Search Tree Applications

Transcription

1 Search Tree Applications 7.1

2 Line Segment Intersection: Detection 7.2

3 Horizontal-Vertical Line Segment Intersection Given: set L H of horizontal line segments; set L V of vertical line segments. Does any line segment in L V intersect a line segment in L H? 7.3

4 Horizontal-Vertical Line Segment Intersection h i U Q S g f e V W d c R a T b 7.4

5 Horizontal-Vertical Line Segment Intersection Brute force algorithm: 1 foreach vertical line segment σ V L V do 2 foreach horizontal line segment σ H L H do 3 if σ H σ V then 4 q σ H σ V ; 5 return (q); 6 end 7 end 8 end 9 return ( No intersection ); 7.5

6 Sort Horizontal Lines h i U Q S g f e V W d c R a T b 7.6

7 1 L V vertical line segments; Sort Horizontal Lines 2 L H horizontal line segments sorted by increasing y; 3 foreach vertical line segment σ V L V do 4 σ H first line segment in L H with σ H.y σ V.miny; 5 while (σ H NIL) and (σ H.y σ v.maxy) do 6 if σ H σ V then 7 q σ H σ V ; 8 return (q); 9 end /* Get next horizontal segment in L H */ 10 σ H σ H.Next; 11 end 12 end 13 return ( No intersection ); 7.7

8 Line Sweep Algorithm h i U Q S g f e V W d c R a T b 7.8

9 Line Sweep Algorithm 1 L V vertical line segments sorted by increasing x; 2 L H horizontal line segments sorted by increasing y; 3 foreach vertical line segment σ V L V do 4 σ H first line segment in L H with σ H.y σ V.miny; 5 while (σ H NIL) and (σ H.y σ v.maxy) do 6 if σ H σ V then 7 q σ H σ V ; 8 return (q); 9 end /* Get next horizontal segment in L H */ 10 σ H σ H.Next; 11 end 12 end 13 return ( No intersection ); 7.9

10 Line Sweep Algorithm h i U Q S g f e V W d c R a T b 7.10

11 Tree Minimum Greater Than or Equal to Find minimum key greater than or equal to

12 Tree Minimum Greater Than or Equal to function TreeMinGE(T, K) /* Return node with min key greater than or equal to K */ 1 u NIL; 2 v T.root; 3 while (v NIL) do 4 if (K v.key) then 5 u v; 6 v v.left; 7 else 8 v v.right; 9 end 10 end 11 return (u); 7.12

13 Horizontal-Vertical Line Segment Intersection 1 T.Init(); 2 L endpoints of all line segments sorted by x-coordinate; 3 foreach point p L do 4 if (p is the left endpoint of a horizontal segment σ H ) then 5 Create new tree node u with 6 u.key = σ H.y and u.data.lineseg = σ H ; 7 TreeInsert (T, u); 8 else if (p is the lower endpoint of a vertical segment σ V ) then 9 u TreeMinGE (T, p.y); 10 if (u NIL) and (u.y σ V.maxy) then 11 return (u.data.lineseg σ V ); 12 else if (p is the right endpoint of a horizontal segment σ H ) then 13 u tree node representing σ H ; 14 TreeDelete (T,u); 15 end 16 end 17 return ( No intersection ); CSE

14 Line Segment Intersection: Reporting 7.14

15 Report All HV Line Segment Intersections Given: set L H of horizontal line segments; set L V of vertical line segments. Report all the intersections of line segments in L H and L V. 7.15

16 Report All HV Line Segment Intersections h i U Q S g f e V W R d c a T b 7.16

17 Report All HV Line Segment Intersections Brute force algorithm: 1 foreach horizontal line segment σ H L H do 2 foreach vertical line segment σ V L V do 3 if σ H σ V then 4 q σ H σ V ; 5 Report intersection point q; 6 end 7 end 8 end 7.17

18 Running Times of Reporting Let I = number of intersections. Desired running time is Θ(f(n) + I) where f(n) is small (e.g., f(n) Θ(n log 2 (n)).) 7.18

19 Report: Line Sweep Algorithm h i U Q S g f e V W R d c a T b 7.19

20 1 T.Init(); Report All HV Line Segment Intersections 2 L endpoints of all line segments sorted by x-coordinate; 3 foreach point p L do 4 if (p is the left endpoint of a horizontal segment σ H ) then 5 Create new tree node u with 6 u.key = σ H.y and u.data.lineseg = σ H ; 7 u.data.lineseg σ H ; 8 TreeInsert (T, u); 9 else if (p is the lower endpoint of a vertical segment σ V ) then 10 Report line segments in T intersecting σ V ; 11 else if (p is the right endpoint of a horizontal segment σ H ) then 12 u tree node representing σ H ; 13 TreeDelete (T,u); 14 end 15 end 7.20

21 Report in Range Report all nodes of the following tree with keys in range [8, 25]: (Range [8, 25] includes 8 and 25.)

22 Report in Range: Running Time procedure TreeRangeReport(x, k min, k max ) 1 if (x NIL) then 2 if (k min x.key) then 3 TreeRangeReport(x.left, k min, k max ); 4 end 5 if (k min x.key k max ) then print x.key; 6 if (x.key k max ) then 7 TreeRangeReport(x.right, k min, k max ); 8 end 9 end h = tree height I = number of nodes reported. Running time: 7.22

