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COMPUTER SCIENCE Algorithms and Data Structures
创建日期：2024-06-26 14:41:31
标签： COMPUTER
COMPUTER SCIENCE Algorithms and Data Structures

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COMPUTER SCIENCE Algorithms and Data Structures


Instructions: For this homework, you should work in groups of 2{4 people. Only one member of the group should submit on Canvas. Specify in your submission the members of your group. The submission must be typed in LaTeX using the provided template.

Grading: We will give a 5% grade bonus to students taking COMP 418. The homework will be graded based on correctness, completeness, and quality of writing.

Notation & remarks: For strings x; y 2 A∗, we say that x is a prefix of y, and write x ≤ y, if xz = y for some z 2 A∗. For example, ab is a prefix of abcde because ab · cde = abcde. Notice that if x ≤ y then there is a unique extension z 2 A∗ such that xz = y. We write x−1y to denote this unique extension. For a string x 2 A∗, we write jxj to denote its length. For all strings x; y 2 A∗, it holds that jx · yj = jxj + jyj. String concatenation satisfies the following left cancellation property:for all x; y; z 2 A∗, xy = xz implies that y = z.

For sets X and Y , we write X ! Y to denote the set of all functions from X to Y .

Guidelines:

• Do not use ellipses in your definitions or your proofs. Instead, use recursive definitions and induction.

• Provide a careful justification of every step in your proofs.

Problem 1 [10 points]

Let A be a set. Show that the prefix relation ≤ on A∗ is a partial order (reflexive, antisymmetric and transitive).

Solution for 1

Fill me in.

Problem 2 [70 points]

In the lecture of February 18 we discussed denotations (functions) of type A+ ! B and of type A+ ! B+. Recall that a function f : A+ ! B describes a streaming computation in the following way: for an input history x 2 A+, the value f(x) is the last output item after consuming x. We call f an incremental denotation, because it gives us the output increment. A function ’ : A+ ! B+ can sometimes describe a streaming computation in the following way: for an input history x 2 A+,the value ’(x) is the overall output history after consuming x. We call ’ a cumulative denotation,because it gives us the cumulative (overall) output.

Remark: In this problem, we are only concerned with streaming computations that produce exactly one output item per step (i.e., for each input item).

Part A [10 points]

Define an injective function lift from the set A+ ! B to the set A+ ! B+. That is, lift : (A+ ! B) ! (A+ ! B+) and lift is injective. The function lift should convert an incremental denotation to a cumulative denotation. Provide a proof that lift is injective.

Solution for 2A

Fill me in.

Part B [30 points]

For all sets A and B, find a subset S(A; B) ⊆ (A+ ! B+) so that lift is a one-to-one correspondence(bijection) from A+ ! B to S(A; B). Provide a proof that lift is a bijection.


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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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