Tea as ed

1
Te???a ?as???? ??µ?? ?ed?µ????
??d?s???te? ?a???? ???st??, ?sa?a??d??
??a??s??? e-mail makri_at_ceid.upatras.gr,
tsak_at_ceid.upatras.gr ??das?a??a ?e?t??a
1300-1500 ?200
2
?e????af? t?? ?a??µat??

• ?? µ???µa ape????eta? se ?s??? f??t?t?? ?????? ?a
  ap??t?s??? ßas???? ???se?? ??a t?? pe????? t??
  d?µ?? ded?µ???? ?a? ??a t?? a??????µ???? te??????
  p?? ????? p??ta?e? ??a t?? ap?d?t??? s?ed?as? ?a?
  ???p???s? t???. 
• ?? µ???µa ap?te?e? s????e?a t?? p??pt???a???
  µa??µ?t?? ??µ?? ?ed?µ????, ???????µ??,
  ???????µ??e? ??µ?? ?a? pa???e? st??? f??t?t?? ta
  ?at?????a ef?d?a ??a t?? s?ed?as? p???p?????
  d?µ?? ded?µ???? p?? µp????? ?a ß???? ???s? se
  d??f??e? efa?µ????.

3
??t??e?µe?a p?? ?a??pt??ta?

• 1.           ???t??a ?p?????sµ?? ?a? µet?????
  ???????? ?a? ??????? p???p????t?ta?
• 2.        ??a???????? ??µ?? ?ed?µ????
• 3.        ???????µ?? ?a? ??µ??  ?ed?µ???? se
  p??ß??µata d?a?e???s?? de?d????? ??af?µ?t??.
• 4.          ????? ???te?a??t?ta? ?a?
  ??t???a??µe?a ???t?a
• 5.         ???????µ?? ???ta??? ?a? ??µ??
  ?a??µat?? st? RAM ???t??a ?p?????sµ??.

4
??ad??ast???

• ???tas? (p??f?????)
• ???as?a
• ?a???s?as?
• G?apt? ??af???
• ?e????? ?a?µ??
• ?µ??????sµa a? ßa?µ?? e??tas?? gt 5

5
??sta ???as???

• Data Structures for Tree Manipulation
• D. Harel and R.E. Tarjan. Fast Algorithms for
  finding nearest common ancestors. SIAM J.
  Computing , 13(2)338-355, 1984.
• A.K. Tsakalidis. The Nearest Common Ancestor in a
  Dynamic Tree, Acta Informatica 25, 37-54 (1988).
• S. Alstrup and M. Thorup, Optimal Pointer
  Algorithms for Finding Nearest Common Ancestors
  in Dynamic Trees, Journal of Algorithms, 35(2)
  169-188 (2000)
• A.L. Buchsbaum, H. Kaplan, A. Rogers and J.R.
  Westbrook, Linear-time pointer machine algorithms
  for lca's, mst verification, and dominators, In
  Annual ACM Symposium on the Theory of Computing
  (STOC), 30, 1998.
• R. Cole and R. Hariharan, Dynamic lca queries on
  trees, In Annual ACM-SIAM Symposium on Discrete
  Algorithms (SODA), 10, 1999.
• Rajamani Sundar, Robert Endre Tarjan Unique
  Binary Search Tree Representations and
  Equality-testing of Sets and Sequences STOC 1990
  18-25
• Kurt Mehlhorn, R. Sundar, Christian Uhrig
  Maintaining Dynamic Sequences under Equality
  Tests in Polylogarithmic Time. Algorithmica
  17(2) 183-198 (1997).
• Richard Cole, Ramesh Hariharan Tree Pattern
  Matching to Subset Matching in Linear Time. SIAM
  J. Comput. 32(4) 1056-1066 (2003)

