Priority Search Trees Keys are distinct ordered pairs (x i, y i ). Basic operations.  get(x,y) … return element whose key is (x,y).  delete(x,y) … delete and return element whose key is (x,y).  insert(x,y,e) … insert element e, whose key is (x,y). Rectangle operations.

minXinRectangle(x L,x R,y T ) Return element with min x-coordinate in the rectangle defined by the lines, x= x L, x= x R, y = 0, y = y T, x L <= x R, 0 <= y T. I.e., return element with min x such that x L <= x <= x R and 0 <= y <= y T. xLxL xRxR yTyT

maxXinRectangle(x L,x R,y T ) Return element with max x-coordinate in the rectangle defined by the lines, x= x L, x= x R, y = 0, y = y T, x L <= x R, 0 <= y T. I.e., return element with max x such that x L <= x <= x R and 0 <= y <= y T. xLxL xRxR yTyT

minYinXrange(x L,x R ) Return element with min y-coordinate in the rectangle defined by the lines, x= x L, x= x R, y = 0, y = infinity, x L <= x R. I.e., return element with min y such that x L <= x <= x R. xLxL xRxR

enumerateRectangle(x L,x R,y T ) Return all elements in the rectangle defined by the lines, x= x L, x= x R, y = 0, y = y T, x L <= x R, 0 <= y T. I.e., return all elements such that x L <= x <= x R and 0 <= y <= y T. xLxL xRxR yTyT

Complexity O(log n) for each operation except for enumerateRectangle, where n is the number of elements in the tree. Complexity of enumerateRectangle is O(log n + s), where s is the number of elements in the rectangle.

Applications – Visibility Dynamic set of semi-infinite vertical line segments.  Vertical lines with end points (x i,infinity) and (x i,y i ). (2,1) (3,4) (4,2) (5,6) (6,3) Opaque/translucent lines.

Translucent Lines Eye is at (x,y). Priority search tree of line end points. enumerateRectangle(x, infinity, y). (2,1) (3,4) (4,2) (5,6) (6,3) 0 y x infinity

Opaque Lines Eye is at (x,y). Priority search tree of line end points. minXinRectangle(x, infinity, y). (2,1) (3,4) (4,2) (5,6) (6,3) 0 y x infinity

Bin Packing First fit. Best fit. Combination.  Some items packed using first fit.  Others packed using best fit. Memory allocation.

Combined First And Best Fit Bins are numbered 0, 1, …, n – 1. Capacity of each is c. Initialize priority search tree with the pairs (c, j), 0 <= j < n.

First Fit minYinXrange(neededSize, infinity) bin index available capacity neededSize

Best Fit minXinRectangle(neededSize, infinity, infinity) bin index available capacity neededSize

Online Intersection Of Linear Intervals Intervals are of the form [i,j], i < j. [i,j] may, for example represent the fact that a machine is busy from time i to time j. Answer queries of the form: which intervals intersect/overlap with a given interval [u,v], u < v.  List all machines that are busy at any time between u and v.

Example Machine a is busy from time 1 to time 3. Interval is [1,3]. Machines a, b, c, e, and f are busy at some time in the interval [2,4]. 1 e 13 a 24 c 46 b 36 f 57 d 2

Example Interval [i,j] corresponds to the pair (x,y), where x = j and y = i. 1 e 13 a 24 c 46 b 36 f 57 d 2 ae f b c d enumerateRectangle(u, infinity, v). enumerateRectangle(2, infinity, 4).

Compare With Segment Tree Min and max interval end points must be known in advance to define the root range of the segment tree. Search interval size is 1. Must break larger search intervals into unit intervals and remove duplicate responses.

Interval Containment List all intervals [i,j] that contain the interval [u,v].  [i,j] contains [u,v] iff i <= u <= v <= j. 1 e 13 a 24 c 46 b 36 f 57 d 2 [u,v] = [5,6]

Interval Containment Interval [i,j] corresponds to the pair (x,y), where x = j and y = i. 1 e 13 a 24 c 46 b 36 f 57 d 2 ae f b c d enumerateRectangle(v, infinity, u). enumerateRectangle(6, infinity, 5).

Intersecting Rectangle Pairs (A,B), (A,C), (D, G), (E,F) Online interval intersection. A F C D E B G

Algorithm Examine horizontal edges in sorted y order.  Bottom edge => insert interval into a priority search tree.  Top edge => report intersecting segments and delete the top edge’s corresponding bottom edge. A F C D E B G

Complexity Examine edges in sorted order.  Bottom edge => insert interval into a priority search tree.  Top edge => report intersecting segments and delete the top edge’s corresponding bottom edge. O(n log n) to sort edges by y, where n is # of rectangles.  Insert n intervals … O(n log n).  Report intersecting segments … O(n log n + s).  Delete n intervals … O(n log n).

IP Router Table Longest-prefix matching  10*, 101*, 1001*  Destination address d =  Longest matching-prefix is 101* Prefix is an interval  d is 5 bits => 101* = [10100, 10111] = [20,23] 2 prefixes may nest but may not have a proper intersection

IP Router Table p(d) = prefixes that match d. Use online interval intersection mapping. p(d) = enumerateRectangle(d,infinity,d) d

IP Router Table lmp(d) = [maxStart(p(d)), minFinish(p(d))] minXinRectangle(d,infinity,d) finds lmp(d) except when >1 prefixes have same finish point. d

IP Router Table Remap finish points so that all prefixes have different finish point. f’ = 2 w f – s + 2 w – 1, w = length of d f’ is smaller when s is bigger d

IP Router Table Complexity is O(log n) for insert, delete, and find longest matching-prefix. d