Dynamic trees (Steator and Tarjan 83)

1. Dynamic trees (Steator and Tarjan 83)

2. Operations that we do on the trees Maketree(v) w = findroot(v) (v,c) = mincost(v) addcost(v,c) link(v,w,r(v,w)) cut(v) findcost(v)

3. a b c d e f Lets start with paths Assume for a moment that each tree T in the forest is a path. Lets represent each path with a balanced binary search tree. d b e f e a c a b c d

4. Findroot(v) Go up to the root and then down on the leftmost path

5. 1 4 3 2 7 a b c d e f Mincost(v) With every vertex x we record cost(x) = the cost of the edge (x,p(x)) We also record with each vertex x mincost(x) = minimum of cost(y) over all descendants y of x. 7,1 d 3,1 4,4 b e 1,1 2,2 f e a c a b c d

6. Mincost(p) Go down from the root following the minimum cost, when there are ties prefer a right child.

7. 1 4 3 2 7 a b c d e f Addcost(p,c) Rather than storing cost(x) and mincost(x) we will store cost(x) = cost(x) - cost(p(x)) min(x) = cost(x) - mincost(x) Addcost(p,c) : Increase cost(x) of the root by c 7,7,6 d 3,-4,2 4,-3,0 b e 1,-2,0 2,-1,0 f e a c a b c d

8. 3 1 4 3 2 7 7,7,6 d a b c d e f 8,8,7 3,-4,2 4,-3,0 h b e 2,-6,0 1,-2,0 2,-1,0 f e i g a c g h a b c d Link(p,q,c), cut(v) Translates directly into catenation and split of binary search trees 8 1 i h g

9. 3 1 4 3 2 7 a b c d e f Link(p,q,c), cut(v) Translate directly into catenation and split of binary search trees 8 1 i h g 7,7,6 d 3,-4,2 4,-3,0 b e 8,8,7 1,-2,0 2,-1,0 f e h a c 2,-6,0 i a b g c d g h

10. 3 1 4 3 2 7 a b c d e f i Link(p,q,c), cut(v) Translate directly into catenation and split of binary search trees 8 1 i h g 7,7,6 d 3,-4,2 4,-3,0 b e 8,8,7 1,-2,0 2,-1,0 f e h a c 2,-6,0 i a b g c d g h

11. 3 1 4 3 2 7 a b c d e f 8,5,7 i Link(p,q,c), cut(v) Translate directly into catenation and split of binary search trees 8 1 i h g 7,7,6 d 3,-4,2 4,-3,0 3,0,1 b e 8,8,7 1,-2,0 2,-1,0 f e h a c 2,-6,0 i a b g c d g h

12. Rebalancing We have to update cost(x) and min(x) through rotations w v v C A w b b B C A B cost’(v) = cost(v) + cost(w) cost’(w) = -cost(v) cost’(b) = cost(v) + cost(b)

13. Update min: w v v C A w b b B C A B min’(w) = max{0, min(b) - cost’(b), min(c) - cost(c)} min’(v) = max{0, min(a) - cost(a), min’(w) - cost’(w)}

14. 3,3,2 1 4 7 3 2 b a b c d e f 7,4,5 3,0,1 d 2,-5,0 8,5,7 4,-3,0 i i 1,-2,0 h e a c 2,-6,0 f e i a b g c d g h 8 1 3 i h g 7,7,6 d 3,-4,2 4,-3,0 3,0,1 b e 8,5,7 2,-1,0 f e 1,-2,0 h a c 2,-6,0 i a b g c d g h

15. Operations that we can do on paths w = findroot(v) (v,c) = mincost(p) addcost(p,c) link(p,q,c) (p,q,c) = cut(v) findcost(v)

16. d f g i l j Actual tree a b c e h k p q m n o r s t u v w

17. f d i g j l Path decomposition a Partition T into disjoint paths b c Represent the paths as before e h k p q m n o r s t For each root of a path keep its “dashed” parent and the cost of the dashed edge u v w

18. a b c d d f f e g i i g h l j j l k p q m n o r s t u v w Change the path decomposition via Splicing a Splice(p)  b c p e p h k p q m n o r s t u v w

19. a a b c b e c h f f f d d d e k i i g i g g p q a m n o h j j l j l l b r s c k t p q m e n o u h v r s k w p q m t n o r s u t v u v w w Expose(v): Put v and the root on the same path expose(v) 

20. Tree operations We can do them via the path operations and expose: findroot(v) : findrootp(expose(v)) mincost(v) : mincostp(expose(v)) addcost(v) : addcostp(expose(v)) link(v,w,c) : linkp(path(v),expose(w),c)

21. e r g v v g g m d j j w m m z t r r s s u s t t u u Link (example) e r g m d w z t u s

22. Tree operations We can do them via the path operations and expose: findroot(v) : findrootp(expose(v)) mincost(v) : mincostp(expose(v)) addcost(v) : addcostp(expose(v)) link(v,w,c) : linkp(path(v),expose(w),c) cut(v) : cutp(expose(v))

23. a b c f d d f e i g i g h j l l j k p q m n o r s t u v w Cut (example) Cut(h)  a b c e h k p q m n o r s t u v w

24. Tree operations We can do them via the path operations and expose: findroot(v) : findrootp(expose(v)) mincost(v) : mincostp(expose(v)) addcost(v) : addcostp(expose(v)) link(v,w,c) : linkp(path(v),expose(w),c) cut(v) : cutp(expose(v)) findcost(v) : findcostp(expose(v))

25. Analysis Expose may do many splices  Cost of expose may be large ? But on average it will be ok. We will show that in a sequence of m operations we will not do more then O(mlogn) splices This gives an overall time bound of O(mlog2n)

26. Analysis (Cont) Define an edge (v, p(v)) as heavy if size(v) > size(p(v))/2, the edge is light otherwise •  • At most one heavy edge enters any vertex • At most log n light edges on any path

27. d f g i j l Heavy Heavy Example a b c e h k p m n o r s t u v w

28. Define the splice to be heavy/light according to the type of the dashed edge which turns to be solid • Count separately light and heavy splices • No more than O(mlog n) light splices • How many heavy splices ?

29. Bound on the number of dashed heavy edges • A link cannot create any • A cut may create at most log n • A light splice may create 1 • An expose may create 1  O(mlog n)