Introduction to Algorithms Massachusetts Institute of Technology Professors Erik D. Demaine and Charles E. Leiserson.

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1 Introduction to Algorithms Massachusetts Institute of Technolog Professors Erik D. Demaine and Charles E. Leiserson October 24, J/18.410J Handout 16 Problem Set 5 MIT students: This problem set is due in lecture on Monda, October 31, The homework lab for this problem set will be held 2 4 P.M. on Sunda, October 30, Reading: Chapter 14 and Skip List Handout. Both eercises and problems should be solved, but onl the problems should be turned in. Eercises are intended to help ou master the course material. Even though ou should not turn in the eercise solutions, ou are responsible for material covered in the eercises. Mark the top of each sheet with our name, the course number, the problem number, our recitation section, the date and the names of an students with whom ou collaborated. Please staple and turn in our solutions on 3-hole punched paper. You will often be called upon to give an algorithm to solve a certain problem. Your write-up should take the form of a short essa. A topic paragraph should summarize the problem ou are solving and what our results are. The bod of the essa should provide the following: 1. A description of the algorithm in English and, if helpful, pseudo-code. 2. At least one worked eample or diagram to show more precisel how our algorithm works. 3. A proof (or indication) of the correctness of the algorithm. 4. An analsis of the running time of the algorithm. Remember, our goal is to communicate. Full credit will be given onl to correct solutions which are described clearl. Convoluted and obtuse descriptions will receive low marks. Eercise 5-1. Do Eercise on page 307 of CLRS. Eercise 5-2. Do Eercise on page 310 of CLRS. Eercise 5-3. Do Eercise on page 317 of CLRS. Eercise 5-4. Do Problem 14.2 on page 318 of CLRS.

2 2 Handout 16: Problem Set 5 Problem 5-1. Skip Lists and B-trees Intuitivel, it is easier to find an element that is nearb an element ou ve alread seen. In a dnamic-set data structure, a finger search from to is the following quer: given the node in the data structure that stores the element, and given another element, find the node in the data structure that stores. Skip lists support fast finger searches in the following sense. (a) Give an algorithm for finger searching from to in a skip list. Your algorithm should run in O(lg(2+ rank() rank() )) time with high probabilit, where rank() denotes the current rank of element in the sorted order of the dnamic set. When we sa with high probabilit we mean high probabilit with respect to m = 2 + rank() rank(). That is, our algorithm should run in O(lg m) time with probabilit 1 1/m α, for an α 1. Assume that the finger-search operation is given the node in the bottommost list of the skip list that stores the element. To support fast finger searches in B-trees, we need two ideas: B + -trees and level linking. Throughout this problem, assume that B = O(1). A B + -tree is a B-tree in which all the kes are stored in the leaves, and internal nodes store copies of these kes. More precisel, an internal node p with k + 1 children c 1, c 2,..., c k+1 stores k kes: the maimum ke in c 1 s subtree, the maimum ke in c 2 s subtree,..., the maimum ke in c k s subtree. (b) Describe how to modif the B-tree SEARCH algorithm in order to find the leaf con taining a given ke in a B + -tree in O(lg n) time. (c) Describe how to modif the B-tree INSERT and DELETE algorithms to work for B + - trees in O(lg n) time. A level-linked B + -tree is a B + -tree in which each node has an additional pointer to the node immediatel to its left among nodes at the same depth, as well as an additional pointer to the node immediatel to its right among nodes at the same depth. (d) Describe how our B + -tree INSERT and DELETE algorithms from part (c) can be modified to maintain level links in O(log n) time per operation. (e) Give an algorithm for finger searching from to in a level-linked B + -tree. Your algorithm should run in O(lg(2 + rank() rank() )) time. These ideas suggest a connection between skip lists and level-linked trees. In fact, a skip list is essentiall a randomized version of level-linked B + -tree. (f) Describe how to implement a deterministic skip list. That is, our data structure should have the same general pointer structure as a skip list: a sequence of one or

3 Handout 16: Problem Set 5 3 ( 2, 3) m 1 = 3 (3, 3) ( 2, 3) m 6 = 3 m 1 = 3 (3, 3) m 6 = 3 (1, 1) m 4 = 4 (1, 1) m 4 = 4 ( 1, 1) (2, 1) m 2 = 2 m 8 = 2 ( 1, 1) m 2 = 2 (2, 1) m 8 = 2 min = 1 Figure 1: In this eample of eight points, if Figure 2: When min = 1, T (1) = f(p i ) = m i, then F (S) = 25. {p 3, p 4, p 5, p 6, p 7, p 8 } and F (T (1)) = 20. more linked lists with pointers between nodes in adjacent lists that store the same ke. The SEARCH algorithm should be identical to that of a skip list. You will need to modif the INSERT operation to avoid the use of randomization to determine whether a ke should be promoted. You ma ignore DELETE for this problem part. Problem 5-2. Fun with Points in the Plane It is 3 a.m. and ou are attempting to watch lectures on video, looking for hints for Problem Set 5. For some odd reason, possibl because ou are fading in and out of consciousness, ou start to notice a strange cloud of black dots on an otherwise white wall in our room. Thus, instead of watching the lecture, our subconscious mind starts tring to solve the following problem. Let S = {p 1, p 2,..., p n } be a set of n points in the plane. Each point p i has coordinates ( i, i ) and has a weight m i (a real number representing the size of the dot). Let f(p) = f(,, m) be an arbitrar function mapping a point p with coordinates (, ) and weight m to a real number, computable in O(1) time. For a subset T of S, define the function F (T ) to be the sum of f(p i ) over all points in T, i.e., F (T ) = f(p). p T For eample, if f(p i ) = m i, then F (S) is the sum of the weights of all n points. This case is depicted in Figure 1. Our goal is to compute the function F for certain subsets of the points. We call each subset a quer, and for each quer T, we want to calculate F (T ). Because there ma be a large number of queries, we want to design a data structure that will allow us to efficientl answer each quer.

