Dynamic Algorithms Multiple Choice Test

Transcription

1 3226 Dynami Algorithms Multiple Choie Test Sample test: only 8 questions 32 minutes (Real test has 30 questions 120 minutes) Årskort Name Eah of the following 8 questions has 4 possible answers of whih exatly one is orret. For eah question, you may selet one or more answers by heking the orresponding boxes. Your test is graded as follows: If you selet only the orret answer, you reeive 2 points. If you selet 2 answers, one of whih is orret, you reeive 1 point. If you selet 3 answers, one of whih is orret, you reeive 0.41 points. if you selet no answer or all answers, you reeive 0 point. If you selet only one answer, and it is wrong, you reeive points. If you selet 2 answers, that are both wrong, you reeive -1 point. If you selet 3 answers, that are all wrong, you reeive points. Note that perfet answers yield a sore of 16 points and random guessing may give you a negative sore. For the logial basis of this sore system, see Frandsen and Shwartzbah, A singular hoie for multiple hoie [

2 Question 1 Consider the six-oloring sheme from Mehlhorn-Sundar-Uhrig applied to the sequene of numbers 13 = , 22 = , 6 = 110 2, 22 = , 26 = In one oloring step one reahes initial after one step What oloring is reahed after an additional step? a b d

3 3226 Question 2 What is the union-split-find data struture a A fully dynami algorithm for maintaining onneted omponents in an undireted graph b An extension to the union-find data struture onsisting in a split operation that allows the removal of a single element from one of the sets An extension to the union-find data struture that makes it equivalent to a van Emde Boas tree. d An extension to the union-find data struture onsisting in a split operation that allows an undo of the latest union operation Question 3 What is the time needed per operation in Eppstein et al.s algorithm for maintaining a minimum spanning forest in a dynami plane graph? a O(log 2 n) b O(log n) O(log 2 n log n) d O( n) Question 4 What is the ut-value tehnique for onstruting dynami algorithms for algebrai problems suh as prefix omputation? a One onsiders a prefix iruit and defines a subset S of the iruit nodes to be a ut if the output is determined by the values at the ut nodes. Suppose there is a ut S suh that eah output node depends on at most d(n) nodes in the ut and eah input node influenes at most d(n) nodes in the ut then one may derive a solution for dynami prefix omputation of omplexity O(d(n)). b One interprets a prefix iruit as a flow graph and defines a ut to be a subset V of the iruit nodes suh that V ontains all the input nodes and no output nodes. If the value of eah node in V an be omputed from d(n) other nodes and if the values of eah node in V influenes the values of at most d(n) nodes outside V then we may derive a solution for dynami prefix omputation of omplexity O(d(n)). One onsiders a prefix iruit and defines a subset S of the iruit nodes to be a ut if the output is determined by the values at the ut nodes. Suppose there is a ut S suh that the value of eah ut node an be omputed from at most d(n) input nodes and eah ut node influenes at most d(n) output nodes then one may derive a solution for dynami prefix omputation of omplexity O(d(n)). d One interprets a prefix iruit as a flow graph and defines a ut to be a subset V of the iruit nodes suh that V ontains all the output nodes and no input nodes. If the value of eah node in V an be omputed from d(n) nodes outside V and if the values of eah node in V influenes the values of at most d(n) other nodes then we may derive a solution for dynami prefix omputation of omplexity O(d(n)).

4 Example Consider the following data type for maintaining balane information in a string of parentheses init(s). Initialise input to the string s of length n. hange(i, ),1 i n. Replae the i th entry in s with parenthesis. query. Deide whether s is perfetly balaned. It is implemented by a binary tree data struture that is illustrated in the figure for the string s ="[[{}}[]]". [ [ { } } [ ] ] The data kept at the internal nodes 1,...,7 for the speifi string "[[{}}[]]" is given by the table 4, 5, 6, 7 true,true,true,true "##","{}","##","##" "[[","##","#[","##", "##","##","}#","]]" 2, 3 true, true "####","#[]#" "[[##","####" "####","}##]" 1 false "[[##}##]" "########" "########" Question 5 This question refers to the previous example desribing a binary tree based data struture for maintaining parenthesis balaning information. In the example no details were given on how the data strings at the internal nodes of the binary tree are represented. Whih of the following representations of the strings mathed at node are part of a solution allowing hange and balane query operations within time O(log O(1) n)? a A string s ="[[##}##]" is divided in the middle into 2 strings s 1 ="[[##" and s 2 =reverse("}##]"). Then blanks # are removed and the two strings are kept in a Mehlhorn-Sundar-Uhrig data struture for dynami sequenes. b A string s ="[[##}##]" is transformed into a string onsisting of just one kind of parentheses t ="((##)##)". Then blanks # are removed and the resulting string is kept in a data struture speialized to keep balaning information for just 1 kind of parenthesis. A string s ="[[##}##]" is transformed into a string onsisting of just one kind of parentheses t ="((##)##)". Without removing blanks # the resulting string is kept in a data struture speialized to keep balaning information for just 1 kind of parenthesis. d A string s ="[[##}##]" is divided in the middle into 2 strings s 1 ="[[##" and s 2 =reverse("}##]"). Without removing blanks the two strings are kept in a Mehlhorn- Sundar-Uhrig data struture for dynami sequenes.

5 3226 Question 6 This question refers to the previous example desribing a binary tree based data struture for maintaining parenthesis balaning information. In the example no details were given on how the data strings at the internal nodes of the binary tree are represented. One may map suh a string to a matrix using a homomorphism h and making omputations modulo a random prime. Whih of the following suggestions for h is part of a randomized solution for the parenthesis balaning problem allowing hange and balane query operations within time O(log O(1) n)? a x # [ ] { } ( ) b x ( # [ ] { } ) d x # [ ] { } ( x # [ ] { } ( ) )

6 Question 7 This question refers to the previous example desribing a binary tree based data struture for maintaining parenthesis balaning information. Consider a hange(5, { ) operation applied to the example resulting in the string "[[{}{[]]" What data should be kept at nodes 6,3, and 1 after the hange? a 6, 3, 1 true, false, false "##","{[]]","#[#####]" "{[","####","[###{###" "##","####","########" b 6, 3, 1 true, false, false "##","{[]]","########" "{[","####","[[######" "##","####","########" 6, 3, 1 true, true, true "##","#[]#","########" "{[","{###","[[######" "##","###]","########" d 6, 3, 1 true, true, true "##","#[]#","#[#####]" "{[","{###","[###{###" "##","###]","########" Question 8 The Frandsen-Miltersen-Skyum implementation of the dynami word problem for group-free monoids makes use of van Emde Boas Trees. How many van Emde Boas trees are needed for a monoid M satisfying the seond ase of the Krohn-Rhodes deomposition theorem, i.e. M \ {1} = a = {a, a 2, a 3,...,a k = a k+1 } a 2 b 0 1 d k