Asymptotic Enumeration of Compacted Binary Trees with Height - PowerPoint PPT Presentation

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). × 5 + 2 4 2 × 1 1 3 3 1 1 y y (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). × 5 + 2 4 2 × 1 1 3 3 1 1 3 y (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). × 5 + 2 4 2 × 1 1 3 3 1 1 3 3 (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). × 5 + 2 4 2 4 1 1 3 3 1 1 3 3 (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). × 5 6 2 4 2 4 1 1 3 3 1 1 3 3 (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)), (6 , ( − , 2 , 4)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). 7 5 6 2 4 2 4 1 1 3 3 1 1 3 3 (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)), (6 , ( − , 2 , 4)), (7 , ( − , 5 , 6)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). 7 × 5 6 − + 2 4 2 4 × × 1 1 3 3 1 1 3 3 x y (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)), (6 , ( − , 2 , 4)), (7 , ( − , 5 , 6)) Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Motivation – Efficiently store redundant information Example Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents ( x 2 − y 2 )( x 2 + y 2 ). 7 × 5 6 − + 2 4 2 4 × × 1 1 3 3 1 1 3 3 x y (1 , ( x , 0 , 0)), (2 , ( × , 1 , 1)), (3 , ( y , 0 , 0)), (4 , ( × , 3 , 3)), (5 , ( − , 2 , 4)), (6 , ( − , 2 , 4)), (7 , ( − , 5 , 6)) Definition Compacted tree is the DAG computed by this procedure. Michael Wallner | LaBRI | 24.05.2018 3 / 36

Compacted Binary Trees | Creating a compacted tree Compacted trees Important property: Subtrees are unique Efficient algorithm to compute compacted tree Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG) Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to n C √ log n , where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees Michael Wallner | LaBRI | 24.05.2018 4 / 36

Compacted Binary Trees | Creating a compacted tree Compacted trees Important property: Subtrees are unique Efficient algorithm to compute compacted tree Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG) Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to n C √ log n , where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees Reverse question How many compacted trees of (compacted) size n exist? Michael Wallner | LaBRI | 24.05.2018 4 / 36

Compacted Binary Trees | Creating a compacted tree Compacted trees Size of a compacted tree: number of internal nodes Number of compacted trees of size n : c n Michael Wallner | LaBRI | 24.05.2018 5 / 36

Compacted Binary Trees | Creating a compacted tree Compacted trees Size of a compacted tree: number of internal nodes Number of compacted trees of size n : c n Figure: All compacted binary trees of size n = 0 , 1 , 2. Michael Wallner | LaBRI | 24.05.2018 5 / 36

Compacted Binary Trees | Creating a compacted tree Compacted trees Size of a compacted tree: number of internal nodes Number of compacted trees of size n : c n Figure: All compacted binary trees of size n = 0 , 1 , 2. Example (Compacted binary trees) size n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 c n 1 1 3 15 111 1119 14487 � 2 n � 1 n ! ≤ c n ≤ · n ! n + 1 n Hence, c n = O ( n !4 n n − 1 / 2 ). Michael Wallner | LaBRI | 24.05.2018 5 / 36

Compacted Binary Trees | Creating a compacted tree Goals of this talk Goals 1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s) Michael Wallner | LaBRI | 24.05.2018 6 / 36

Compacted Binary Trees | Creating a compacted tree Goals of this talk Goals 1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s) Methods 1 Recurrence relations 5 Differential equations 2 Bijections 6 Singularity analysis 3 Generating functions 7 Chebyshev polynomials 4 Symbolic method 8 Guess and prove Michael Wallner | LaBRI | 24.05.2018 6 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree Idea Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Michael Wallner | LaBRI | 24.05.2018 7 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree Idea Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness Michael Wallner | LaBRI | 24.05.2018 7 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree Idea Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness Procedure 1 Take a binary tree of size n 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Michael Wallner | LaBRI | 24.05.2018 7 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree – Example Procedure 1 Take a binary tree of size n (called spine ) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Michael Wallner | LaBRI | 24.05.2018 8 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree – Example Procedure 1 Take a binary tree of size n (called spine ) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Valid compacted tree Michael Wallner | LaBRI | 24.05.2018 8 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree – Example Procedure 1 Take a binary tree of size n (called spine ) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Valid compacted tree Invalid compacted tree Michael Wallner | LaBRI | 24.05.2018 8 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree – Example Procedure 1 Take a binary tree of size n (called spine ) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Valid compacted tree Invalid compacted tree We call the underlying binary tree from step 1 the spine . Michael Wallner | LaBRI | 24.05.2018 8 / 36

Compacted Binary Trees | Creating a compacted tree Building a compacted tree from a binary tree – Example Procedure 1 Take a binary tree of size n (called spine ) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order NOT violating uniqueness Valid compacted tree Invalid compacted tree We call the underlying binary tree from step 1 the spine . This spine is associated to 3 valid compacted trees. Michael Wallner | LaBRI | 24.05.2018 8 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 2 2 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 3 2 2 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 3 4 2 2 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 3 4 2 2 4 4 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 3 4 2 2 13 4 4 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 1 3 4 6 6 2 2 13 4 4 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 31 1 3 4 6 6 2 2 13 4 4 Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | Creating a compacted tree A bigger example We take a binary tree of size 8. 31 1 3 4 6 6 2 2 13 4 4 In total, this spine corresponds to 1 · 3 · 4 · 13 · 31 = 4836 compacted trees. Michael Wallner | LaBRI | 24.05.2018 9 / 36

