Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search. Pawel Gawrychowski, Shay Mozes, Oren Weimann. Max?

Transcription

1 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search Pawel Gawrychowski, Shay Mozes, Oren Weimann Max?

2 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search

3 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search M ik + M jl M il + M jk k l i j

4 Submatrix Maximum Queries in Partial Monge Matrices are Equivalent to Predecessor Search undefined

5 Submatrix Maximum Queries in Partial Monge Matrices are Equivalent to Predecessor Search M ik + M jl M il + M jk undefined

6 Submatrix Maximum Queries in Partial Monge Matrices are Equivalent to Predecessor Search M ik + M jl? M il + M jk undefined

7 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search Max?

8 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search

9 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search O(n polylogn) space requires Ω(loglogn) query-time [Patrasçu, Thorup STOC 06] Given n integers in {1,2,, n 2 } 1 x 2 3 x x 9... x x n 2

10 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search [Kaplan,Mozes, Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] [Gawrychowski, Mozes, W. ICALP 15]

11 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search n x n Monge: [Kaplan,Mozes, Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] [Gawrychowski, Mozes, W. ICALP 15] Space O(n log n) O(n) O(n) Query O(log 2 n) O(log n) Θ(loglogn)

12 Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search n x n Monge: [Kaplan,Mozes, Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] [Gawrychowski, Mozes, W. ICALP 15] Space O(n log n) O(n) O(n) Query O(log 2 n) O(log n) Θ(loglogn) n x n partial: Space O(n log n α(n)) O(n) O(n) Query O(log 2 n) O(log n α(n)) Θ(loglogn)

13 Application si Shortest [Kaplan,Mozes,Nussbaum,Sharir paths in planar SODA 12] graphs

14 Application I Shortest paths in planar graphs

15 Application I Shortest paths in planar graphs A piece

16 Application I Shortest paths in planar graphs i l j k j i l k M ik + M jl M il + M jk

17 Application II: Largest empty rectangle Input: a set of n points Query: find largest empty rectangle containing a point

18 Submatrix Maximum Queries Max?

19 Today: Subcolumn Maximum Queries Max

20 Even easier: entire-column ranges Max

21 Even easier: entire-column ranges The rows of the column maxima increase monotonically

22 Even easier: entire-column ranges The rows of the column maxima increase monotonically

23 Even easier: entire-column ranges Enough to compute list of breakpoints (predecessor search) O(n) time SMAWK [Shor,Moran,Aggarwal,Wilber,Klawe 1987] The rows of the column maxima increase monotonically

24 Even easier: entire-column ranges Enough to compute list of breakpoints (predecessor search) O(n) time SMAWK [Shor,Moran,Aggarwal,Wilber,Klawe 1987] O(n α(n)) time for partial matrices [Klawe, Kleitman 1990] The rows of the column maxima increase monotonically

25 Subcolumn Maximum Queries Max

26 The tree of breakpoints [Kaplan,Mozes,Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] Each node computes the breakpoints of its submatrix By merging the breakpoints of its two children (overall O(n log n) time and space)

27 The tree of breakpoints [Kaplan,Mozes,Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] Each node computes the breakpoints of its submatrix By merging the breakpoints of its two children (overall O(n log n) time and space)

28 The tree of breakpoints [Kaplan,Mozes,Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] Each node computes the breakpoints of its submatrix By merging the breakpoints of its two children (overall O(n log n) time and space)

29 The tree of breakpoints [Kaplan,Mozes,Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] A subcolumn query j

30 The tree of breakpoints [Kaplan,Mozes,Nussbaum,Sharir SODA 12] [Gawrychowski, Mozes, W. ICALP 14] A subcolumn query is covered by O(log n) canonical nodes. In each canonical node, find predecessor(j) in its breakpoints. j O(log n loglog n) time O(log n) via fractional cascading

31 Our Tree Each node stores breakpoints of every suffix/prefix of rows

32 Our Tree Each node stores breakpoints of every suffix/prefix of rows

33 Our Tree - Query A subcolumn query only two predecessor(j) searches j LCA

34 Our Tree - Construction

35 Our Tree - Construction

36 A subcolumn query Our Tree - Construction j LCA

37 Our Tree A subcolumn query = 2 x predecessor queries on a root-to-leaf path = 2 x weighted ancestor queries = O(1) predecessor queries in O(loglogn) time - Construction j LCA m size of this tree = m (each row adds one breakpoints and kills some others) size of all trees = O(nlogn) space

38 Improving the space from O(n log n) to O(n)

39 Improving the space Theorem [Gawrychowski, Mozes, W. ICALP 14] Given a logn-by-n Monge matrix, there is a O(logn) space data structure that answers entire-column queries in O(1) time. Mega Row entries fetched in O(1) time using the above Theorem. n logn Mega Row logn Mega Row logn Mega Row logn Mega Row

40 Given n integers in {1,2,, n 2 }: Lower Bound x x x x x x n 2 linear-time reduction n n Max? Two challenges: Matrix has to be Monge Implicit O(n polylog n) rep. fetching entries in O(1) time

41 Lower Bound n n n n n x x x x x x n 2 linear-time reduction

42 Lower Bound n n n n n x x x x x x n 2 linear-time reduction

43 Partial Monge matrices

44 Partial Monge matrices The rows of the column maxima increases monotonically

45 Partial Monge matrices

46 Partial Monge matrices Decomposition I: Reduction to staircase matrices

47 Partial Monge matrices Decomposition I: Reduction to staircase matrices Decomposition II: Cover staircase matrices by full matrices Much more

48 Open Problems

49 Open Problems Shortest paths in planar graphs 49

50 Open Problems Shortest paths in planar graphs m-by-n staircase 50

51 Open Problems Shortest paths in planar graphs m-by-n staircase O(log 2 n) O(log 2 n) O(log 2 n) In the beginning all rows are deactivated activate a row and add k to all its entries delete column report minimum active entry [Fakcharoenphol Rao, 2006] 51

52 Open Problems Shortest paths in planar graphs m-by-n staircase In the beginning all rows are deactivated O(log 2 n) activate a row and add k to all its entries O(log 2 n) delete column O(log 2 n) report minimum active entry [Fakcharoenphol Rao, 2006] Find the O(m) breakpoints in linear time 52

53 Open Problems Shortest paths in planar graphs m-by-n staircase In the beginning all rows are deactivated O(log 2 n) activate a row and add k to all its entries O(log 2 n) delete column O(log 2 n) report minimum active entry [Fakcharoenphol Rao, 2006] Find the O(m) breakpoints in linear time O((m+n)α(n))) [Klawe Kleitman, 1990] O(mlogn) [Gawrychowski, Mozes, W. ICALP 14] 53

54 Arigato!