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3 Segment Trees - motivation.  When one wants to inspect a small portion of a large and complex objects. Example: –GIS: windowing query in a map - given a detailed map contains enormous amount of data, the system has to determine efficiently the part of the map corresponding to a specified region (window).

4 Semi dynamic segment trees.  Let I = { [x 1, x 1 ’ ], [x 2, x 2 ’ ], , [x n, x n ’ ] } be a set of n intervals.  Let p 1, p 2, , p m be the list of distinct intervals endpoints, sorted from left to right. The elementary intervals are defined to be : (- , p 1 ), [p 1, p 1 ], (p 1, p 2 ), [p 2, p 2 ],  (p m,  ) p1p1 p2p2 pmpm

5 Constructing a segment tree  A balanced binary tree T. The leaves of T correspond to the elementary intervals (ordered from left to right). The elementary interval in a leaf  is denoted Int(  )  The internal nodes of T correspond to the intervals that are the union of elementary intervals: Int(v) is the union of intervals of its two children.

6 Constructing a segment tree- cont.  Each node or leaf v in T stores the interval Int(v) and a canonical set I(v)  I of intervals. This set contains the intervals [x, x’]  I s.t Int(v)  [x, x’] and Inv(Parent(v))  [x, x’].  Lemma: A segment tree on a set of n intervals uses O(nlogn) storage.

7 Segment tree- example s1s1 s2s2 s5s5 s3s3 s4s4 S5S5 S 2, S 5 S1S1 S1S1 S1S1 S3S3 S4S4 S3S3 S4, S3S4, S3 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7

8 Lemma - proof T is a balances binary tree  its height is O(logn). Claim: any interval [x, x’] is stored in the set I(v) for at most two nodes at the same depth of T. proof: Let v1, v2, v3 be three nodes from left to right in same depth. Suppose [x, x’] is at v1 and v3.  [x, x’] spans the interval from left point of Int(v1) to right point of Int(v3), v2 lies between v1, v3  Int(parent(v2))  [x, x’].

9 Lemma - proof. Continue  Any interval is stored at most twice at a given depth of T.  The total amount of storage is O(nlogn). v1v1 v2v2 v3v3

10 Algorithm: Query Segment Tree(V,q x )  Input: the root of a segment tree and a query point q x.  Output: All intervals in the tree containing q x.  Report all intervals in I(v).  If v is not a leaf –then if q x  Int(lc(v)) –then QuerySegmentTree(lc(v), qx) –else QuerySegmentTree(rc(v), qx)

11 Algorithm: Query Segment Tree(V,q x ) - Complexity.  Visit O(logn) nodes in total.  At each node v spend O(1+k v ) time. The intervals containing a query point q x are reported in O(logn+k) time, where k is the number of reported intervals.

12 Algorithm: Constructing a segment tree.  Sort the endpoints of the intervals and get the elementary intervals.  Construct a balanced binary tree on the elementary intervals and determine Int(v) for each v. (bottom-up manner).  Determine the canonical subsets : insert the intervals one by one to T.

13 Algorithm: Insert Segment Tree(V,[x,x’])  Input: the root of a segment tree and an interval.  Output: The interval will be stored in tree. If Inv(v)  [x, x’] then store [x, x’] at v. else if Int(lc(v))  [x, x’]   then InsertSegmentTree(lc(v), [x, x’]) if Inv(rc(v))  [x, x’]   then InsertSegmentTree(rc(v), [x, x’])

14 Algorithm: Insert Segment Tree(V,[x,x’]) - Complexity.  An interval is stored at most twice at each level.  There is at most one node at every level whose interval contains x (the same for x’).  Hence, We visit at most 4 nodes per level. The insertion time is O(logn). Construction time is O(nlogn).

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16 Introduction  Generalization of union-find problem: copy operation is also supported.  The structure: A bipartite graph with two node sets V 1, V 2 and edges between them.  V 1 form the sets and V 2 form the elements. An edge between v 1  V 1 and v 2  V 2 indicates that v 2 is a member element of the set v 1.

17 Notations  Let  = {S 1,  S m } be a collection of sets.   = {x 1, ,x n } be a collection of elements.  The sets in  are subsets of .  Let  x be the collection of sets that contain an element x.

18 The structure Consists four ingredients:  Set nodes - .  Element nodes - .  Normal nodes - connect sets to elements.  Reversed nodes - connect elements to sets. Whenever there is a path from a set node to an element node, the corresponding element is in the corresponding set.

19 The structure - continue. The following invariant are maintained:  A set node has one outgoing edge.  An element node has one incoming edge.  A normal node has one incoming edge, and at least two outgoing edges.  A reversed node has one outgoing edge and at least two incoming edges.  No edge may go from a reversed node to another reversed node.

