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Unformatted text preview: g UNIVERSITE
_ EUNIVERSITY Department of Computer Science and Engineering case 2011: Fundamentals of Data Structures (Fall 2004) Instructors: G. Turpin and N. Vlajic Final Examination
December 16, 2004 ' Instructions: may be used. There are 8 questions. The points for each question are given in square brackets. next to the
question title. The overall maximum score is 100. Answer each question in the space provided. If you need to continue an answer onto the back
of a page, clearty indicate that and label the continuation with the question number. Question Points
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SECTION (circleone): 7 lo [20
8 I10 Total 1. True/False Choice [10 points] Indicate whether each of the following statements is true or false. No marks will be deducted for QR.“ incorrect answers. The space requirement of a vectorbased implementation of an illbalanced binary tree 2 Insertion in an AVL tree is "commutative". That is. inserting
x and then y into an AVL trees leaves the same tree as
inserting y and then _x. 6mm ow. L i .4 . The fifth smallest element of a min—heap stored in a
complete binary tree will have a depth of four or less. 53i— '  nary search tree with n elements can be converted into
a heap in O(n) time. The worst case running time of 'insert’ operation in _a_hash
table with chaining can be as low as 0(1). n addressin works well even keys than can be stored directly in the hash table. QKDSECK add r easing.
Let G=(V,E) be a weighted graph and let M be a UPS
spanning tree of G. The path in M between' any pair of
vertices v1 and v2 must be their shortest path in G. A graph algorithm with 8(mn) running time is asymptotically better than an algorithm with 9(nn) running time for
a ‘free tree‘ graph with n veﬁic nd ed es. {rice idea: conneded {mega
he aides (I? [1 point] [1 point] [1 point] [1 point] [1 point] \\d—B\\ f... 2. Algorithm Running Time [6 points] (a) [2 points]
What is the execution running time of add 8 n) (see the code below). as a function of n (n 2 0).
(Express the running time using big—O notation, you do not need to provide an exact analysis. int add(a,b) . i
if (a==0) return b; ff a 5" 0 else if§a<0) return add(a+1, bl); {g 0 Q We!" ) else return may} .emw
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(b) [2 points] f '1 I +£«ea1 ou can OCR)
Consider the function add from Question 2.a again. and xpress the execution runni 9 time of “Mn
add(n,8). cult £st
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_______________. (c) [2 points] .
What is the running time of the following algorithm. as a function of n. (Express the running time
using bige notation; you do not need to provide an exact analysis.) Assume n 2 1. int i=1;
int sum=0;
while (iinj) {
for (int_.'L=_n2;_J'_>__1i j=j/2){ [0 if; he, "'
sum++; "'— Z/o n
} 3 ’
 i += 2; f
} ______ W9(2’°7M) a W?”
a QM MADAME MOOM ’3'  In»; a I
3. Implementation [14 points] (a) [7 POMS] Palindromes are words or phrases that appear the same whether read backwards or
forwards, e.g. “eye”, “madam”, etc. Complete the Boolean method, isPalindromeQS(Queue q). which takes a queue of
Characters as its input variable, and returns true if the text contained in the given queue is a
palindrome. The method uses only one additional data structure. namely a Stack, to
accomplish its main function. 37 M
3 g boolean isPalindromeqsiQueue q) {
K
M“ Stack s = new Stack“;
JQUZ ;E L'
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3, Push L q.&eqoeuc03} 3 4% is; (catcle 3 q, deaoeue CY: 444115 Va viable is. (same! I‘UL
“WM am Swab. mommy L we (and acid .‘i—dbslqclc
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'3=°) Hung. move see/2 some QJLLB “L  (a! v iFQL 5 POQQ) equals (at de‘IUMCBBBi C. . F . {mire 2" fé—U—‘(n Faisal fig?» % discard MM: ‘ F TDD? “Orr (b) [7 points] Fill in the blanks below in the resursive countDlvisible method. countDivfsiQLe takes three Qosiﬁve integers, a, b. and Q. and returns how many integers between a and b (inclusive) are
divisible by c. For example, countDivisible(9, 18, 5) returns 2 since 10,15 are divisible by 5. (Hint: x % y = 0
if x is divisible by y.) at 1‘5 5 ilDilKB
public int countDivisible(int a, int b, int c) {
if ((oL==Lb+0))/¥;b Ma 4 i 6 yet “LEE
return 0; COM/ago! ‘K/ "9 (.01 l 1'5 i1?)
