1 / 40

Notation 2

Notation 2. Let A be a sequence of n ordered pairs of real numbers (a 1 , w 1 ), (a 2 , w 2 ), …, (a n , w n ), w i > 0 . A segment A(i, j) is a consecutive subsequence (a i , w i ), (a i+1 , w i+1 ), …, (a j , w j ).

reilly
Download Presentation

Notation 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Notation 2 • Let A be a sequence of n ordered pairs of real numbers (a1, w1), (a2, w2), …, (an, wn), wi > 0. • A segment A(i, j) is a consecutive subsequence (ai, wi), (ai+1, wi+1), …, (aj, wj). • The width w(i, j) of segment A(i, j) is wi+wi+1+ …+ wj. • The sum s(i, j) of segment A(i, j) is ai+ai+1+ …+ aj. • The density d(i, j) of segment A(i, j) is s(i, j) / w(i, j).

  2. Notation 1 • A segment A(i,j) is feasible if its width satisfies ℓ ≦ w(i, j) ≦ u. • The rank of an element in a set P of the real numbers, r(x, P) = | {y | y  P, y  x} |

  3. Density Selection Problem (DSP) 2 • Input: • a sequence A of n ordered pairs of real numbers (a1, w1), (a2, w2), …, (an, wn), wi > 0 • two nonnegative real numbers ℓ, u with ℓ u • a positive integer k. • Output: • The segment A(i*, j*) such that the rank of its density d(i*, j*) in the set of densities of all possible feasible segments is k. The total number of feasible segment is (u - ℓ) n = O(n2)

  4. Density Selection Problem 1 An Example • Input: • A=3, -4, -2, 5, wi = 1 • ℓ= 2, u = 4 • k = 3 • Output: • Densities of all feasible segments: • The k smallest feasible segments: d(1, 2) = -1/2 d(2, 3) = -3, d(3, 4) = 3/2, d(1, 2) = -1/2, d(2, 4) = -1/3, d(1, 3) = -1, d(1, 4) = 1/2, d(i*, j*) = d(1, 2) = -1/2, d(2, 3) = -3, d(1, 3) = -1, d(3, 4) = 3/2,

  5. Generalization of Three Well-known Problems

  6. Previous Results

  7. Transforming DSP to Geometric Slope Selection Problem (GSSP) 5 • Define a set L = {l1, l2 …, ln} of n lines in R2, where li : y = xi x - yi = w(1, i) x – s(1, i) • li andli will intersect at point pij with abscissa xij = (yj -yi)/(xj - xi) = s(i+1, j)/w(i+1, j) = d(i+1, j) • An intersection point of two lines li, lj is feasible if ℓ ≦ xj - xi = w(i+1, j)≦u.

  8. Transforming DSP to Geometric Slope Selection Problem 4 • Input: • a set L = {l1, l2 …, ln} of n lines in R2, where li : y = xi x - yi = w(1, i)x – s(1, i) • two nonnegative real numbers ℓ, u with ℓ u • a positive integer k. • Output: • The feasible intersection point such that the rank of its abscissa in the set of all possible feasible intersection points is k.

  9. DSP Transform to GSSP 2An Example • Input: • A=3, -4, -2, 5 • ℓ= 2, u = 4, wi = 1 • k = 3 • Output: the 3rd feasible density d(1, 2) = -1/2 Transform DSP to GSSP • Input: • l0 ,l1 ,…, l5where li : y = w(1, i) x – s(1, i) = i x - s(1, i) • ℓ= 2, u = 4, wi = 1 • k = 3 • Output: the 3rd feasible intersection point p0,2 = -1/2

  10. ℓ0 : y= 0 ℓ1: y = x-3 ℓ2: y = 2x+1 ℓ3: y = 3x+3 ℓ4: y = 4x-2 DSP Transform to GSSP 1 An Example 5 ℓi: y=w(1,i) x-s(1,i) ℓj: y=w(1,j) x-s(1,j) When ℓi intersects ℓj, we have x = s(i+1,j)/w(i+1,j) = d(i+1, j) p0,2 = -1/2 0 • ℓ = 2, u = 4, k = 3 An intersection point is feasible if ℓ ≦ xj - xi = w(i+1, j)≦ u. -5 Find the k-th abscissa over all feasible intersection points (solid blackpoints) -5 0 5 10

  11. Comparing DSP with Traditional Selection Problem

  12. Main Ideas for Solving DSP: Prune-and-Search 4 • Using three techniques to do Prune-and-Search • Random Sampling: Randomly generate n intersection points such that their abscissae lie in a given interval [dl, dr] • Counting mode for Density Range Query Problem Count the total number of feasible intersection points in a given interval [dl, dr]. • Reporting mode for Density Range Query Problem Enumerate feasible intersection points in a given interval [dl, dr].