20 The structure - continue.  No edge may go from a normal node to another normal node.  There is one unique path from a set node to an element node iff the element is in the set.  Any path in the union-copy structure from a set node to an element node consists of an alternating sequence of normal and reversed nodes. The union copy structure is acyclic.

21 The structure - example S1S1 S2S2 S3S3 S4S4 S5S5 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 set normal reversed element

22 Defining the operations.  Set-create(S) - create a new empty set in  with name S.  Set-insert(S i,x j ) - (defined when x j  S i ). make S i  S i  {x j }.  Set-destroy(S i ) - remove S i from .  Set-find(S i ) - report all elements in S i.  Set-union(S i,S j ) - (defined when S i, S j disjoint). S i  S i  S j and S j .  Set-copy(S i,S j ) - (defined when S j is empty). S j  S i.  Element-create(x) - create an element x in  and  x  .

23 Defining the operations. Continue  Element-insert(x i, S j ) - (define when x i  S j )  xi  xi  {S j }.  Element-destroy(x i ) - remove x i from  and from all its sets.  Element-find(x i ) - report all sets that contain x i.  Element-union(x i,x j ) - (defined when no set contains them both).  S  xj, S  S  {x i }\{x j }, hence  xi becomes  xi  xj and  xj .  Element-copy(x i,x j ) - (defined when x j in no set). Puts x j in all sets containing x i.

24 Set-insert(S 4,x 2 ) S1S1 S2S2 S3S3 S4S4 S5S5 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8

25 S3S3 Set-destroy(S 3 ) S1S1 S2S2 S3S3 S4S4 S5S5 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8

26 Set-union(S 4,S 1 ) S1S1 S2S2 S3S3 S4S4 S5S5 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8

27 Set-copy(S 5,S 6 ) S1S1 S2S2 S3S3 S4S4 S5S5 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 S6S6

28 The structure - notations.  N-UF is a union-find structure represents the normal nodes. Sets in N-UF  normal nodes. Elements in N-UF  outgoing edges.  R-UF is a union-find structure represents the reversed nodes. Sets in N-UF  reversed nodes. Elements in R-UF  incoming edges.

29 Operations N-UF structure supports  N-union(v,w) - Unite the edges of the v and w to become the edges of v. w looses its edges.  N-find(e i ) - Return the normal node of e i.  N-add(v,e i ) - Add edge e i to node v.  N-delete(e i ) - Delete e i from its origin.  N-enumerate(v) - Return all outgoing edges of v.

30 The operations. Set-find(S i )  if out(S i )  nil then Traverse(out(S i ) ) Traverse(e)  case type(dest(e)) of –element: report the element as an answer. –normal: for all e’  N-enumerate(dest(e)) do Traverse(e’ ) –reversed: Traverse(out(R-find(e)))

31 Set-union(S i, S j )  if out(Si)  nil or type(child(S i ))  normal then exchange S i and S j.  if out(Sj)  nil then ready.  else if type(child(S i ))  normal and type(child(S j ))  normal then N-union(child(S i ), child(S j ))  else if type(child(S i ))  normal then N-add(child(S i ), out(S j ))  else // both are reversed or elements. make a normal node v. N-add(v, out(S i ) ) ; N-add(v, out(S j ) ) child(S i )  v ; out(S j )  nil

32 Set-union - examples. SiSi SjSj SiSi SjSj SiSi SjSj SiSi SjSj SiSi SiSi SjSj SjSj v

33 Set-copy(S i, S j )  if out(S i )  nil then ready.  else if type(child(S i ))=reversed then R-add(R-find(out(S i )), out(S j ))  else // type(child(S i ))=normal or element. make a reversed node v. child(v)  child(S i ) R-add(v, out(S i )) R-add(v, out(S j ))

34 Set-copy - examples. SiSi SiSi SjSj SiSi SiSi SjSj v

35 Set-insert(S i, x j )  Make a new edge e  if out(S i )  nil then out(S i )  e  else if type(child(S i ))  normal then N-add(child((S i ), e)  else // type(child(S i )) is reversed or element. Make a normal node v N-add(v, out(S i )) ; N-add(v, e) child(S i ) = v  // now e has an origin.  if in(x j )  nil then in(x j )  e  else if type(parent(x j ))  reversed then R-add(parent(x j ), e)

36 Set-insert(S i, x j ) - continue  else // type(parent(x j )) is normal or set. Make a reversed node w. R-add(w, in(x j ) ); R-add(w, e) parent(xj)  w  // now e has a destination. SiSi SiSi xjxj e v w xjxj

