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Euler tour trees

Euler tour trees. 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). 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.

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Euler tour trees

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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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