Approximation: Theory and Algorithms
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1 Approimation: Theor and Algorithms Edit Distance Compleit, Upper and Lower Bounds Nikolaus Augsten Free Universit of Bozen-Bolzano Facult of Computer Science DIS Unit 8 April, 009 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
2 Outline Ke Roots and Left-Most Leaf Descendants Eample: Tree Edit Distance Computation Conclusion Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
3 Ke Roots and Left-Most Leaf Descendants The Edit Distance Algorithm I/II tree-edit-dist(t, T ) td[.. T,.. T ] : empt arra for tree distances; l = lmld(root(t )); kr = kr(l, leaves(t ) ); l = lmld(root(t )); kr = kr(l, leaves(t ) ); for = to kr do for = to kr do forest-dist(kr [], kr [], l, l, td); We need the following algorithms: lmld(i): computes an arra with the left-most leaf descendants of all descendants of a node i kr(l, lc): given the arra l = lmld(i) of left-most leaf descendants, and the number lc of leaf descendants of i, compute all ke roots of the subtree rooted in i Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
4 Ke Roots and Left-Most Leaf Descendants Computing the Left-Most Leaf Descendants lmld(v, l) foreach child c of v (left to right) do l lmld(c, l); if v is a leaf then l[id(v)] id(v) else c first child of v; l[id(v)] l[id(c )]; return l; Input: root node v of a tree T, empt arra l[.. T ] Output: arra l, l[i] is the left-most leaf descendent of node T[i] Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
5 Ke Roots and Left-Most Leaf Descendants Computing the Ke Roots kr(l, lc) kr[..lc]: empt arra; visited[ ]: boolean arra of size l, init with false; k kr ; i l ; while k do if not visited[l[i]] then kr[k- -] i; visited[l[i]] true; i- -; return sort(kr); Input: l[.. T ]: l[i] is the left-most leaf descendent of node T[i] lc = leaves(t) is the number of leaves in T Output: arra kr[.. leaves(t) ] with ke roots sorted b node ID Note: Loop condition is correct, as k i (the number of ke roots is eactl the number of leaves, and it will alwas be filled when all nodes are traversed) Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
6 Eample Trees and Edit Costs T T d f e f c e a c b d a b Eample: Edit distance between T and T. ω ins = ω del = ω ren = 0 for identical rename, otherwise ω ren = Each of the following slide is the result of a call of forest-dist(). Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
7 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j 0 0 l [i] = l [d i ] and l [j] = l [d j ] 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
8 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j 0 l [i] = l [d i ] and l [j] = l [d j ] 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
9 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j 0 l [i] = l [d i ] and l [j] = l [d j ] 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
10 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d j d i 0 l [i] = l [d i ] and l [j] = l [d j ] 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
11 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d j d i 0 0 l [i] = l [d i ] and l [j] = l [d j ] 0 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
12 Eecuting the Algorithm (/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d j d i 0 l [i] = l [d i ] and l [j] = l [d j ] 0 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
13 Eecuting the Algorithm (7/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j 0 l [i] = l [d i ] and l [j] = l [d j ] 0 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
14 Eecuting the Algorithm (8/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j 0 l [i] = l [d i ] and l [j] = l [d j ] 0 0 Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, 009 / 0
15 Eecuting the Algorithm (9/9) l l kr kr i = kr [] = l [i] = j = kr [] = l [j] = temporar arra fd: d i d j l [i] = l [d i ] and l [j] = l [d j ] Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
16 Summar Conclusion Tree Edit Distance Computing Ke Roots and Left-Most Leaf Descendants Tree Edit Distance Eample Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
17 What s Net? Conclusion Compleit of the Tree Edit Distance Lower/Upper Bounds for the Tree Edit Distance Nikolaus Augsten (DIS) Approimation: Theor and Algorithms Unit 8 April, / 0
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