6

• Persistence
• J. R. Driscoll, N. Sarnak, D. Sleator, and R.
  Tarjan. Making data structures persistent. J. of
  Computer and System Science, 3886-124, 1989.
• J. Driscoll, D. Sleator, and R. Tarjan. Fully
  persistent lists with catenation. Journal of the
  ACM , 41(5)943-959, 1994.
• L. Buchsbaum and R. E. Tarjan. Confluently
  persistent deques via data structural
  bootstrapping. J. of Algorithms , 18513-547,
  1995.
• R. Sundar A. L. Buchsbaum and R. E. Tarjan. Data
  structural bootstrapping, linear path
  compression, and catenable heap ordered double
  ended queues. SIAM J. Computing ,
  24(6)1190-1206, 1995.
• H. Kaplan and R. E. Tarjan. Persistent lists
  with catenation via recursive slow-down. In
  Proceedings of the 27th Annual ACM Symposium on
  Theory of Computing , pages 93-102. ACM Press,
  1995.
• H. Kaplan and R. E. Tarjan. Purely functional
  representations of catenable sorted lists. In
  Proceedings of the 28th Annual ACM Symposium on
  Theory of Computing , pages 202-211. ACM Press ,
  1996.
• A. Fiat, H. Kaplan, Making Data Structures
  Confluently Persistent, ACM SODA 2001.

7

• Search Trees and Priority Queues
• A.K. Tsakalidis, AVL-trees for localized search.
  Information and Control , 67173-194, 1985.
• R. Fleischer, A simple balanced search tree with
  O(1) worst-case update time. International
  Journal of Foundations of Computer Science,
  7137-149, 1996
• G. S. Brodal. Finger Search Trees with Constant
  Insertion Time. In Proc. 9th Annual ACM-SIAM
  Symposium on Discrete Algorithms, pages 540-549,
  1998.
•  M. A. Bender and M. Farach-Colton. The Level
  Ancestor Problem Simplified. LATIN, pages
  508-515, 2002.
• M. A. Bender, R. Cole, E. Demaine, M.
  Farach-Colton, and J. Zito. Two Simplified
  Algorithms for Maintaining Order in a List.
  Proceedings of the 10th European Symposium on
  Algorithms (ESA), pages 152-164, 2002.
• Daniel Dominic Sleator and Robert Endre Tarjan.
  Self-adjusting binary search trees. Journal of
  the ACM , 32(3)652-686, July 1985.
• John Iacono, Alternatives to Splay Trees with
  O(logn) Worst-Case Access Times, ACM/SIAM SODA
  2001, 516-522.
• Mihai Badoiu and Erik D. Demaine, A Simplified
  and Dynamic Unified Structure,  in Proceedings of
  the 6th Latin American Symposium on Theoretical
  Informatics (LATIN 2004), Lecture Notes in
  Computer Science, volume 2976, Buenos Aires,
  Argentina, April 5-8, 2004, pages 466-473.
• D.E. Demaine, D. Harmon, J. Iacono, M. Patrascu,
  (2004), Dynamic Optimality? Almost, IEEE Symp. on
  the Foundations of Computer Science, 45th, Rome,
  Italy, Oct. 17?19, pp. 484?490.

8

• RAM Algorithms
• M.L. Fredman and D.E. Willard. Surpassing the
  information theoretic bound with fusion trees.
  Journal of Computer and System Sciences , 
  47424-436, 1993.
• M.L. Fredman and D.E. Willard. Trans-dichotomous
  Algorithms for Minimum Spanning Trees and
  Shortest Paths, Journal of Computer and System
  Sciences , 48533-551, 1994.
• H. Gabow, R. Tarjan A Linear-Time Algorithm for
  a Special Case of Disjoint Set Union Journal of
  Computer and System Sciences 30(209-220) 1985 .
• Arne Andersson, Mikkel Thorup, Dynamic Ordered
  Sets with Exponential Search Tree, ACM STOC 2000,
  pp.335-342.
• A. Andersson. Faster deterministic sorting and
  searching in linear space, Proc. 37th FOCS, pages
  135?141, 1996.
• Yijie Han Improved fast integer sorting in
  linear space. SODA 2001 793-796
• Yijie Han Deterministic sorting in O(nloglogn)
  time and linear space. J. Algorithms 50(1)
  96-105 (2004)
• Ran Mendelson, Mikkel Thorup, Uri Zwick
  Meldable RAM priority queues and minimum directed
  spanning trees. SODA 2004 40-48
• Ran Mendelson, Robert Endre Tarjan, Mikkel
  Thorup, Uri Zwick Melding Priority Queues. SWAT
  2004 223-235