23 1 T.Init(); Report All HV Line Segment Intersections 2 L endpoints of all line segments sorted by x-coordinate; 3 foreach point p L do 4 if (p is the left endpoint of a horizontal segment σ H ) then 5 Create new tree node u with 6 u.key = σ H.y and u.data.lineseg = σ H ; 7 u.data.lineseg σ H ; 8 TreeInsert (T, u); 9 else if (p is the lower endpoint of a vertical segment σ V ) then 10 TreeRangeReport (T.root, σ V.miny, σ V.maxy); 11 else if (p is the right endpoint of a horizontal segment σ H ) then 12 u tree node representing σ H ; 13 TreeDelete (T,u); 14 end 15 end 7.23

24 Line Segment Intersection: Counting 7.24

25 Count HV Line Segment Intersections Given: set L H of horizontal line segments; set L V of vertical line segments. For each σ V L V, count number of intersections with lines in L H. 7.25

26 Count HV Line Segment Intersections h i U Q S g f e V W R d c a T b 7.26

27 1 count 0; 2 T.Init(); Count HV Line Segment Intersections 3 L endpoints of all line segments sorted by x-coordinate; 4 foreach point p L do 5 if (p is the left endpoint of a horizontal segment σ H ) then 6 Create new tree node u; 7 u.key σ H.y; 8 u.data.lineseg σ H ; 9 TreeInsert (T, u); 10 else if (p is the lower endpoint of a vertical segment σ V ) then 11 count count+ CountNumIntersections (T, σ V ); 12 else if (p is the right endpoint of a horizontal segment σ H ) then 13 u tree node representing σ H ; 14 TreeDelete (T,u); 15 end 16 end 7.27

28 Binary Search Tree: Counting For each node u in the binary search tree, add a field u.size which stores the number of nodes in the subtree rooted at u

29 Count Nodes Greater Than or Equal to function TreeCountGE(x, K) /* Return number of nodes with keys greater than or equal to K in subtree rooted at x */ 1 count 0; 2 v x; 3 while (v NIL) do 4 if (v.key K) then 5 count count + 1; 6 if (v.right NIL) then count count + v.right.size; 7 v v.left; 8 else 9 v v.right; 10 end 11 end 12 return (count); 7.29

30 Count Number of Nodes in Range function TreeRangeCount(x, k min, k max ) /* Return number of nodes with keys in range [k min, k max ] in subtree rooted at x */ 1 v x; 2 while (v NIL) and (v.key [k min, k max ]) do 3 if (v.key k min ) then v v.right; 4 else if (v.key k max ) then v v.left; 5 end 6 if (v NIL) then 7 count L TreeCountGE(v.left, k min ); 8 count R TreeCountLE(v.right, k max ); 9 return (1 + count L + count R ); 10 else return (0); 7.30

31 1 count 0; 2 T.Init(); Count HV Line Segment Intersections 3 L endpoints of all line segments sorted by x-coordinate; 4 foreach point p L do 5 if (p is the left endpoint of a horizontal segment σ H ) then 6 Create new tree node u; 7 u.key σ H.y; 8 u.data.lineseg σ H ; 9 TreeInsert (T, u); 10 else if (p is the lower endpoint of a vertical segment σ V ) then 11 count count+ TreeRangeCount (T, σ V.miny, σ V.maxy); 12 else if (p is the right endpoint of a horizontal segment σ H ) then 13 u tree node representing σ H ; 14 TreeDelete (T,u); 15 end 16 end 7.31

32 Point Location u G t E F D s q C B A v r 7.32

33 Point Location Given: set S rect of non-intersecting, axis-parallel rectangles; set S point of points. Report the rectangle containing each point. 7.33

34 Data Structures: Applications 7.34

35 Data Structures Stack; Heap (HeapSort, Prim s MST algorithm, Dijkstra s shortest path algorithm); Red-black tree (Line segment intersection); Hash table (Duplicate elements, counting number of elements); Union-find structure (Connected components). 7.35

36 Exercise: Deep Thought - Highest Priority You are in charge of the supercomputer Deep Thought. Deep Thought has an Input List of programs which it needs to process; Every minute new programs are being added to the Input List by scientists from across the world; Each program has a priority number from 1 to ; Each time Deep Thought completes a program, it selects the next program with highest priority from the Input List and works on that program. Give a data structure for quickly adding new programs to the list and selecting the next program for Deep Thought to process. 7.36

37 Exercise: Deep Thought - Count Competitive scientists always want to know how many other programs in the Input List have priority less than their program, and how many have priority greater than their program. Every minute new programs are being added to the Input List by scientists from across the world; Each program has a priority number from 1 to ; Each time Deep Thought completes a program, it selects the next program with highest priority from the Input List and works on that program. Give a data structure which allows a scientist to quickly query how many programs in the Input List have higher priority and how many programs have lower priority. 7.37

38 Exercise: Deep Thought - Wait Because Deep Thought is such a valuable resource, some scientists want to hog Deep Thought for their research. For this reason, you instituted a rule that after a scientist has submitted a program, she must wait 24 hours until she can submit another program. Each scientist has a 128 bit password which she must submit with her program; Each program has a submission time which is the time the program was added to the Input List. Give a data structure for quickly determining if the scientist who submitted a program has waited 24 hours from the last time she submitted a program. 7.38

39 Exercise: Deep Thought - Median Priority Some selfish scientists have been assigning high priority to their programs to get them processed first. To punish such selfishness, Deep Thought no longer selects the program with highest priority from the Input List. Instead, Deep Thought selects the program with median priority. Every minute new programs are being added to the Input List by scientists from across the world; Each time Deep Thought completes a program, it selects the next program with median priority from the Input List and works on that program. Let n be the number of programs in the Input List. The median priority is an integer I such that n/2 elements have priority less than or equal to I and n/2 elements have priority greater than or equal to I. Give a data structure for quickly adding new programs to the list and selecting the next program for Deep Thought to process. 7.39