  13. Main Ideas for Solving DSP: Prune-and-Search 3 • To find a subinterval [dl', dr'] • containing the k-th feasible point d* = d(i*, j*) • containing only a specific ratio of feasible points in [dl, dr] and a specific ratio of points in [dl, dr]. • To prune the set of points outside [dl', dr']. dl d* dr dl' dr'

  14. Main Ideas for Solving DSP: Prune-and-Search 2 • to check whether [dl', dr'] containing k-th feasible pt d* • k2 = # of feasible points in [dl, dr'] • k1 = # of feasible points in [dl, dl'] • d* in [dl', dr']  k1 < k ≦ k2 • to check whether [dl', dr'] containing a specific ratio of feasible pts in [dl, dr] and a specific ratio of pts in [dl, dr]. dl d* dr dl' dr'

  15. Main Ideas for Solving DSP: Prune-and-Search 1 • Repeat the algorithm in [dl', dr']until:[dl'', dr''] contains O(n) feasible intersection points and d* lies in [dl'', dr''] • Reporting mode for Density Range Query Problem • Enumerate all feasible intersection points in [dl'', dr''] by the Reporting subroutine of DRQP • Select from those feasible points the k-th smallest point d* by using any standard selection algorithm. dl'' dl d* dr dl' dr'' dr'

  16. Density Range Query Problem: Reporting Mode 3 • Input: • a sequence A of n ordered pairs of real numbers (a1, w1), (a2, w2), …, (an, wn), wi > 0 • two nonnegative width bounds ℓ, u with ℓ u • two density bounds dl, dr with dl dr • Output: • Enumerate all feasible segments A(i, j) such that their densities d(i, j) satisfy dl  d(i, j)  dr.

  17. Example:DRQP Reporting Mode 2 • Input: • A = 1, 3, 5, 7, 9, 11 , wi = 1 • ℓ = 2, u = 4 • dl = 4.5, dr = 7.5 • Output: s(2, 4), s(3, 5), s(2, 5) s(3, 4), 1 segment of width 6 2 segments of width 5 4 feasible segments of width 3 5 feasible segments of width 2 3 feasible segments of width 4 6 segments of width 1 0 1 2 3 4 5 6 7 8 9 10 11 12

  18. Density Range Query Problem: Reporting Mode 1 • Lemma 1: The reporting version of the Density Range Query Problem can be solved in O(n) space and O(n log m + h) time, where m is min{(u-ℓ)/wmin, n} and h is the output size. • Proof: Reduce to n range queries of the form R= [u, w] x (-∞,v] using dynamic priority search tree.

  19. Density Range Query Problem: Counting Mode 3 • Input: • a sequence A of n ordered pairs of real numbers (a1, w1), (a2, w2), …, (an, wn), wi > 0 • two nonnegative width bounds ℓ, u with ℓ u • two density bounds dl, dr with dl dr • Output: • Find the number of feasible segments A(i, j) such that their densities d(i, j) satisfy dl  d(i, j)  dr.

  20. Example: DRQP Counting Mode 2 • Input: • A = 1, 3, 5, 7, 9, 11 • ℓ = 2, u = 4 • dl = 4.5, dr = 7.5 • Output: 4 0 1 2 3 4 5 6 7 8 9 10 11 12

  21. Density Range Query Problem: Counting Version 1 • Lemma 2:The counting version of the Density Range Query Problem can be solved in O(n) space and O(n log m) time, where m is min{(u-ℓ)/wmin, n} • Proof: Reduce to n rank queries to order-statistics trees of size m.

  22. Random Sampling 3 • Lemma 3: [Dillencourt et al. ‘92] • Given a set L = {l1, l2 …, ln} of n lines in R2 and an interval [dl, dr]  We can randomly generate n intersection points in O(n log n) time to obtain a random sampling S such that the abscissae of all intersection points of S are in [dl, dr]. • Proof: By merge sort technique.