Compacted Binary Trees | A recurrence relation A recurrence relation Michael Wallner | LaBRI | 24.05.2018 10 / 36

Compacted Binary Trees | A recurrence relation A recurrence for compacted binary trees Counting formula Let n , p ∈ N , then n � γ n +1 , p = for n ≥ 1 , γ i , p γ n − i , p + i , i =0 Helps us to efficiently compute c n Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i Michael Wallner | LaBRI | 24.05.2018 11 / 36

Compacted Binary Trees | A recurrence relation A recurrence for compacted binary trees Counting formula Let n , p ∈ N , then n � γ n +1 , p = for n ≥ 1 , γ i , p γ n − i , p + i , i =0 γ 0 , p = p + 1 , Helps us to efficiently compute c n Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i Michael Wallner | LaBRI | 24.05.2018 11 / 36

Compacted Binary Trees | A recurrence relation A recurrence for compacted binary trees Counting formula Let n , p ∈ N , then n � γ n +1 , p = for n ≥ 1 , γ i , p γ n − i , p + i , i =0 γ 0 , p = p + 1 , γ 1 , p = p 2 + p + 1 . Helps us to efficiently compute c n Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i Michael Wallner | LaBRI | 24.05.2018 11 / 36

Compacted Binary Trees | A recurrence relation A recurrence for compacted binary trees Counting formula Let n , p ∈ N , then n � γ n +1 , p = for n ≥ 1 , γ i , p γ n − i , p + i , i =0 γ 0 , p = p + 1 , γ 1 , p = p 2 + p + 1 . We are interested in c n = γ n , 0 . Helps us to efficiently compute c n Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i Michael Wallner | LaBRI | 24.05.2018 11 / 36

Compacted Binary Trees | A recurrence relation Relaxed compacted binary trees Drop the condition of uniqueness of the subtrees, i.e. c n ≤ r n . Michael Wallner | LaBRI | 24.05.2018 12 / 36

Compacted Binary Trees | A recurrence relation Relaxed compacted binary trees Drop the condition of uniqueness of the subtrees, i.e. c n ≤ r n . 1 4 6 6 2 2 4 4 Michael Wallner | LaBRI | 24.05.2018 12 / 36

Compacted Binary Trees | A recurrence relation Relaxed compacted binary trees Drop the condition of uniqueness of the subtrees, i.e. c n ≤ r n . 1 4 6 6 2 2 4 4 In total, this spine corresponds to 1 · 3 · 4 · 4 2 · 6 2 = 6912 relaxed trees. (Recall, that the same spine corresponds to 4836 compacted trees.) Michael Wallner | LaBRI | 24.05.2018 12 / 36

Compacted Binary Trees | A recurrence relation Relaxed compacted binary trees of size 3 Michael Wallner | LaBRI | 24.05.2018 13 / 36

Compacted Binary Trees | A recurrence relation Relaxed compacted binary trees of size 3 The relaxed tree of size 3 which is not a compacted tree compacted tree binary tree relaxed tree Reason: subtrees not unique Michael Wallner | LaBRI | 24.05.2018 13 / 36

Compacted Binary Trees | A recurrence relation A recurrence for relaxed compacted binary trees Counting formula Let n , p ∈ N , then n � δ n +1 , p = δ i , p δ n − i , p + i , for n ≥ 0 , i =0 δ 1 , p = p 2 + p + 1 . ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ δ 0 , p = p + 1 , ❤ We are interested in r n = δ n , 0 . Michael Wallner | LaBRI | 24.05.2018 14 / 36

Compacted Binary Trees | A recurrence relation A recurrence for relaxed compacted binary trees Counting formula Let n , p ∈ N , then n � δ n +1 , p = δ i , p δ n − i , p + i , for n ≥ 0 , i =0 δ 1 , p = p 2 + p + 1 . ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ δ 0 , p = p + 1 , ❤ We are interested in r n = δ n , 0 . Recursion still too complicated. Michael Wallner | LaBRI | 24.05.2018 14 / 36

Compacted Binary Trees | A recurrence relation A recurrence for relaxed compacted binary trees Counting formula Let n , p ∈ N , then n � δ n +1 , p = δ i , p δ n − i , p + i , for n ≥ 0 , i =0 δ 1 , p = p 2 + p + 1 . ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ δ 0 , p = p + 1 , ❤ We are interested in r n = δ n , 0 . Recursion still too complicated. Example (Relaxed binary trees) size n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 1 1 3 15 111 1119 14487 c n r n 1 1 3 16 127 1363 18628 Michael Wallner | LaBRI | 24.05.2018 14 / 36

Compacted Binary Trees | Operations on trees Operations on trees Michael Wallner | LaBRI | 24.05.2018 15 / 36

Compacted Binary Trees | Operations on trees Bounded right height We restrict to a subclass of relaxed binary trees: bounded right height . Michael Wallner | LaBRI | 24.05.2018 16 / 36

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