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CS6515 Exam 1 Prep with verified answers
Knapsack |!|without |!|repetition |!|- |!|correct |!|answer |!| ✔ k(0) |!|= |!| 0 for |!|w |!|= |!| 1 |!|to |!|W: |!|if |!|w_j |!|>w: |!|k(w,j) |!|= |!|k(w, |!|j |!|- |!|1) |!|else: |!|K(w,j) |!|= |!|max{K(w, |!|j |!|-1),K(w |!|- |!|w_j, |!|j |!|-1) |!|+ |!|v_i} knapsack |!|with |!|repetition |!|- |!|correct |!|answer |!| ✔ knapsack |!|repeat(w_i....w_n, |!|w_i... |!|w_n, |!|B) k(0) |!|= |!| 0 for |!|i |!|= |!| 1 |!|to |!|n |!|if |!|w_i |!|<= |!|b |!|& |!|k(b) |!|<v_i |!|+ |!|K(b-w_i) |!|then |!|k(b) |!|= |!|v_i |!|+ |!|K(b-w_i) Longest |!|Increasing |!|Subsequence |!|- |!|correct |!|answer |!| ✔ LIS(a_1.... |!|a_n) for |!|i |!|= |!| 1 |!|to |!|n |!|L(i) |!|= |!| 1 |!|for |!|j |!|= |!| 1 |!|to |!|n |!|- |!|if |!|a_j |!|< |!|a_i |!|& |!|L(i) |!|< |!| 1 |!|+ |!|L(j) |!|L(i) |!|= |!| 1 |!|+ |!|L(j) max |!|= |!| 1 for |!|i |!|= |!| 2 |!|to |!|n |!|if |!|L(i) |!|> |!|L(max) |!|then |!|max |!|= |!|i |!|return(L(max)) longest |!|common |!|subsequence |!|algo |!|- |!|correct |!|answer |!| ✔ LCS(X,Y) for |!|i |!|= |!| 0 |!|to |!|n: |!|L(i, |!|0) |!|= |!| 0 for |!|j |!|= |!| 0 |!|to |!|n: |!|L(0,j) |!|= |!| 0
for |!|i |!|= |!| 1 |!|to |!|n |!|for |!|j |!|= |!| 1 |!|to |!|n |!|if |!|X_i |!|== |!|Y_j: |!|L(i,j) |!|= |!|L(i |!|- |!|1, |!|j |!|- |!|1) |!|+ |!| 1 |!|else: |!|L(i,j) |!|= |!|max(L(i |!|- |!|1, |!|j),L(i,j-1) |!|return(L(i,j) longest |!|common |!|substring |!|- |!|correct |!|answer |!| ✔ what |!|is |!|Big |!|oh |!|of |!|LCS? |!|- |!|correct |!|answer |!| ✔ O(n**2) what |!|is |!|big |!|Oh |!|of |!|longest |!|common |!|substring? |!|- |!|correct |!|answer |!| ✔ O(mn) what |!|is |!|important |!|to |!|remember |!|when |!|calculating |!|longest |!|common |!|subsequence |!|as |!|opposed |!| to |!|substring? |!|- |!|correct |!|answer |!| ✔ substring |!|is |!|very |!|diagonal |!|and |!|plus |!| 1 subsequence |!|you |!|have |!|to |!|use |!|a |!|max |!|function Perform |!|longest |!|common |!|substring |!|and |!|subsequence |!|on: "abcdaf" "3bcdf" What |!|are |!|the |!|recurrences? |!|write |!|down |!|the |!|algorithm |!|and |!|results |!|- |!|correct |!|answer |!| ✔ if |!|you |!|are |!|presented |!|with |!|a |!|coins |!|(1,5,6,8) |!|and |!|your |!|knapsack |!|is |!|k |!|= |!|11, |!|what |!|is |!|the |!| minimum |!|number |!|of |!|coins, |!|what |!|would |!|be |!|the |!|computational |!|time, |!|and |!|what |!|is |!|the |!| recurrence? |!|- |!|correct |!|answer |!| ✔ https://www.youtube.com/watch?v=Y0ZqKpToTic how |!|to |!|calculate |!|longest |!|increasing |!|substring. |!|Try |!|it |!|using |!|the |!|following: 541208563 |!|- |!|correct |!|answer |!| ✔ It |!|should |!|only |!|require |!|two |!|separate |!|loops: one |!|to |!|loop |!|to |!|go |!|through |!|to |!|calculate |!|the |!|values
