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Algorithmic Solutions for Computational Problems: A Comprehensive Overview

Explore computational problems, algorithm design, efficiency analysis, and optimality in computer science. Learn from examples like Fibonacci numbers and powering problems. Dive into complexities such as Landau's Big-Oh notation and asymptotic growth.

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Algorithmic Solutions for Computational Problems: A Comprehensive Overview

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  1. „Introduction to Algorithms“ References • Cormen/Leiserson/Rivest/Stein: Introduction to Algorithms • Knuth: The Art of Computer Programming • Sipser: Introduction to the Theory of Computation Course page http://kaist.theoryofcomputation.asia/18cs300 E-Learning: http://klms.kaist.ac.kr Homework: weekly assignments, individually, hand-written; random grading + programming in ELICEhttp://kaist.elice.io/ Pledge for integrity and academic honesty to be signed and submitted! No email requests, please!

  2. „Introduction to Algorithms“ Syllabus 1. Computational problems and algorithmic solutions • Virtues of Computer Science • Specification • Algorithm design • Efficiency analysis • Optimality, Example: Repeated Squaring • Effect of Different Primitives: Fibonacci Numbers • Recurrences and the Master Theorem • Polynomial Multiplication: Long, Karatsuba, Toom, Cook • Matrix Multiplication: open problem

  3. 1. Computational problems and algorithmic solutions “Virtues” of ComputerScience • problem specification • formal semantics • algorithm design • and analysis (correctness, efficiency) • optimality proof

  4. 1. Computational problems and algorithmic solutions specification • Find me a girlfriend/boyfriend. • Does God exist? • How many angels can dance on the head of a pin? • What is the best movie of all time? • How many twin primes are there? • Is „no“ the only correct answer to this question? • What is the smallest positive integer that cannot be described in English using 100 characters?

  5. 1. Computational problems and algorithmic solutions semantics • Find me a girlfriend/boyfriend. • Does God exist? • How many angels can dance on the head of a pin? • What is the best movie of all time? Kiss your seat neighbor! Yes! Try! Watch „Carmencita“ (1894) ; let B:=1 For each N=2,…8880980 do Compare movie #Nin IMDb to movie #B: If you like #N better than #B, let B:=N. Return B.

  6. 1. Computational problems and algorithmic solutions algorithm vs. code mov esi, offset list top: mov edi, esi inner: mov eax, [edi] mov edx, [edi+4] cmp eax, edx jle no_swap mov [edi+4], eax mov [edi], edx no_swap: add edi, 4 cmp edi, list_end - 4 jb inner add esi, 4 cmp esi, list_end - 4 jb top • primitive operations • their semantics • their costs # vowels list vowels = ['e', 'a', 'u', 'o', 'i'] # sort the vowels vowels.sort() # print vowels print('Sorted list:', vowels)

  7. 1. Computational problems and algorithmic solutions pseudo-code bubbleSort ( A[n] ) while n≠0 do newn := 0 for i := 1 to n-1 do if A[i-1] > A[i] then swap(A[i-1], A[i]) newn := i endif endfor n := newn endwhile • primitive operations • their semantics • their costs „Premature optimization is the root of all evil“ (D.Knuth)

  8. 1. Computational problems and algorithmic solutions asymptoticefficiency • Algorithm analysis, as opposed to empirical program evaluation „Premature optimization is the root of all evil“ (D.Knuth)

  9. 1. Computational problems and algorithmic solutions asymptotic growth Landau Big-Oh notation f = O(g) • constant 17 • logarithmic log(n) • square root √n • linear n • quasilinear n·log(n) • quadratic n² • cubic n³ • polynomial c·nc • superpolynomial nlog(n) • exponential 2n • doubly exponential 22n • tetration 22…2ⁿ

  10. 1. Computational problems and algorithmic solutions Optimality Example Powering Problem: Fix some monoid M. Input:XM and nN. Output:Xn. Cost: #multiplications • Xn = X· X · … · X : n-1 multiplications • Recursive algorithm: compute • Xn= (X n/2)2 if n even • Xn= (X(n-1)/2)2 · Xif n odd • #multiplications T(1) =0, T(n)  T(n/2)+2, • Proof by induction: T(n)  2·log2(n) T(n)  T(n/2)+1 T(n)  T(n/2)+2

  11. 1. Computational problems and algorithmic solutions Optimality Example Powering Problem: Fix some monoid M. Input:XM and nN. Output:Xn. Cost: #multiplications Xn = en·ln(X) 3 operations! Devised & analysed algorithm using ≤2log2n multiplications Lemma a) Any algorithm starting with input X, and using only multiplications, produces only monomials in X as (intermediate) results. b) After ℓ multiplications, all intermediate results (monomials) have degree n≤2ℓ. So computing Xn requires ℓ≥log2n multiplications.

  12. 1. Computational problems and algorithmic solutions Example: FibonacciNumbers F0=0 F1=1 Fn = Fn-1 + Fn-2 FibIter(n) if n=0 return 0; fib := 1; fibL := 0; while n>1 do tmp:=fibL; fibL := fib; fib := fibL+tmp; n := n-1 ; end return fib; ≥ 2n/2 −1, n≥1 FibRek(n) if n=0 return 0; if n=1 return 1; return FibRek(n-1)+FibRek(n-2); k = · · = k := n-1

  13. 1. Computational problems and algorithmic solutions Example: FibonacciNumbers F0=0 F1=1 Fn = Fn-1 + Fn-2 φ := (1+√5)/2Golden Ratio ≥ 2n/2 −1, n≥1 Prove by induction: Fn=(φn-(-1/φ)n) / √5 φn = exp(n·ln φ) Five different algorithms, from exponential to constant “time”! k = · · = k := n-1

