Representing Multidimensional Trees. David Brown, Colin Kern, Alex Lemann, Greg Sandstrom Earlham College

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1 Representing Multidimensional Trees David rown, Colin Kern, lex Lemann, Greg andstrom Earlham College 1

2 Introduction Presentation Layout Dave: n Introduction to Multidimensional Trees and Grammars lex: Representing Multidimensional Trees Colin: Chomsky Normal Form (CNF) Transformation for Multidimensional Grammars. 2

3 n Example Context-Free Grammar (CFG) G : Σ, V,, P Σ : {(, )} V : {} : P : {, (), ()} Initial ( ) () ( ) ( ( ) ) () ( ( ) ) ( ) () ( ( ) ) ( ) ( ) () 3

4 n Example CFG as a et of Local Trees Rule 1 ( ) Rule 2 ( ) Rule 3 (2) (1) ( (3) (3) ) (1) (3) ( ) ( ) ( ) 4

5 n Example Tree-djoining Grammar (TG) Σ: V: : { a, b, c } { } I: α1: c {β1, β2} { } { } : a β1: {β1, β2} β2: b {β1, β2} { } a 5 { } b

6 n Example TG Derivation a Γ: γ u γ u b γ l + β b β γ l b b c a 6

7 n Example Multidimensional Grammar Σ: V: : { a, b, c } { I,, } I P: a b a b I c 7

8 Multidimensional Derivation and 2D Yield a I c a a b b b b b b a b b c 8

9 n Infinite Hierarchy Equivalent to Weir s Control Language Hierarchy x y z x y z x y z x y z... 9

10 Representing Multidimensional Trees David rown, Colin Kern, lex Lemann, and Greg andstrom Earlham College 10

11 Left-child Right-sibling Form C D C D E F E F X X C G C D E F H I D E F G H I 11

12 ExLeft and ExUp Definitions ExLeft X Y X Y ExUp X X Y Y 12

13 uilding a 2d tree using ExUp and ExLeft ExUp(F, ) = t1 ExLeft(t1, ) = f1 ExUp(E, ) = t2 C D ExLeft(t1, f1) = f2 ExUp(D, ) = t3 ExLeft(t3, ) = f3 E F ExUp(C, f2) = t4 ExLeft(t4, f3) = f4 ExUp(, ) = t5 ExLeft(t5, t4) = f5 ExUp(, f5) = t6 ExLeft(t6, ) = final tree 13

14 Unified Constructor for Tree-ordered Forest Definition X X f2 f2 f1 f1 14

15 uilding a 2d tree using the unified constructor C D E F T (F,, ) = f1 T (E, f1, ) = f2 T (D,, ) = f3 T (C, f3, f2) = f4 T (, f4, ) = f5 T (,, f5) 15

16 uilding a 3d tree using the unified constructor T (J,,, ) = f3 T (I, f3,, ) = f5 T (H,, f5, ) = f6 Then we create f4: f10 C ~ f9 f8 ~ D ~ F ~ f4 ~ ~ ~ H ~ f7 G ~ f1 ~ E ~ f2 ~ ~ I ~ f5 ~ ~ J f3 f6 ~ ~ T (G,,, ) = f1 T (F, f1,, ) = f4 nd the last child of D is f2: T (E,,, ) = f2 Next we combine them in D: T (D, f2, f4, f6) = f7 nd then we can continue to build the rest of the tree: T (C, f7,, ) = f8 T (,, f8, ) = f9 T (,,, f9) = f10 16

17 Tree Definition bstract Data Type lgebra C++ Class Flat Form 17

18 bstract data type realized as a C++ class template<class label_type> class forest { public: forest(label_type label, vector<forest*> links) void set_label(label_type new_label); void set_link(std::size_t link_number, forest* new_link); label_type get_label( ) const; tree* get_successor(std::size_t link_number) const; private: label_type label; vector<forest*> link; } 18

19 Flat form representation C D E F T (F,, ) = f1 F (, ) T (E, f1, ) = f2 E(F (, ), ) T (D,, ) = f3 D(, ) T (C, f3, f2) = f4 C(D(, ), E(F (, ), )) T (, f4, ) = f5 (C(D(, ), E(F (, ), )), ) T (,, f5) (, (C(D(, ), E(F (, ), )), )) 19

20 CNF Transformation for Multidimensional Grammars David rown, Colin Kern, lex Lemann, Greg andstrom Earlham College 20

21 Chomsky Normal Form C C D C C D C C D 21

22 X C D E X X X X X X X X D C E 22

23 n example factoring of a 3d tree C D E I C D E I F G H D D F G H 23

24 The entire factoring of the tree C D C D E I D τ D E C D E I F G H D D F G H D D τ D E I D F G E τ E I G G τ H 24

25 4-dimensional local tree of a grammar being factored into 2-branching local trees τ E τ F D C E F E C D E 25

26 The transformation algorithm: Traverse the tree in a depth-first method. For every node, check each link for a successor that breaks the 2-branching definition. For every such successor, split the tree into two trees: tree with the subtree rooted at the successor removed, with the current node renamed to a unique label. The subtree rooted at the successor with the current node at the root. Use τ nodes to fill out the nodes of a new tree if the dimension is less than the dimension of the original tree. Repeat on each factored tree until no more factors are created. 26

27 2 dimensions 3 dimensions 4 dimensions C D E C C D # of nodes = # of dimensions

28 C D C D E X X X X X X X X X 28

29 N 3 (0) = 1 N 3 (d) = N 3 (d 1) 2 N 3 (d 1) + 2 We can solve this recursion: N 3 (d) = Ω(k (2(d 1)) ) The growth is hyper-exponential in the dimension 29

30 The 2-branching trees show linear growth in the dimension. The 3-branching trees show hyper-exponential growth in the dimension. 30

31 Theorem 1 For a full d-dimensional, n-branching local tree, the number of local trees in the factored form required is equal to the total number of 1-dimensional links. While the number of local trees in the grammar grows by a factor that is hyper-exponential in the dimension, the growth is optimal in the sense that it differs only by a constant factor from the growth in the number of nodes in arbitrary branching local trees as a function of the dimension. 31

32 Definition 1 Tree-ordered Forests is an (empty) (i, d)-forest for all 0 i d If t 1, t 2,..., t d are, respectively, (0, d)-, (1, d)-,..., (d 1, d)-forests and X Σ then T (X, t 1, t 2,..., t d ) is a (j, d)-forest for all 0 j i, where i is the smallest dimension such that t i is not empty, or d if all t k are empty. Here each t k is the successor of the new node labeled X in the kth dimension. Nothing else is a tree-ordered forest. 32

Even More on Dynamic Programming

Algorithms & Models of Computation CS/ECE 374, Fall 2017 Even More on Dynamic Programming Lecture 15 Thursday, October 19, 2017 Sariel Har-Peled (UIUC) CS374 1 Fall 2017 1 / 26 Part I Longest Common Subsequence