 12⟩ ⟨9, 11⟩ ⟨17, 10⟩⟨3, 9⟩ ⟨16, 8⟩⟨10, 7⟩ ⟨22, 6⟩⟨15, 4⟩⟨12, 3⟩⟨1, 2⟩ ⟨11, 1⟩

⟨19, 21⟩⟨4, 18⟩

⟨14, 5⟩

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Applications Priority queue Randomized searching (treaps) Range and dominance searching RMQ (Range Maximum Query) LCA (Least Common Ancestor) Integer sorting Memory management Suffix trees ...

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From RMQ to LCA

2, 22, 9, 18, 12, 17, 15, 13, 11, 7, 1, 3, 5, 4, 8, 10, 19, 21, 16, 20, 6

22 201917 16151312 11 109 87 6432 1

2118

59, 18, 12, 17, 15, 13, 11, 7, 1

17151312 119 71

18

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From LCP array to suffix tree$I$IPPI$ISSIPPI$ISSISSIPPI$MISSISSIPPI$PI$PPI$SIPPI$SISSIPPI$SSIPPI$SSISSIPPI$

0 1 1 4 0 0 1 0 2 1 3$ I M... P S

$ P... SSI

P... S...

I$ PI$ I SI

P... S...

P... S...

1

23

4

56

78 91011

12

121185211097463

01140010213

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History Static setting

O(n) construction time, provided elements already sorted Randomized

Random priority values – treaps O(log n) expected height O(log n) expected update time

Other distributions yield O(√n) or even O(n) height (Devroye 1994) Dynamic and deterministic

???

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Our result Dynamic Cartesian tree

Supports insertion Supports weak deletion Maintains actual tree structure between each operation O(log n) amortized time per operation

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Solution outline Combinatorial analysis

How many tree elements change due to n insertions? Notion of entropy is exploited

Auxiliary structure for accessing tree Needed to quickly access tree elements which need to change Based on the interval tree

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Insertion ⟨2, 22⟩ ⟨21, 20⟩⟨18, 19⟩⟨6, 17⟩ ⟨20, 16⟩⟨7, 15⟩⟨8, 13⟩⟨5, 12⟩ ⟨9, 11⟩ ⟨17, 10⟩⟨3, 9⟩ ⟨16, 8⟩⟨10, 7⟩ ⟨22, 6⟩⟨15, 4⟩⟨12, 3⟩⟨1, 2⟩ ⟨11, 1⟩

⟨19, 21⟩⟨4, 18⟩

⟨14, 5⟩

⟨13, 14⟩

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Insertion – worst case⟨1, 16⟩

⟨7, 4⟩⟨8, 2⟩

⟨17, 15⟩⟨16, 13⟩⟨15, 11⟩⟨14, 9⟩⟨13, 7⟩⟨12, 5⟩⟨11, 3⟩⟨10, 1⟩

⟨2, 14⟩⟨3, 12⟩⟨4, 10⟩⟨5, 8⟩⟨6, 6⟩

⟨9, 17⟩

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Analysis – main idea Inserting new elements does not require comparing y coordinates of existing points In turn, deleting points does Conclusion: insertions reduce tree information content ... so information entropy can be used as a potential function in an amortized analysis

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>> >

>>>

>

>>

>>

Insertion revisited

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Insertion reversed (deletion)

?????

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Formally... Tree T induces partial order ≺T on nodes

Defined by the heap condition Partial order ≺T has ℒ(T) linear extensions

Linear extensions are permutations satisfying the order, i.e. P[i] ≺T P[j] ⇒ i < j We define missing entropy: ℋ(T)=log ℒ(T)

Information needed to sort nodes given tree topology

> > >>> >>>>A B CD E IG JF H

A B H J F G I D E CH A J F G I D B E CA J D F H G I B E CD J I A H F G E B CH J D A F G I E B CA D H F G J I B E C. . .

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Missing entropy Can be zero ℋ(T)=0 Or can be up to ℋ(T)=O(n log n) When an insertion affects k edges, ℋ(T) increases by at least Ω(k)

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So what now? Amortized number of edge modifications is O(log n) per insertion into an initially empty tree

Node modifications are always constant But how to access the edges to modify?

Without increasing the complexity

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Implementation overview Companion interval tree stores tree edges Edges in Cartesian tree are either disjoint or nested

So the interval tree has additional properties Operations are tailored to the special case of the Cartesian tree

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Insertion once again

1. Find parent2. Edges affected

4. Shrink k3a. Delete 2

3b. Insert 31. Find parent2. Edges affected3a. Delete 23b. Insert 34. Shrink k

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Action implementations1. Find parent

Uses the interval tree as a search tree2. Edges affected Special kind of stabbing query3. Insert and delete Standard interval tree operations4. Shrink Emulating using inserts and deletes would yield O(k∗log n) Amortized argument based on the fact that shrinking edge travels down

O(log n)O(log n+k)

O(1)∗O(log n)k∗O(1)

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Summary Scheme for maintaining a Cartesian tree under insertion and week deletion

Amortized O(log n) time per update At any moment the actual tree structure is accessible

Solution components Combinatorial analysis of tree behavior Auxiliary data structure

First result on dynamic Cartesian trees in a non-stochastic setting

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Further work

Strong deletion Applications

For example RMQ

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Thank you!

Questions?