What |!|are |!|the |!|components |!|of |!|the |!|master |!|theorem |!|and |!|when |!|can |!|you |!|use |!|it? |!|- |!|correct |!| answer |!| ✔ In |!|terms |!|of |!|the |!|master |!|theorem, |!|what |!|would |!|be |!|a |!|result |!|that |!|would |!|provide |!|O(nlogn) |!| time? |!|- |!|correct |!|answer |!| ✔ Consider |!|w_16. |!|For |!|what |!|power |!|k |!|is |!|(ω_16)^k |!|= |!|-1? |!|- |!|correct |!|answer |!| ✔ (ω_16)^8; This |!|is |!|because |!|it |!|adds |!|180*(7/8) |!|to |!|(w_16)^ Also, |!|zero |!|exponent |!|makes |!|0, |!|that |!|would |!|be |!|(w_16)^ Consider |!|the |!|n-th |!|roots |!|of |!|unity |!|for |!|n |!|=16. |!|WHat |!|is |!|ω_16 |!|in |!|polar |!|coordinates? |!|- |!|correct |!| answer |!| ✔ At |!|point |!|0, |!|polar |!|coordinates |!|are |!|(r, |!|theta) |!|== |!|(1, |!|2pie/2), |!|if |!|n |!|= |!|16, |!|then |!|it |!| becomes
- |!|(1, |!|2pie/2)^16 |!|== |!|(1,2pie/16) |!|== |!|(1, |!|pie/8) Consider |!|ω_16. |!|For |!|what |!|power |!|k |!|is |!|(ω_16)^k |!|= |!|-ω_16? |!|- |!|correct |!|answer |!| ✔ it |!|is |!|(ω_n)^(1+ (n/2)), |!|in |!|the |!|case |!|of |!|ω_16, |!|n/2 |!|= |!|8, |!|(ω_16)^(1+8) |!|= |!|(ω_16)^ Consider |!|ω_16. |!|For |!|what |!|power |!|k |!|(ω_16)^-1 |!|= |!|ω16^k? |!|In |!|other |!|words, |!|for |!|what |!|k |!|is |!|ω |!|* |!|(ω_16)^k |!|= |!|1? |!|- |!|correct |!|answer |!| ✔ k |!|== |!| 15 This |!|is |!|because |!|(ω_16)^16 |!|== |!|(ω_16)^ For |!|what |!|power |!|k |!|is |!|(ω_16)^k |!|= |!|(ω_8)^2? |!|- |!|correct |!|answer |!| ✔ k |!|== |!| 4 In |!|terms |!|of |!|radians, |!|what |!|is |!|the |!|imaginary |!|axis? |!|- |!|correct |!|answer |!| ✔ what |!|is |!|the |!|significance |!|of |!|z |!|= |!|a |!|+ |!|bi |!|- |!|correct |!|answer |!| ✔ it |!|can |!|describe |!|a |!|point |!|in |!|a |!| cartesian |!|plane How |!|do |!|you |!|convert |!|the |!|cartesian |!|coordinates |!|(a,b) |!|to |!|polar |!|coordinates? |!|- |!|correct |!|answer |!| ✔ (a,b) |!|= |!|rcos(theta), |!|rsin(theta)
what |!|is |!|euler's |!|formula? |!|- |!|correct |!|answer |!| ✔ cos(theta) |!|+ |!|isin(theta) |!|= |!|e^(i*theta) what |!|degree |!|is |!|-1 |!|in |!|terms |!|of |!|polar |!|coordinates? |!|what |!|are |!|the |!|polar |!|coordinates? |!|- |!|correct |!| answer |!| ✔ 180 |!|degrees |!|(positive |!|starts |!|on |!|the |!|right |!|side)
- |!|(1, |!|pie)
- |!|What |!|is |!|the |!|radian |!|coordinates |!|which |!|degree |!|= |!|0?
- |!|How |!|would |!|we |!|add |!| 180 |!|degrees |!|to |!|it? |!|- |!|correct |!|answer |!| ✔ - |!|(r,theta)
- |!|(r, |!|theta) |!|* |!|(1,pie) what |!|is |!|meant |!|by |!|roots |!|of |!|unity? |!|- |!|correct |!|answer |!| ✔ 1 |!|to |!|be |!|exact
- |!|What |!|numbers |!|can |!|be |!|used |!|when |!|n |!|= |!| 4 |!|for |!|the |!|roots |!|of |!|unity?