37 Set-destroy(S i )  if out(S i )  nil then case type(child(S i )) of element: in(child(Si))  nil reversed : Restore(out(S i ) ) normal: for all e  N-enumerate(child(S i )) do if type(dest(e))  element then in(dest(e))  nil else //type(dest(e)) is reversed. Restore(e)  remove S i

38 Set-destroy(S i ) - Restore(e)  v  R-find(e)  R-delete(e) from v  if v has only one incoming edge e’ then if type(org(e’))  set or type(child(v))  element then out(v)  e’ else // org(e’) and child(v) types are normal. w  N-find(e’) for all e’’  N-enumerate(child(v)) do N-add(w, e’’ ) remove child(v) remove v

39 Set-destroy(S i ) - examples SiSi v SiSi v e’ w

40 The analysis Theorem Given a collection  of m sets and collection  of n elements, a union copy structure has the following performance: set-create O(1)elm-create O(1) set-insert O(1)elm-insert O(1) set-destroy O(1+k  (F N (n)+F R (m)))elm-destroy O(1+k  (F R (m)+F N (n))) set-findO(1+k  F R (m))elm-findO(1+k  F N (n)) set-unionO(U N (n))elm-unionO(U R (m)) set-copyO(F R (m)) elm-copyO(F N (n))

41 Theorem - proof Any normal node has out degree  n. Any reversed node has in degree  m.  The operations N-find (R ), N-enumerate (R ) operate on sets of size O(n) ( O(m) ). Proof for set-find: If there are k answers, then there are O(k) normal nodes and reversed nodes that have been visited. N-enumerate is linear with the degree of the node  the total time spent on normal nodes is O(k). The time to visit a reversed node is O(F R (m)).  At most O(1+k  F R (m)) time is taken.

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43 Dynamic segments trees - motivation  With semi-dynamic data structures, new segments may only inserted if their endpoints are chosen from a restricted universe.  When using dynamic segments trees segments may be inserted and deleted freely. Split and concatenate operations are also provided.

44 Defining the operations.  Create(T) - (defined if T does not exist yet). Creates an empty tree T.  Insert(T,[x 1,x 2 ]) - (defined if [x 1,x 2 ]  T). [x 1,x 2 ] is inserted to T.  Delete(T,[x 1,x 2 ]) - (defined if [x 1,x 2 ]  T). [x 1,x 2 ] is deleted from T.  Stabbing-query(T,y) - all segments [x 1,x 2 ] for which x 1  y  x 2 are reported.

45 Defining the operations. Continue  Concatenate(T 1,T 2,T) - (defined if the right endpoint of segment in T 1 is less than the left endpoint of segment in T 2, and T is empty. It makes T  T 1  T 2.T 1 and T 2 are returned empty.  Split(T,T 1,T 2,y) - (Defined if all segments [x 1,x 2 ] in T either y  x 1 or x 2  y, and T 1, T 2 are empty). It makes T 1  {[x 1,x 2 ]; x 2  y},T 2  {[x 1,x 2 ]; y  x 1 } T is returned empty.

46 Weak segment tree - definition.  A w.s.t for a set S of segments consists of a binary tree T, with ordered elementary intervals in its leaves. Each node represents a subset of S, s.t  s  S we have: –s is represented exactly once on every path from the root to a leaf, of which the elementary interval in contained in s. –s is not represented on any other path from the root to a leaf. A traditional segment tree is a w.s.t.

47 Weak segment tree - example. I 1,I 2 I 1 =[1,3] I 2 =[2,3] (-,1) [1,1] (1,2) [2,2] (2,3) [3,3] (3,-) I1I1 I2I2 I1I1 I2I2 I2I2 I 1,I 2 I1I1 I1I1 I1I1

48 The union copy structure for dynamic segment trees  Every node in T corresponds to a set in the union-copy structure.  Every segment corresponds to an element.  The union-copy structure has O(n) sets and O(n) elements.

49 The union copy structure for dynamic segment trees. Continue  We take for the N-UF a union-find structure. (O(1) time for N-union).  For R-UF we take a structure in which R-find takes O(1) (UF(i) structure of La Poutre). Allows : R-find in O(1), R-union in O(  (n)) amortized.

50 The structure. The dynamic segment tree consists of:  A week segment tree noted ST T (RB tree). Every node is augmented with an extra pointer to a set node of the union-copy structure.  A union-copy structure.  A dictionary tree (RB tree) noted D T, storing the set S of segments in its leaves order on increasing (lexicography) left endpoint. Every leaf stores a segment which is an element node of the union-copy structure.