  23. Monte Carlo Random Sampling 5 • Input: • a set L = {l1, l2 …, ln} of n lines in R2, where li : y = xix - yi = w(1, i) x – s(1, i). • two nonnegative real numbers ℓ, u with ℓ u • two density bounds dl, dr with dl dr • Output: • Randomly generate a set S of n intersection pts such that their abscissae lie in [dl, dr] and contain M to 3M feasible intersection points, where M = (n/2)(Nf/N), Nf = # feasible intersection pts in [dl, dr] N = # intersection pts in [dl, dr]

  24. Monte Carlo Random Sampling 4 • We can show that in a random sampling of n intersection points the probability of the number F of feasible intersection points lying between M and 3M, is no less than ½, i.e., Proba.{M≦F≦3M} ≧½, where F is # feasible intersection pts in S.

  25. Monte Carlo Random Sampling 3 An Example Randomly generate n = 6 pts such that their abscissaelie in [dl, dr]= [3.5, 8.5]. They may contain only 1 feasible pt < M = 1.7, where M = (n/2)(Nf/N), Nf=12, N= 21. • Input: • A = 1, 3, 5, 7, 9, 11 • ℓ = 2, u = 4 • dl = 3.5, dr = 8.5 • Output: s(3, 3), s(5, 5),s(3, 4),s(1, 5) , s(2, 6) , s(1, 6) 0 1 2 3 4 5 6 7 8 9 10 11 12

  26. Monte Carlo Random Sampling 2 An Example Randomly generate n = 6 pts such that their abscissaelie in [dl, dr]= [3.5, 8.5]. They may contain 6 feasible pts > 3M = 5.1, where M = (n/2)(Nf/N), Nf=12, N= 21. • Input: • A = 1, 3, 5, 7, 9, 11 • ℓ = 2, u = 4 • dl = 3.5, dr = 8.5 • Output: s(2, 3), s(3, 4), s(4, 5), s(2, 4) , s(3, 5) , s(2, 5) 0 1 2 3 4 5 6 7 8 9 10 11 12

  27. Monte Carlo Random Sampling 1 An Example Randomly generate n = 6 pts such that theirabscissaelie in [dl, dr]= [3.5, 8.5]. They contain M = 1.7 to 3M = 5.1 feasible pts, where M = (n/2)(Nf/N), Nf=12, N= 21. • Input: • A = 1, 3, 5, 7, 9, 11 • ℓ = 2, u = 4 • dl = 3.5, dr = 8.5 • Output: s(2, 4), s(3, 5), s(2, 5) s(3, 4), 0 1 2 3 4 5 6 7 8 9 10 11 12

  28. Monte Carlo Random Sampling 3 • Lemma 4:The Monte Carlo Random Sampling can be solved in O(n) space and expected O (n log n) time. • Proof: • Monte Carlo random sampling subroutinewith success probability p ≧1/2 • By Chebyshev’s inequality.

  29. Using Monte Carlo Random Sampling to Contract Search Interval2 • Assume we have obtained a random sampling S' containing M to 3M feasible points, S' = {s1, s2, …,sF}, where M ≦ F ≦ 3M • For each si in S', it has a probability k/Nf such that it is smaller than or equal to d*. • Consider such an event a "success" in performing F independent Bernoulli trials, each with a probability k/Nf. s4 s1 s2 s3 sF dl d* dr

  30. Using Monte Carlo Random Sampling to Contract Search Interval1 • Let Xi be a random variable, • attaining value 1 with probability p= k/Nf • attaining value 0 with probability q= 1 - k/Nf • X = X1 + X2 +… + XF # sample feasible pts smaller than or equal tod*. •  = F p = F k/Nf ,  = (F p(1-p))1/2 ≦ F1/2/2. • The -th smallest element sin S' should be a good approximation for the k-th smallest feasible intersection point d*. • Let l' = max {1,  - t F1/2/2} and r' = min {F,  + t F1/2/2} for some constant t to be determined later. • Select l'-th smallest element dl' and the r'-th smallest element dr' in S' by any standard selection algorithm to obtain a subinterval [dl' ,dr']. sr' s sl' dl dl' d* dr' dr

  31. Properties of Subinterval Obtained by Monte Carlo Random Sampling3 • Using the Monte Carlo Random Sampling, wecan find a subinterval [dl',dr'] such that • d*[dl', dr'] with probability p≧1/2. • it contains O(Nf/M1/2) feasible pointswith probability p≧1/2 and O(N/M1/2) intersection pointswith probability p≧1/2.