- |!|Can |!|you |!|describe |!|why |!|that |!|is |!|the |!|case? |!|- |!|correct |!|answer |!| ✔ 1,-1,i, |!|-i At |!|what |!|point |!|does |!|the |!|roots |!|of |!|unity |!|repeat: |!|essential, |!|what |!|does |!|n |!|equal |!|in |!|order |!|to |!|go |!| 360 |!|degrees? |!|- |!|correct |!|answer |!| ✔ 8, |!|after |!| 8 |!|it |!|repeats why |!|is |!|omega |!|increased |!|in |!|the |!|exponent |!|in |!|order |!|to |!|traverse |!|through |!|the |!|roots |!|of |!|unity? |!|- |!|correct |!|answer |!| ✔ theta |!|= |!|2pie/n, |!|and |!|omega_n |!|= |!|(1, |!|2pie/n). |!|Multiply |!|them |!|together |!|to |!| increase |!|each |!|step what |!|is |!|the |!|point |!|on |!|the |!|roots |!|of |!|unity |!|circle |!|(using |!|omega) |!|where |!|it |!|is |!|zero |!|degrees? |!|- |!| correct |!|answer |!| ✔ w^0_n |!|== |!| 1 why |!|is |!|the |!|point |!|before |!|ω^(n-1) |!|right |!|before |!|w^n? |!|- |!|correct |!|answer |!| ✔ This |!|describes |!|the |!| step |!|(whether |!|it |!|is |!|in |!|4ths, |!|8ths, |!|or |!|16ths, |!|right |!|before |!|reaching |!|radian |!|2piei/n, |!|whether |!|is |!| the |!|degree |!|of |!|"0") what |!|is |!|another |!|way |!|to |!|express |!|-ω_16?? |!|- |!|correct |!|answer |!| ✔ ω^9_16, |!|this |!|is |!|because |!|it |!| moves |!|1/16 |!|into |!|the |!|lower |!|left |!|quartile
You |!|need |!|to |!|get |!|the |!|polynomial |!|multiply, |!|what |!|do |!|you |!|do? |!|- |!|correct |!|answer |!| ✔ simply |!| multiply, |!|and |!|list |!|from |!|coefficient, |!|to |!|x |!|ascending |!|exponent what |!|are |!|the |!|different |!|types |!|of |!|filtering? |!|- |!|correct |!|answer |!| ✔ mean |!|filtering |!|= |!|1/(2m+1) gaussian |!|filtering: |!|f |!|= |!|1/z(e^(-m^2), |!| gaussian |!|blur: |!|2-dim what |!|are |!|the |!|two |!|representations |!|of |!|A(x)? WHich |!|is |!|more |!|convenient |!|for |!|multiplying |!|polynomials? |!|- |!|correct |!|answer |!| ✔ 1.) |!|coefficients 2.) |!|values: |!|A(x_1), |!|A(x_2)....A(x_n) Values |!|are |!|more |!|efficient |!|to |!|do |!|so What |!|is |!|more |!|efficient |!|in |!|terms |!|of |!|complexity? |!|Polynomial |!|Multiplication |!|with 1.) |!|values |!|or 2.) |!|coefficients |!|- |!|correct |!|answer |!| ✔ - |!|values values |!|== |!|O(n) |!|time |!|(from |!| 1 |!|-> |!|2n) coefficients |!|== |!|O(n^2) |!|time What |!|is |!|the |!|solve |!|time |!|for |!|FFT |!|- |!|correct |!|answer |!| ✔ O(nlogn) what |!|is |!|the |!|difference |!|between: 1.) |!|((ω_n)^(j)) |!|and 2.) |!|((ω_n)^((n/2)+j)) |!|- |!|correct |!|answer |!| ✔ it |!|is |!|the |!|oppose |!|point. |!| (ω_n)^(n/2) |!|== |!|-ω_n so (ω_n)^((n/2)+j |!|== |!|(-ω_n)^j
When |!|calling |!|FFT |!|with |!|the |!|"even" |!|and |!|"odd" |!|partitions |!|of |!|your |!|function |!|(lets |!|say |!|"A"), |!| what |!|happens |!|to |!|"ω"? |!|- |!|correct |!|answer |!| ✔ It |!|is |!|squared Why |!|do |!|we |!|square |!|"ω" |!|in |!|FFT? |!|- |!|correct |!|answer |!| ✔ Why |!|is |!|omega |!|squared? |!|- |!|correct |!|answer |!| ✔ WHat |!|is |!|the |!|FFT |!|algorithm |!|of |!|a |!|single |!|function? |!|- |!|correct |!|answer |!| ✔ FFT(A,ω) if |!|n |!|= |!| 1 |!|then |!|return |!|a_ else |!|let |!|Aeven |!|= |!|[a_0, |!|a_2... |!|a_(n-2)] |!|let |!|Aodd |!|= |!|[a_1, |!|a_3... |!|a_(n-1)] |!