51 Why do we use a w.s.t? S2S2 S2S2 S1S1 S1S1 v u After rotation right at v node u holds S 2 – contradicts the definition of a w.s.t! u v S1S1 S2S2 S2S2 S1S1 S1S2S1S2 u v S2S2 S1S1 S1S2S1S2 S1S2S1S2

52 The solution The auxiliary operation Down(  ). Empty a set S  by shifting  ’s elements to both its children:  Set-create(S’)  Set-copy (S ,S’)  Set-union(S lchild(  ),S’)  Set-union(S rchild(  ),S  )  Set-destroy(S’). // remove the empty set S’. This operation will take O(1)!

53 Insert ([x 1,x 2 ],T)  Add [x 1,x 2 ] to D T.  NewEndpoint(x 1,[x 1,x 2 ])  NewEndpoint(x 2,[x 1,x 2 ])    The node in ST T where the paths to x 1 and to x 2 are split.  for  1 on path from lchild(  ) to leaf of x 1 do if x 1 is in left subtree of  1 then Set-insert(S rchild(  1), [x 1,x 2 ])  for  2 on path from rchild(  ) to leaf of x 2 do if x 2 is in left subtree of  2 then Set-insert(S lchild(  2), [x 1,x 2 ])

54 NewEndpoint (x,[x 1,x 2 ])    The leaf that contains x in ST T.  if Int(  ) is an open interval (a,b) then Int(  )  [x,x] add (a,x) as a leaf  ’ to ST T add (x,b) as a leaf  ’’ to ST T Set-create(S  ’ ) ; Set-create(S  ’’ ) Set-copy(S , S  ’ ) ; Set-copy(S , S  ’’ ) m(  )  0 ;  Set-insert(S ,[x 1,x 2 ]) ; m(  )  m(  ) +1

55 Insert ([x 1,x 2 ],T) - example I 1,I 2 I 1 =[1,3] I 2 =[2,3] (-,1) [1,1] (3,-) I1I1 I2I2 Insert ( [1.5,2.5] ) [2,2] (1.5,2) (1,1.5) [1.5,1.5] [3,3] (2.5,3) (2,2.5) [2.5,2.5] I 3 =[1.5,2.5] I1I1 I3I3 I3I3 I3I3 I3I3 m(  )=1 (1,2) (2,3)

56 Delete([x 1,x 2 ],T)    The leaf in D T that contains [x 1,x 2 ]  Element-destroy(  )  delete [x 1,x 2 ] from D T  Remove-endpoint(x 1 )  Remove-endpoint(x 2 )

57 Remove-endpoint(x)    The leaf in ST T that contains x.  m(  )  m(  ) - 1  if m(  ) = 0 then // let Int(leaf left  )=(a,x) // let Int(leaf right  )=(x,b) Int(  )  (a,b) delete from ST T the leaf left of  delete from ST T the leaf right of 

58 [1.5,1.5] Delete ([x 1,x 2 ],T) - example I 1,I 2 I 1 =[1,3] I 2 =[2,3] (-,1) [1,1] (3,-) I1I1 I2I2 Delete ( [1.5,2.5] ) (1.5,2) (1,1.5) (2.5,3) [2.5,2.5] I 3 =[1.5,2.5] I1I1 I3I3 I3I3 I3I3 I3I3 m(  )=0 (1,1.5) (1.5,2) I3I3 I3I3 I3I3 I3I3 [2.5,2.5] [1.5,1.5] (1,2) (2.5,3) [2,2] (2,3) [3,3] (2,2.5)

59 Concatenate(T 1,T 2,T)  if T 1 is empty then T  T 2 ; ready  if T 2 is empty then T  T 1 ; ready   1  rightmost leaf of ST T1 // let Int(  1 ) = (x,  )   2  leftmost leaf of ST T2 // let Int(  2 ) = (- ,y)  Int(  1 ) = (x,y)  delete  2 from ST T2  concatenate ST T1 and ST T2 to ST T  concatenate D T1 and D T2 to D T

60 Stabbing-query(T,y)  for all nodes  on the path ST T to y do Set-find(S  )

61 The analysis Lemma1: The operations insert and concatenate take O(logn) time on a dynamic segment tree that stores n segments, and this operations increase the space of the structure by at most O(logn).

62 The analysis - Continue. Lemma2: Suppose the union copy-structure (as we chose) has O(nlogn) edges. Then O(n) Element-destroy operations take O(nlogn  (n)) time.

63 The analysis - Continue. Theorem: The operations insert and concatenate on the dynamic segment tree as described above for storing a set of n segments each take O(logn) amortized time. The delete operations take O(logn  (n)) amortized time, and stabbing queries take O(logn+k) time. The structure requires O(nlogn) space and can be constructed in O(nlogn) time.