4 4 Handout 16: Problem Set 5 First we consider queries that restrict the coordinate. In particular, consider the set of points whose coordinates are at least min. Formall, let T ( min ) be the set of points T ( min ) = {p i S i min }. We want to answer queries of the following form: given an value min as input, calculate the value F (T ( min )). Figure 2 is an eample of such a quer. In this case, min = 1, and the points of interest are those with coordinate at least 1. (a) Show how to modif a balanced binar search tree to support such a quer in O(lg n) time. More specificall, the computation of F (T ( min )) can be performed using onl a single walk down the tree. You do not need to support updates (insertions and deletions) for this problem part. (b) Consider the static problem, where all n points are known ahead of time. How long does it take to build our data structure from part (a)? (c) In total, given n points, how long does it take to build our data structure and answer k different queries? On the other hand, how long would it take to answer k different queries without using an data structure and using the naïve algorithm of computing F (T ( min )) from scratch for ever quer? For what values of k is it asmptoticall more efficient to use the data structure? (d) We can make this data structure dnamic b using a red-black tree. Argue that the augmentation for our solution in part (a) can be efficientl supported in a red-black tree, i.e., that points can be inserted or deleted in O(lg n) time. Net we consider queries that take an interval X = [ min, ma ] (with min ma ) as input instead of a single number min. Let T (X) be the set of points whose coordinates fall in that interval, i.e., T (X) = {p i i [ min, ma ]}. See Figure 3 for an eample of this sort of quer. We claim that we can use the same dnamic data structure from part (d) to compute F (T (X)). (e) Show how to modif our algorithm from part (a) to compute F (T (X)) in O(lg n) time. Hint: Find the shallowest node in the tree whose coordinate lies between min and ma. Finall, we generalize the static problem to two dimensions. Suppose that we are given two intervals, X = [ min, ma ] and Y = [ min, ma ]. Let T (X, Y ) be the set of all points in this rectangle, i.e., T (X, Y ) = {p i i X and i Y }. See Figure 4 for an eample of a two-dimensional quer.

5 Handout 16: Problem Set 5 5 ( 2, 3) m 1 = 3 (3, 3) m 6 = 3 ( 2, 3) m 1 = 3 ma = 2.5 (3, 3) m 6 = 3 (1, 1) m 4 = 4 (1, 1) m 4 = 4 ( 1, 1) (2, 1) m 2 = 2 m 8 = 2 ( 1, 1) m 2 = 2 (2, 1) m 8 = 2 ma = 3.5 min = 1 ma = 1 min = 1 min = 2 Figure 3: When X = [1, 3.5], T (X) = Figure 4: For X = [1, 3.5] and Y = {p 3, p 4, p 6, p 7, p 8 } and F (T (X)) = 14. [ 2, 2.5], T (X, Y ) = {p 3, p 4, p 8 } and F (T (X, Y )) = 8. (f) Describe a data structure that efficientl supports a quer to compute F (T (X, Y )) for arbitrar intervals X and Y. A quer should run in O(lg 2 n) time. Hint: Augment a range tree. (g) How long does it take to build our data structure? How much space does it use? Unfortunatel, there are problems with making this data structure dnamic. (h) Eplain whether our argument in part (d) can be generalized to the two-dimensional case. What is the worst-case time required to insert a new point into the data structure in part (f)? (i) Suppose that, once we construct the data structure with n initial points, we will per form at most O(lg n) updates. How can we modif the data structure to support both queries and updates efficientl in this case? Completel Optional Parts The remainder of this problem presents an eample of a function F that is useful in an actual application and that can be computed efficientl using the data structures ou described in the previous parts. Parts (j) through (l) outline the derivation of the corresponding function f(p i ). The remainder of this problem is completel optional. Please do not turn these parts in! As before, consider a set S = {p 1, p 2,..., p n } of n points in the plane, with each point p i having coordinates ( i, i ) and a weight m i. We want to compute the ais that minimizes the moment of

6 6 Handout 16: Problem Set 5 inertia of the points in the set. Formall, we want to compute a line L in the plane that minimizes the quantit n ( 2 m i d(l, p i ), i=1 where d(l, p i ) is the distance from point p i to the line L. If m i = 1 for all i, we can think of this ais as the orientation of the set. (j) One parameterization of a line in the plane is to describe it using a pair (ρ, θ), where ρ is the distance from the origin to the line and θ is the angle the line makes with the ais. It can be shown that the distance between a point (, ) and a line L parameterized b (ρ, θ) is sin θ cos θ + ρ. We defined the orientation of the set of points S as the line L = (ρ, θ) that minimizes the function n f (ρ, θ) = m i ( i sin θ i cos θ + ρ) 2. i=1 Show that setting αf = 0 gives us the constraint αρ where M 1 sin θ M 1 cos θ + M 0 ρ = 0, n n n M 0 = m i, M 1 = m i i, M 1 = m i i. i=1 i=1 i=1 (k) Show that setting αf = 0 and using the constraint from part (j) leads to the equation αθ where 2 (M 0 M M 1 M 1 ) tan 2θ = M0 (M 2 M 2 ) + M 2 M, n n n 2 2 M = m i i i, M 2 = m i i, M 2 = m i i i=1 i=1 i=1. (l) Give the function f (p i ) that makes the orientation problem a special case of the prob lem we just solved. Hint: The function f (p i ) is a vector-valued function.