Fundamental Algorithms
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1 Fudametal Algorithms Chapter 2b: Recurreces Michael Bader Witer 2014/15 Chapter 2b: Recurreces, Witer 2014/15 1
2 Recurreces Defiitio A recurrece is a (i-equality that defies (or characterizes a fuctio i terms of its values o smaller argumets. Examples: Time complexity of MergeSort: { c1 for 1 T MS ( = ( 2T MS 2 + c2 for 2 or, give i Θ-otatio: { Θ( for 1 T MS ( = ( 2T MS 2 + Θ( for 2 iequality for the BFPRT algorithm (to fid the media: { Θ( for C T BF ( ( T ( BF TBF O( for > C Chapter 2b: Recurreces, Witer 2014/15 2
3 The Substitutio Method step 1: guess the type of the solutio step 2: fid the respective parameters, ad prove that the resultig fuctio satisfies the recurrece (e.g. by iductio Example: (MergeSort recurrece { c1 for 1 T MS ( = ( 2T MS 2 + c2 for 2 1. guess solutio: T ( = a log 2 + b 2. determie the correct values for the parameters a ad b Note: Is recurrece formula correct? Should t it be 2T MS ( 2 + c2? Chapter 2b: Recurreces, Witer 2014/15 3
4 Solvig the MergeSort Recurrece via Substitutio For 1: T MS ( = c 1 T MS (1 = a 1 log 2 (1 + b 1! = c 1 ca oly be true, if b := c 1 For > 1: T MS ( = 2T MS ( 2 + c2 isert T MS ( = a log 2 + c 1 ito equatio: a log 2 + c 1 = 2 ( a 2 log ( c1 2 + c2 a log 2 + c 1 = a (log c 1 + c 2 0 = a + c 2 a = c 2 therefore: T MS ( = c 2 log 2 + c 1 Chapter 2b: Recurreces, Witer 2014/15 4
5 The Recursio-Tree Method (or Iteratio Method Geeral Steps: 1. draw a tree of all recursive fuctio calls 2. state the local costs for each ode (fuctio call of the tree 3. sum up the costs of all odes o each level of the tree Possible Results: a sum of costs-per-level that ca be added up easily a easier recurrece for the costs-per-level a good guess for the substitutio method Example: MergeSort recurrece Chapter 2b: Recurreces, Witer 2014/15 5
6 The Master Theorem Prerequisites: costats a 1, b > 1 (a, b R; a fuctio f ( recurrece give by T ( = at ( b + f (, T (1 Θ(1 The, T ( ca be bouded asymptotically as follows: 1. if f ( O ( log b a ɛ for some ɛ > 0, the T ( Θ ( log ba 2. if f ( Θ ( log b a, the T ( Θ ( log ba log 3. if f ( Ω ( log a+ɛ b for some ɛ > 0, ad if af ( b cf ( for some costat c < 1 ad all > 0, the T ( Θ (f ( Proof: see textbook (Corme et al. Chapter 2b: Recurreces, Witer 2014/15 6
7 The Master Theorem Remarks Iterpretig the Master Theorem: for f ( = 0, i.e., T ( = at ( b, a solutio is T0 ( = log b a at 0 ( b ( logb a log b a = a = a = log b a = T 0 ( b a The master theorem compares the o-recursive part of the costs, f (, with this solutio T 0 ( case 1: f ( O (T 0 ( ɛ, costs of recursio domiate, ad T ( Θ ( log ba case 2: f ( Θ (T 0 (, costs are balaced, ad T ( Θ ( log ba log case 3: f ( Ω (T 0 ( ɛ, costs f ( domiate, ad T ( Θ (f ( Chapter 2b: Recurreces, Witer 2014/15 7
8 The Master Theorem Remarks (2 Floor ad Ceil: if i at ( b the fractio b occurs as b or b, the theorem still holds situatios as i T ( ( 2 + T 2 (compare MergeSort recurrece are also covered 2T ( 2 the master theorem will cover may (but ot all divide-ad-coquer recurreces Techicalities of the Theorem: case 1: f ( O (T 0 ( is ot sufficiet: f ( eeds to be polyomially smaller tha T 0 ( case 3: f ( Ω (T 0 ( is ot sufficiet: f ( eeds to be polyomially larger tha T 0 ( Chapter 2b: Recurreces, Witer 2014/15 8
9 The Master Theorem Examples MergeSort: T ( = 2T ( 2 + f (, where f ( Θ( a = 2 ad b = 2, therefore T 0 ( = log 2 2 = case 2 applies: f ( Θ(, therefore T ( Θ( log( Expesive Merge: T ( = 2T ( 2 + f (, but f ( Θ(2 agai a = 2 ad b = 2, thus T 0 ( = log 2 2 = ow f ( Ω( 1+ɛ for ay 0 < ɛ < 1 ad a f (/b = c 2 for ay 1 2 < c < 1 therefore case 3 applies: T ( Θ(f ( = Θ( 2 Odd-Eve MergeSort: T ( = 2T ( 2 + f (, with f ( Θ( log still a = 2 ad b = 2, thus T 0 ( = log 2 2 = ow, f ( Ω(, but f ( Ω( 1+ɛ for ay ɛ > 0 thus, the Master theorem does ot apply Chapter 2b: Recurreces, Witer 2014/15 9
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