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Analysis of Algorithms
创建日期：2024-04-08 17:19:16
标签： A1
Analysis of Algorithms

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Analysis of Algorithms

Homework 5

1. (a) Consider the following chain of six matrices: A1, A2, A3, A4, A5, and A6, where A1 is 5 × 10, A2 is
10× 3, A3 is 3× 12, A4 is 12× 5, A5 is 5× 50, and A6 is 50× 6. Find an optimal parenthesization of this
matrix-chain. Show both the table containing the optimal number of scalar operations for all slices and the
choice table.
(b) Prove using the strong form of induction that for any n ∈ N, if n ≥ 1 then a full parenthesization of an
n-element expression has n− 1 pairs of parentheses.
2. (a) CLRS 15.3-1
(b) Draw the recursion tree for the merge-sort algorithm on an input sequence of length 16. Explain why
memoization fails to speed up a good divide-and-conquer algorithm like merge-sort.
(c) Consider a variant of the matrix-chain multiplication problem in which the goal is to parenthesize the
sequence of matrices so as to maximize the number of scalar multiplications. Does this problem exhibit
optimal substructure?
3. (project) Recall Fn the recurrence that defines the Fibonacci numbers. Write a function fibDyn that computes
Fibonacci numbers which implements the naive recurrence via dynamic programming.
4. Consider the 0-1 knapsack problem in CLRS chapter 16.
(a) Write functional pseudo-code for a recursive solution to the variation on the 0-1 knapsack problem that
computes the maximum value that can be placed in the knapsack. The first parameter is a sequence of items
(i.e., value-weight pairs), and the second parameter is the knapsack weight capacity. Note that weights are
natural numbers.
(b) (project) Give a dynamic programming solution to the 0-1 knapsack problem that is based on the previous
problem; this algorithm should take and return the same values as the functional pseudo-code above.
Implement this algorithm and call it knapsack.
(c) (project) Implement a function called knapsackContents that takes the same values but returns a
sequence of items that maximize the knapsack’s value.
5. Consider the problem of neatly printing a paragraph on the screen (or on a printer). For the project portion,
put all functions in the file printPar. The input text is a sequence S of n words (represented as strings) of
lengths `1, . . . , `n (measured in characters). The input bound M is the maximum number of characters a line
can hold. The key to neatly printing a paragraph is to identify in the text sequence the lines of the paragraph so
that new-lines can be placed at the end of each line. We can formalize the notion of the “badness” of a line as
the number of extra space characters at the end of the line or∞ if the bound M is exceeded. We can formalize
the notion of the “badness” of a paragraph as the badness of the worst (i.e., maximum) line of the paragraph not
1
including the last line. Thus to identify the lines for a neat paragraph, we seek to minimize the badness of the
paragraph.
(a) If a given line contains words i through j, and we leave exactly one space between words, the number of
extra space characters at the end of the line is M − j+ i−∑jk=i `k. Write functional pseudo-code for the
function e(S,M, i, j) that computes the number of extra space characters at the end of a line.
(b) (project) Write a procedure extraSpace that implements e.
(c) Use the function e to write functional pseudo-code for the function bl(S,M, i, j) that computes line bad-
ness.
(d) (project) Write a procedure badnessLine that implements bl.
(e) Write functional pseudo-code for the recursive function mb(S,M) that computes the minimum paragraph
badness (using slicing). The base case must be that the sequence S is the last line (and not that S is empty).
(f) Write functional pseudo-code for the recursive function mb′(S,M, i) where mb′(S,M, i) = mb(S[i :],
M). The base case must be that the sequence characterized by S and i is the last line (and not that S is
empty).
(g) (project) Write a recursive procedure minBad that implements the function mb′. It should take three
parameters: S, M , and i a slicing index.
(h) (project) Write a procedure minBadDynamic that implements the functionmb′ using dynamic program-
ming. It should take only two parameters: S, and M .
(i) (project) Write a procedure minBadDynamicChoice that implements the function mb′ using dynamic
programming. In addition to returning mb(S,M), it should also return the choices made. Then write a
procedure printParagraph which takes two parameters: S and M that displays the words in S on
the screen using the choices of minBadDynamicChoice. What is the asymptotic running time of your
algorithm?
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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