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Computing polynomials with few multiplications

Computing polynomials with few multiplications. 973311 張弘暐 973320 江宗翰 指導教授 : 張經略. 作者 : Shachar Lovett ECCC(Electronic Colloquium on Computational Complexity). 目的. 改進計算多項式的時間複雜度 將 upper bound 接近 lower bound. n 項多項式 degree ≤ d. Lower bound : 原先 Upper bound : 新 Upper bound : .

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Computing polynomials with few multiplications

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  1. Computing polynomials with fewmultiplications 973311 張弘暐 973320 江宗翰 指導教授:張經略

  2. 作者 :Shachar Lovett • ECCC(Electronic Colloquium on Computational Complexity)

  3. 目的 • 改進計算多項式的時間複雜度 • 將upper bound 接近 lower bound

  4. n項多項式 degree ≤ d • Lower bound : • 原先Upper bound : • 新Upper bound :

  5. n項多項式 degree ≤ d • 例 :n = 3 , d = 5 2 2 3 2 1 1 1 2 0

  6. monotone( 單調函數) x ≤ y if and only if f (x) ≤ f (y) • M(n,d) = multiplications 1 0 2 1 2 3

  7. 目標 • A 、B 為兩種monotone n項多項式 degree ≤ d 的排列方法 • 找出一組A、B • 使得 A+B 為一種n項多項式 degree ≤ d 的排列方法 • 目標 : |A|、|B|≤

  8. 方法 • 令 A=M(S,d/2) B = M(T,d/2) • |S|=|T|=(n+1)/2 • |S ∩T| = 1

  9. |S|=|T|=(n+1)/2 |S ∩T| = 1 S ∩T T S (n+1)/2

  10. 成果 • A=M(S,d/2) B = M(T,d/2) • |S|=|T|=(n+1)/2 • 代入 M(n,d) = • M((n+1)/2,d/2) = ≤ ≤ 滿足目標 : |A|、|B|≤

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