From: <¥Ñ Microsoft Internet Explorer 5 Àx¦s> Subject: Data Structure Final (20070625) Date: Mon, 25 Jun 2007 12:32:46 +0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C7B724.EF945590" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3138 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C7B724.EF945590 Content-Type: text/html; charset="big5" Content-Transfer-Encoding: quoted-printable Content-Location: http://localhost/jang/courses/cs2351/exam/2007FinalExam.asp?option=withAnswer Data Structure Final (20070625)

Data Structure Final (20070625)

  1. (15%)=20
    1. (3%) If a graph has 6 vertices, with the following degrees: 1, = 1, 2, 2,=20 3, 5. How many edges does this graph have?
      Answer:=20 (1+1+2+2+3+5)/2=3D7. (See page 262, section 6.1.2 of the text.)
    2. (3%) A graph G is represented by the following adjacency lists:=20
      • vertex 0 -> 1 -> 2=20
      • vertex 1 -> 0 -> 3 -> 4=20
      • vertex 2 -> 0 -> 5 -> 6=20
      • vertex 3 -> 1 -> 7=20
      • vertex 4 -> 1 -> 7=20
      • vertex 5 -> 2 -> 7=20
      • vertex 6 -> 2 -> 7=20
      • vertex 7 -> 3 -> 4 -> 5 -> 6
      What is the = sequence=20 of breadth first search starting from vertex 0?
      Answer:=20 0 1 2 3 4 5 6 7 (See page 274, section 6.2.2 of the text.)
    3. (3%) What is the sequence of depth first search of the graph G = in the=20 previous sub-problem, starting from vertex 0?
      Answer: 0=20 1 3 7 4 5 2 6 (See page 274, section 6.2.2 of the text.)
    4. (3%) What is a "stable" sorting algorithm?
      Answer:=20 A stable sorting algorithm will not chagne the relative order of = already=20 sorted elements. In symbol, if ai =3D aj and i = < j,=20 then si < sj after a stable sorting algorithm. =
    5. (3%) Given the weights of 1, 3, 4, 5, 7, 8, 12, construct a = Huffman tree=20 with minimum weighted external path length.
      Answer:=20

  2. (6%) Answer the following short questions about insertion sort.=20
    1. (2%) What is the average and worse case time complexity of = insertion=20 sort of n numbers?
      Answer: O(n2),=20 O(n2)
    2. (2%) Given a random permutation of a sequence from 1 to 6, what = is the=20 sequence that exhibits the worst case behaviour of insertion sort? =
      Answer: 6 5 4 3 2 1
    3. (2%) Is insertion sort stable?
      Answer: = Yes.=20

  3. (9%) Answer the following short questions about quick sort.=20
    1. (2%) What is the average and worse case time complexity of quick = sort of=20 n numbers?
      Answer: O(n2), = O(n*log(n))=20
    2. (2%) Given a random permutation of a sequence from 1 to 6, what = is the=20 sequence that exhibits the worst case behaviour of quick sort? =
      Answer: 1 2 3 4 5 6=20
    3. (3%) How do you avoid the worse case behavior of quick sort? =
      Answer: Use the technique of "median of three" in page = 332, section 7.4 of the=20 text. Or you can do a random permutation before quick sort.=20
    4. (2%) Is quick sort stable?
      Answer: = No.=20

  4. (10%) Suppose that we are using 10 queues as the bins for LSD = radix 10=20 sort. Show the contents of these queues after each of the three passes = of LSD=20 radix sort when the input sequence is [179, 208, 306, 93, 859, 984, = 55, 9,=20 271, 33].
    Answer: See pages 356, 357, 358, Figure 7.16 of section 7.8 of = the=20 textbook.=20

  5. (10%) Consider the following undirected graph:
    1. (5%) Find a minimum-cost spanning tree by Kruskal's algorithm. = (Please=20 give partial results at the end of each iteration.)
      Answer: = (Click to=20 see the answer.)
    2. (5%) Repeat (a) using Prim's algorithm.
      Answer: (Click to = see the=20 answer.)