  32. Properties of Subinterval Obtained by Monte Carlo Random Sampling2 • Lemma 5: • N = # of intersection pts in [dl, dr] • Nf = # of feasible intersection pts in [dl, dr]  For a random choice of F independent feasible pts in [dl, dr] where M≦F≦3M, we can find a subinterval [dl', dr'] such that the probability that the k-th smallest feasible intersection pt d* not lying in [dl', dr'] is at most 2e-t2/2. • We can choose t large enough such that 2e-t2/2≦ ½.

  33. Properties of Subinterval Obtained by Monte Carlo Random Sampling 1 • Lemma 6: • N = # of intersection pts in [dl, dr] • Nf = # of feasible intersection pts in [dl, dr] • For a random choice of F independent feasible pts in [dl, dr] where M≦F≦3M, we can find a subinterval [dl', dr'] such that the probability that it contains at least (t2/(t-1))(Nf/M1/2) feasible pts is at most e-sqrt{M}/2(t-1)and the probability that it contains at least (3t2/2(t-1))(N/M1/2) pts is at most e-sqrt{M}/2(t-1). • We can choose M large enough such that e-sqrt{M}/2(t-1) ≦ ½.

  34. Density Selection Range Query Problem 3 • Input: • a sequence A of n ordered pairs of real numbers (a1, w1), (a2, w2), …, (an, wn), wi > 0 • two nonnegative real numbers ℓ, u with ℓ u • an interval [dl, dr] • a positive integer k. • Output: • The feasible segment A(i*, j*) such that the rank of its density d*[dl, dr] in the set of densities of possiblefeasible segments in [dl, dr] is k.

  35. DSP: A Special Case of DSRQP 2 • The Density Selection Problem is a special case of Density Selection Range Query Problem such that [dl, dr] = (-, ).

  36. Example: DSRQP 1 • Input: • A = 1, 3, 5, 7, 9, 11 • ℓ = 2, u = 4 • dl = 3.5, dr = 8.5 • k = 6 • Output: s(3, 5) 0 1 2 3 4 5 6 7 8 9 10 11 12

  37. Algorithm for Density Selection Range Query Problem 2 • Use Monte Carlo Random Sampling to select F feasible segments S = {s1, s2, …, sF} in [dl, dr]. • Let l' = max {1,  - (t/2)} and r' = min {F,  + (t/2)}, for some t and select the l'-th smallest element dl' and the r'-th smallest element dr' in S, where  is expected value of d* and  is standard deviation. dl dl' dl' dr' d* dr' dr dl' dr' Until d* lies in [dl', dr'] and [dl', dr']contains < Nf'=(t2/(t-1))(Nf/M1/2) feasible pts & < (3t2/2(t-1))(N/M1/2) pts If [dl', dr'] contains ≧(t2/(t-1))(Nf/M1/2)feasible pts or ≧ (3t2/2(t-1))(N/M1/2) pts, then repeat the random experiment. If d* doesn’t lie in [dl', dr'], then repeat the random experiment.

  38. Algorithm for Density Selection Range Query Problem 1 • We first run the algorithm in [dl, dr] until: d* lies in [dl', dr'] and [dl', dr'] contains less than Nf' = (t2/(t-1))(Nf/M1/2) feasible intersection points • We then run the algorithm once more in [dl', dr'] until: d* lies in [dl'', dr''] and [dl'', dr''] contains less than Nf'' = (t2/(t-1))( Nf'/M1/2) =O(n) feasible intersection points • Then, we can enumerate all feasible densities in the interval [dl'', dr''] by the reporting mode of Density Range Query Problem • Then select from those densities theκ-th smallest density d* for an appropriate κby using any standard selection algorithm. dl'' dl d* dr dl' dr'' dr'

  39. Main Theorem • Theorem 1: The Density Selection Problem can be solved in O(n) space and expected (n log n) time.

  40. Conclusion • We have presented a randomized algorithm for the Density Selection Problem that runs in expected (n log n) time. • It is a generalization of three well known problems, • maximum density segment problem, • slope selection problem • selection problem. • How to find a deterministic algorithm for the problem that runs within the same time bound is still open.

More Related