9
??sa??????

• ??µ? ?ed?µ???? ? s???e???µ??? ???p???s? e???
  s???? eµfa????µe??? ?f???µ???? ??p?? ?ed?µ????.
• ?f???µ???? ??p?? ?ed?µ???? e??a? ??a s????? µa??
  µe µ?a s?????? p???e?? sta st???e?a t?? s??????.
• ?a?ade??µata ??µ??
• -- ?e???? (ßas???? p???e?? ? e?sa????/d?a??af?
  st???e??? ?a? ? ??e???? a???e?/de? a???e?)
• -- ????? ???te?a??t?ta? (ßas???? p???e?? ? ???es?
  st???e???, ? e??es? ?a? d?a??af? t?? e?a??st??)

10
???t??a ???a???

• ???t??? ???a??? ?e??t?? (Pointer Machine) ?
  µ??µ? ap?te?e?ta? ap? s?????? e???af??, µe ???e
  e???af? ?a ap?te?e?ta? ap? µ?a s?????? ap? ?e???.
  ???e ?e?? ??e? ??a t?p? (pointer, integer, real)
  ?a? ? p??sp??as? ?e???? ???eta? µe ???s? de??t??.
  
• RAM ???t??? ?p?????sµ?? ? µ??µ? ap?te?e?ta? ap?
  ??a p??a?a ap? ?e???, ?p?? ???e ?e??
  p??spe?a??eta? µe t? d?e????s? t??. ?p???t??µe
  ?t? ???e ?e?? µp??e? ?a pe????e? a???µ???
  a??a??et?? µe?????? ?a? ?t? ?? ßas????
  a???µ?t???? p???e?? ?a? p???e?? de??t?? pa??????
  sta?e?? ????? (uniform cost assumption).
• ?et????? ??t?µ?s?? ?p?d?s??
• ??????, ??????? p???p????t?ta.

11
??????? ????p????t?ta

• ?????s? µ?s?? pe??pt?s?? (average case analysis)
  µ?a p??a??t??? ?ata??µ? t??eta? st?? p???e?? e???
  af???µ???? t?p?? ded?µ???? ?a? t? µ?s? ??st??
  t?? p???e?? ?p??????eta?.
• ?????s? ?e???te??? ?e??pt?s?? (worst case
  analysis) ?p????????ta? p???p????t?te?
  ?e???te??? pe??pt?s?? ??a ???e p????.
• ?????s? ?p?µe??sµ???? ????p????t?ta?
  ?p??????eta? t? s??????? ??st?? µ?a? a???????a?
  p???e?? ?a? ep?µe???eta? t? ??st?? se ?a?eµ?a
  p????.

12
?e?????? ??t?µ?s?? ????p????t?ta?

• ????d?? ?????s?? - ?µes?? ?p?????sµ?? t??
  ??st??? µ?a? a???????a? p???e?? µ?s? e???
  s??d?ast???? t?p??
• ????d?? ??ape??t? Te????µe ?t? ?p???e? ??a?
  t?ape????? ???a??asµ?? s??dedeµ???? µe ???e d?µ?
  ded?µ???? p?? a?a????µe. Se ???e p????
  s?s?et????µe µ?a e????e?a a??????? ?a? ?at??es??
  ???µ?t??.
• ????d?? F?s???? ??a??t??µe se ???e st?µ??t?p?
  t?? d?µ?? µ?a s????t?s? p?? ?aµß??e? ?et????
  p?a?µat???? t?µ?? ?a? ???µ??eta? s????t?s?
  d??aµ????.