  6. (10%)=20
    1. (5%) Please describe Dijkstra's algorithm for the single-source=20 all-destination shortest path problem. (Please clearly define your=20 notation.)
      Answer: (My description is a bit = formal. You=20 can use your own way to explain this algorithm.)
      Let V denotes = the set of=20 all vertices. Let S denote the set of vertices, including the = starting=20 vertex a, whose shortest paths have been found. The distance between = vertices x and y is denoted by w(x, y). The shortest distance from a = to x is=20 denoted by s(a, x).=20
      1. Initially, set S=3D{a}. For every vertex t in V-S, let = s(t)=3Dw(a, t).=20
      2. Select a vertex t in V-S such that t =3D argy = minx in S,=20 y in V-S {s(x)+w(x, y)}=20
      3. Include t in S. Repeat step 2 and 3 until the end vertex is = included=20 in S.
    2. (5%) Please use the algorithm on the following graph to find the = shortest paths from the source node a to all the other nodes. = (Please show=20 the partial result at each iteration.)
      Answer: (You can also use other methods to demonstrate = the=20 operation of this algorithm.)
      Iteration S Seleted vertex Shortest distance
      a b c d e f=20
      1 {a} b 0 1 4 =A1=DB =A1=DB =A1=DB=20
      2 {a, b} c 0 1 3 =A1=DB =A1=DB =A1=DB=20
      3 {a, b, c} e 0 1 3 =A1=DB 4 =A1=DB=20
      4 {a, b, c, e} d 0 1 3 7 4 =A1=DB=20
      5 {a, b, c, e d} f 0 1 3 7 4 9

  7. (10%) Consider the following AOE network:
    1. (3%) Obtain the earliest and latest times for each vertex using = the=20 backward-forward approach.
      Answer: (Earliest time in blue, = latest time=20 in red)
    2. (3%) Obtain the early and late starting times for each activity. =
      Answer:
      Activity Early starting time Late starting time Critical activity=20
      a0 0 9-5=3D4 No=20
      a1 0 6-6=3D0 Yes=20
      a2 5 12-3=3D9 No=20
      a3 6 12-6=3D6 Yes=20
      a4 6 15-3=3D12 No=20
      a5 12 15-3=3D12 Yes=20
      a6 12 19-4=3D15 No=20
      a7 12 16-4=3D12 Yes=20
      a8 15 16-1=3D15 Yes=20
      a9 15 19-4=3D15 Yes=20
      a10 16 21-5=3D16 Yes=20
      a11 19 21-2=3D19 Yes=20
      a12 16 23-4=3D19 No=20
      a13 21 23-2=3D21 No
    3. (2%) List all the critical paths.
      Answer: =
      1. a1, a3, a7, a10,=20 a13=20
      2. a1, a3, a5, a8,=20 a10, a13=20
      3. a1, a3, a7, a9,=20 a11, a13
    4. (2%) Can you reduce the project length by speeding up activity=20 a10 alone? Why or why not?
      Answer: = No, you=20 cannot reduce the project length by speeding up a10 = alone.=20 Besides speeding up a10, you also need to speed up = a9=20 or a11 since they are parallel in two critical paths.=20

  8. (10%)=20
    1. (6%) Explain the operations of any 3 uniform hash functions that = have=20 been described in the textbook.=20
    2. (1%) What is "collision" in accessing a hash table?=20
    3. (3%) What is "overflow" in accessing a hash table? Describe 3 = strategies=20 in handling overflow?

  9. (10%) Consider the following min-max heap:
    1. (2%) Plot the min-max heap after inserting 5 into the original = heap.=20
      Answer:
    2. (2%) Plot the min-max heap after inserting 75 into the original = heap.=20
      Answer:
    3. (3%) Plot the min-max heap after deleting min from the original = heap.=20
      Answer:
    4. (3%) Plot the min-max heap after deleting max from the original = heap.=20
      Answer:
    Please=20 indicate the modified vertieces using double circles when drawing the = final=20 heaps.=20

  10. (10%) Draw the deaps after the operations indicated next:=20
    1. (2%)
      Answer: =
    2. (2%)
      Answer: =
    3. (3%)
      Answer: =
    4. (3%)
      Answer: =
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