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return (muntblvfsmh LQHJ‘DN‘AT: to qé ‘q
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else . , —)(u,i%,§‘)+l
return cauntb'wlsiun GEMINI?) I\/ ll 75 I"? H m lT—“fo
} _) a! / 
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{5'5" emit) {q COUwJ‘J'H 77% 4. AVL Tree [10 points] The following is an AVL tree with node labels A  N used to identify the nodes (the node
labels have noting to do with ordering/searching the tree). Redraw this AVL tree as .it would
look like if the node with label D were deleted. {blower : BEHAMIM Filo K011, /’ 5. Heaps [15 points] The vector show below represents the underlying array of a min—heap. O l 3 9
Inﬂam
. l Suppose the following operations are performed (in this order!) on thegiven heap: f W): RemoveMinO: nsert(38); Insert(15). 233 K‘Fﬁ\
Show the heap (Le. underlying array) as it would look after each operation! //Z%\K_/g/J+ \
Space for rough work is provided on the next page.  W RemoveMinO RemoveMinO @ 0 1 2 3 4 5 6 7 8 9 In.
3d ’32 3C lnsert(38) o 1 2 3 4 5 ' 6 * 7
I 3’2. '9‘? “3% 8 9 tnsert(15)
0 l 2 3 4 5 6 'i 8 9
.
31 3C 3% '57”
Hid/38 see, back if PLO/(Cd!) 1’) 7,? 3% 31 C(14 36: 8‘? 3? 1—4" G) lq’ [email protected]\ / \ p/
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33’ © / \ 6. Hash Table [15 points] (a) [9 points]
Consider a map implemented using a hash table with 7 buckets. Quadratic probing and a hash fu ' = k % 7 Show the table as it would look after each of the following put
insert statements in order. a put(7, "A") . ":1 1*‘u3 _
Paw D ) a+q_6 __é:3
Wham") '55? “’0 3i :2 Q. 3+q=“/ 17% Z.
0 (D) [6 points] ' Assume that you have a hash table with n slots (indexed from 0 to (n—1)). For each of the
following functions h(k), where k is a key. answer two questions: a Is the function acceptable as a hash function. i.e. can it be used to index into the array? 
o If so, is it a good hash function? Why of why not? Z Wigwam/for (b.1) h(k)=</n, wherekandnare egers
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W_K is! au gimme wﬂ‘ 10¢ bj deg—0E,» aim—1% i. Imagine Jim rest.— «m etements Cn2mu3§m+ $19. (b.3) h(k) = (k + Ran om (n)) mod n _
Assume that the function Random(n) returns a random integer between 0 and (n—1) Incluswe. Maﬂbe. This (would resemb’b {05F linear Pm
5C CLOSE”. (23" ‘YQMW} PlaCC/ﬁm FCJH'ULj would
be deﬁant because Metal um W MM ' aw: VIM “bebokﬁfg li search 4441,
ham 4am; “than. gLe 1mm apt—(m 7. Graph [20 points] Assume a graph G=(V,E) with: Ix
container of all vertices V={1,2,3,4,5.6.7.8}, Adiaeent‘o‘ertieeefﬁafasL jg. Adjacenweﬁticesiiﬂaﬂ. K WK
' 4}. MEWS) X Adiaeenwemcesﬁnﬁa}, KNOTE, here AdjacentVertices[k] represents the list of vertices
adjacent to vertex[k]. Answer the following: (a) [4 points]
Write down the adjacency matrix correSponding to G. Merelm ﬁWIOI'Zjilftqg
essoMtt/Lj 4—Wa c: ' Malaya b33111 a I. a MA Courtech l—bt‘ ’   ‘ lat .
[Ni O. 4—hol5‘4 3 L n 1 —) H»u_s Pale its nob necessavg. 5' g ‘ led 9“ W”) ‘ a (; lvalr betrcuc + 4 :2 WW 6/? rl's easier ’59 ‘ SkaO (IQl'eﬂe( 4W“ S'bll'FﬁiS all UMUQL. TL“; chanjeg .51; ﬁg .6 (b) [5 points]
Draw a representation of G using circles for vertices (with vertex number in the circle) and
arrows for edges. ls G directed? Circle the right answer.
Is G connected? Circle the right answer. “mu/c9 connede +o e, no Cb“ M search algorithm DFS(G. v) (
answer these questions. (c) [3 points]
Find the number of connected components in a graph G ’1 I "  / Simpg SUb’n (J.th are.
marked W‘sﬁred gm bar of? tier/412255.
m +ka+ are. ' (d) [3 points]
Determine whether the connected graph G has any cycles. 4°01 {3r 0L cigar— 5w» gave 93: a. po
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8. Data Structures Applied ' [‘l 0 points] Consider the following problem: Given N elements in a data structure (N is odd, and all
elements are distinct). find the middle element in sorted order. (Note. the elements are not
necessarily in sorted order unless the data structure forces them to be.) For example. if the
numbers are: 34 76 87 12 8 96 83. then the middle element in sorted order is 76. lf the elements had been sorted, 76 would be in the middle. For each of the following data structures assume the N elements are already in the data
structure, and 1) describe brieﬂy an efﬁcient algorithm to ﬁnd the middle element in sorted
order (do not write code), 2) give the worst case running time of the algorithm. Your algorithms should be as efﬁcient as possible for full credit. (3) Data structure1: an unsorted array
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