13
?? p??ß??µa t?? ?e?????

• ???sµ??
• ??d? ??µ??
• Comparison Based ??µ?? ?ed?µ????
• Representation Based ??µ?? ?ed?µ????

14
??d? ??µ?? ?ed?µ????

• Implicit ??µ?? ?ed?µ???? (p??a?a? st???e???)
• ???s? ?e??st?ef?µe??? ??st??, ?µµes?
  ??d???p???s? de??t?? -gt ?((logn)2) ????? a??
  ????µ? ?a? p???? e??µ???s?? (Munro)
• O(logn) ????? a?? ????µ? ?a? p???? e??µ???s??
  (Francescinni)
• ??d? ??µ??
• Comparison Based ??µ?? ?ed?µ????
• - Static
• - Dynamic (weight balanced, height balanced)
• Representation Based ??µ?? ?ed?µ????
• - Interpolation Search
• - Tries
• - Hashing

15
?? Weighted Dictionary ???ß??µa

• ????e??
• Access(x,S), O(log(W/w))
• Insert(x,S) O(log(Ww/min(w,w-,w))
• Delete(x,S) O(log(W/min(w,w-))
• Join(S1,S2) O(log((w1w2)/w))
• Split(x,S) O(log(W/w))
• ChangeWeight(x,S,d) ?(log(Wd)/w))
• Amortized complexity a? ???s?µ?p??????? splay
  trees.
• Worst Case Complexity a? ???s?µ?p??????? biased
  trees.

16
Union-Split-Find ???ß??µa

• ??a?e?????µaste µ?a d?aµ???s? t?? s?µpa?t??
  U1,..,nµe t?? ?p?st????? t?? a????????
  p???e??
• Find(x) ep?st?e?e t? ???µa t?? s?????? p??
  pe????e? t? x
• Union(A,B) s?????e?se ta d?? s????a ?,? se ??a
  ??? s?????\
• Split(A,x) d??spase t? ? se d?? ?p?s????a
• Union-Find p??ß??µa, Split-Find p??ß??µa
• ?(a(m,n))minz?1 A(z,4?m/n?)gtlogn
• ?p??
• A(i,0)1, ??a ???e i
• ?(0,x)2x, ??a ???e x
• ?(i1,x1)A(i,A(i1,x)) ??a ???e i,x.
• Interval Union Split Find ???ß??µa
• Upper and Lower Bounded by ?(loglogn)
• vEB Data Structure

17
????? ???te?a??t?ta?

• ?as???? p???e??
• Makequeue
• Insert(i,h)
• Findmin(h)
• Deleremin(h)
• Meld(h1,h2)
• Decreasekey(d,i,h)
• Delete(i,h)
• Heap-ordered(2-3) d??t?a, leftist d??t?a,
  binomial queues, self adjusting heaps, pairing
  heaps, Fibonacci Heaps, Relaxed Heaps.

18
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19
Nearest Common Ancestor

• static off-line version
• static on-line version
• linking roots version
• linking and cutting version
• dynamic trees version

20
?e?????? ???aµ?p???s??

• Te??e?ste µ?a d?µ? µe ????? p??epe?e??as?a?
  Ps(n), ????? e??t?s?? Qs(n), ?a? ????? Ss(n) ?ts?
  ?ste ?? s??a?t?se?? Qs(n), Ps(n)/n, Ss(n)/n ?a
  e??a? a????se?. ???a? ef??t? ? d??aµ?p???s? ?a?
  pa?a???? µ?a? d?µ?? µe ???????? lognQs(n) ???????
  p???p????t?ta , lognPs(n)/n ????? e???se?? ?a?
  ???? Ss(n).