|FFT(Aeven, |!|ω^2) |!|= |!|...s_n |!|FFT(Aodd, |!|ω^2) |!|= |!|...t_n |!|for |!|j |!|= |!|n/2 |!|- |!| 1 |!|r_j |!|= |!|s_j |!|+ |!|ω^j |!|* |!|t_j |!|r_((n/2)+j) |!|= |!|s_j |!|- |!|w^j |!|* |!|t_j In |!|the |!|FFT |!|algorithm, |!|why |!|does |!|"j" |!|in |!|the |!|last |!|section |!|only |!|go |!|to |!|(n/2 |!|- |!|1)? |!|- |!|correct |!| answer |!| ✔ This |!|is |!|because |!|each |!|portion |!|performs |!|half |!|of |!|the |!|function, |!|similar |!|to |!|the |!|root |!| unity |!|circle |!|traversal What |!|is |!|the |!|difference |!|between |!|FFT |!|of |!|one |!|item |!|vs |!|FFT |!|multiplication |!|of |!|two |!|functions? |!|- |!| correct |!|answer |!| ✔ - |!|ω |!|is |!|not |!|squared
- |!|there |!|is |!|no |!|odd |!|even |!|separation
- |!|coefficients |!|produced |!|are |!|n |!|= |!| 0 |!|to |!|2_n |!|-1, |!|same |!|for |!|the |!|only |!|j |!|loop
- |!|lastly, |!|you |!|multiply |!|the |!|coefficients |!|together
- |!|then |!|have |!|to |!|do |!|the |!|inverse |!|FFT
2.) |!|a.) |!|M_n(ω_n)^-1 |!|A |!|= |!|a b.) |!|(1/2) |!| |!|M_n((ω_n)^-1) 3.) |!|a |!|= |!|(1/2)FFT(A, |!|(ω_n)^n-1) What |!|is |!|the |!|direction |!|of |!|roots |!|of |!|unity |!|calculation |!|with |!|FFT |!|vs |!|inverse |!|FFT? |!|- |!|correct |!| answer |!| ✔ inverse |!|FFT |!|== |!|counterclockwise FFT |!|== |!|clockwise For |!|(ω_n)^2, |!|what |!|is |!|its |!|multiplicative |!|inverse? More |!|precisely, |!|for |!|what |!|power |!|k |!|is |!|(ω_n)^k |!|× |!|(ω_n)^2 |!|= |!|1? |!|- |!|correct |!|answer |!| ✔ k |!|= |!|(n-2) It |!|is |!|whatever |!|it |!|takes |!|to |!|get |!|to |!|0, |!|which |!|becomes |!|one so |!|if |!|applied: |!|For |!|(ω_16)^2 |!|*(ω_16)^(16-2) |!|== |!|(ω_16)^16 |!|== |!|(ω_16)^0 |!|== |!| 1 1.) |!|what |!|is |!|the |!|product |!|of |!|the |!|roots |!|of |!|unity? 2.) |!|What |!|about |!|the |!|sum? 3.) |!|What |!|is |!|the |!|sum |!|of |!|Aeven? 4.) |!|what |!|is |!|the |!|sum |!|of |!|Aodd? |!|- |!|correct |!|answer |!| ✔ 1.) |!|if |!|it |!|is |!|even, |!|then |!|it |!|is |!|== |!| 1 if |!|it |!|is |!|odd, |!|then |!|it |!|is |!|== |!|- 2.) |!|The |!|sum |!|is |!|== |!|0. |!|This |!|is |!|because |!|(ω_n)^0 |!|== |!| 1 |!|and |!|(ω_n)^(n/2) |!|== |!|-1. |!|These |!|cancel |!| each |!|other |!|out. |!|So |!|if |!|you |!|go |!|through |!|the |!|whole |!|sequence.. |!|it |!|eventually |!|becomes |!|0.
3.) |!|(n/2) |!|* |!| 1 4.) |!|(n/2) |!|* |!|- Why |!|are |!|off |!|diagonal |!|entries |!|not |!|equal |!|to |!|one? |!|- |!|correct |!|answer |!| ✔ In |!|the |!|original, |!|it |!|is |!| (ω_n)^k |!|* |!|(ω_n)^-k |!|== |!|(ω_n)^0 |!|== |!| 1 However, |!|the |!|off-diagonal |!|entries |!|are: |!|(ω_n)^k |!|* |!|(ω_n)^-j |!|== |!|(ω_n)^(k-j); |!|so |!|result |!|of |!| 1 |!|is |!|definitely |!|not |!|guaranteed in |!|degrees, |!|what |!|are |!|the |!|values |!|of |!|the |!|following: cos(0) cos(90) cos(180) cos(270) cos(360) Do |!|the |!|same |!|for |!|sin() |!|- |!|correct |!|answer |!| ✔ a.) |!| 1 b.) |!| 0 c.) |!|- d.) |!| 0 e.) |!| 1 How |!|to |!|translate |!|i^6 |!|to |!|i |!|or |!|1? |!|- |!|correct |!|answer |!| ✔ If |!|it |!|is |!|less |!|than |!| 4 |!|remember |!|the |!|table: i, |!|-1, |!|-i, |!| 1 if |!|greater, |!|do |!|exponent |!|mod |!|4, |!|and |!|whatever |!|the |!|remainder |!|is
