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Maintain a Forest: Euler Tour Trees Operation

Implement and manage forest operations using Euler tour trees including link, cut, find tree, find min val, change val, add val functions. Utilize search trees for efficient vertex value storage.

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Maintain a Forest: Euler Tour Trees Operation

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  1. Euler tour trees

  2. Maintain a forest under the operations: link(v w) cut(v,w) find-tree(v) find-min-val(T) change-val(v,x) add-val(T,x)

  3. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  4. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  5. p o n (p o) (o n) (n n) (n o) (o o) (o p) (p p) a (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) b m c k l d i (o p) (p p) (p o) (o n) (n n) (n o) (o o) e j f (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) h g

  6. p o n (o p) (p p) (p o) (o n) (n n) (n o) (o o) a (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) b m c k l d i e (o p) (p p) (p o) (o n) (n n) (n o) (o o) (o l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l o) j f h g

  7. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  8. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  9. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  10. p o n a b m c  k l d i e (p o) (o n) (n n) (n o) (o o) (o p) (p p) j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  11. p o n (p o) (o n) (n n) (n o) (o o) (o p) (p p) a (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) b m c k l d i e (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) j f h (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) g

  12. Link + cut • Linked lists would do. • What about finding the list containing a vertex ? • What about vertex values ?

  13. Search trees • Represent the lists as search trees.

  14. p o n a b (p o) (o n) (n n) (n o) (o o) (o p) (p p) m c k l d i e j f (a a) (a b) (b b) (b c) (c d) (d d) (d e) (e e) (e f) (f f) (f g) (g g) (g h) (h h) (h i) (i i) (i k) (k k) (k i) (i l) (l l) (l i) (i j) (j j) (j i) (i h) (h g) (g f) (f e) (e d) (d c) (c c) (c b) (b m) (m m) (m b) (b a) h g

  15. So we can easily do link(v,w) : assume v and w are in different trees cut(v,w) : assume v and w are adjacent in a tree findtree(v) In logarithmic time What about vertex values ? Store with each node the minimum value in its subtree

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