  14. 1. Computational problems and algorithmic solutions recurrences Master Theorem: Let f:N→R be increasing and satisfy the recurrence relation f(n) = a · f(n/b) + O(nd) whenever n is an integral power of b, where a,b≥1 are integers and d>0. Then

  15. 1. Computational problems and algorithmic solutions LongMultiplication Input: coefficients of polynomials A(x)=a0+a1x+a2x²+…+aN-1xN-1 and B(x) of degree <N. Output: coefficients of polynomial C(x) := A(x)· B(x) of degree ≤K:=2N-2. T(N):=#arithmetic operations (multiplications, linear combinat.s) w.l.o.g. 2|N e.g.(-5) or + Recursive algorithm, Distributive law: T(N) = 4· T(N/2) + O(N) "Naîve“ ck= j aj·bk-j (A0(x)+ A1(x)·xN/2) · (B0(x)+ B1(x)·xN/2) = C0(x) + C1(x) · xN/2+ C2(x) · xN C1(x)= A0(x)·B1(x) + A1(x)·B0(x) C2(x)= A1(x)·B1(x) C0(x)= A0(x)·B0(x) C(x)= c0 ,… cN/2-1 cN/2 ,… cN-1 cN-1 ,… c3/2·N-1 c3/2·N-1 ,… c2N-2

  16. 1. Computational problems and algorithmic solutions KaratsubaMultiplication Input: coefficients of polynomials A(x)=a0+a1x+a2x²+…+aN-1xN-1 and B(x) of degree <N. Output: coefficients of polynomial C(x) := A(x)· B(x) of degree ≤K:=2N-2. T(N):=#arithmetic operations (multiplications, linear combinat.s) w.l.o.g. 2|N e.g.(-5) or + Recursive algorithm, based on: Distributive law: T(N) = 4· T(N/2) + O(N) „Karatsuba law“: T(N) = 3· T(N/2) + O(N) (A0(x)+ A1(x)·xN/2) · (B0(x)+ B1(x)·xN/2) = C0(x) + C1(x) · xN/2+ C2(x) · xN C0(x) := A0(x) · B0(x) , C2(x) := A1(x) · B1(x) C1(x) := (A0(x) + A1(x) ) · ( B0(x) + B1(x) )−C0(x) −C2(x)

  17. 1. Computational problems and algorithmic solutions ToomMultiplication o T1 := (A0+2A1+4A2) · (B0+2B1+4B2) T2 := (A0 + A1 + A2) · (B0 + B1 + B2) T3 := (4A0+2A1+A2) · (4B0+2B1+B2) o o w.l.o.g. 3|N C3 = −C0 + ⅓T1−2T2 + ⅙T3 − 3½C4 C2 = 3½C0 −½T1 +5T2 −½T3 + 3½C4 C1 = −3½C0 + ⅙T1 −2T2 +⅓T3 −C4 C0 = A0 · B0 , C4 = A2 · B2 Distributive law: T(N) = 4· T(N/2) + O(N) o o „Karatsuba law“: T(N) = 3· T(N/2) + O(N) „Toom‘s law“: T(N) =5· T(N/3) + O(N) (A0(x)+A1(x)·xN/3 + A2(x)·x2N/3)(B0(x)+B1(x)·xN/3 +B2(x)·x2N/3) = C0(x) + C1(x)·xN/3 + C2(x)·x2N/3 + C3(x)·x3N/3 + C4(x)·x4N/3

  18. A(x)·B(x) A(x)·B(x) | : | A(x)·B(x) A(x)·B(x) A(x)·B(x) | : | A(x)·B(x) 0 0 0 0 1 xx² … xK 1 xx²… xK   :  1 xx² … xK 1 xx² … xK 1 xx² … xK   :  1 x x² … xK 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 K k k K K k k K k K 1. Computational problems and algorithmic solutions Toom-Cook Input: coeff. of A(x)=a0+a1x+a2x²+… of deg<N,B(x) of deg<M Output: coeffients c0,…cK of C(x) := A(x) · B(x), deg(C)≤K:= N+M-2 Long Multiplication: N·Mproducts & linear combinations -1 A(x)· B(x) = C(x) = (1 x x²… xK) Vandermonde Matrix invertible for any distinct fixedx0,…xK c0 c1  | : | cK  = =   Evaluation/Interpolation: K+1products, O(N²+M²)linear comb.s T(N,M) = (n+m-1) · T(N/n,M/m) + O((N+M)·(n+m)) (A0(x) + A1(x)·xN/n+ A2(x)·x2N/n+ ...+ An-1(x)·x(n-1)·N/n)(B0(x) + B1(x)·xM/m+ B2(x)·x2M/m+ ...+ Bm-1(x)·x(m-1)·M/m)

  19. 1. Computational problems and algorithmic solutions Input: entries of nn-matrices A,B Output: entries of C := A +B A1,1 A1,2 B1,1 B1,2 C1,1 C1,2 · = A2,1 A2,2 B2,1 B2,2 C2,1 C2,2 matrix multiplication T(n) = #arithmetic operations  O(n²) O(n³) +18 lin.comb.s 7 multiplications of (n/2)(n/2)-matrizes T1:=(A2,1+A2,2)·B1,1 T2:=(A1,1+A1,2)·B2,2 T3:=A1,1·(B1,2-B2,2) T4:=A2,2·(B2,1-B1,1) T5:=(A1,1+A2,2)·(B1,1+B2,2) T6:=(A2,1-A1,1)·(B1,1+B1,2) T7:=(A1,2-A2,2)·(B2,1+B2,2) C1,1=T5+T4-T2+T7 , C1,2=T3+T2 C2,1=T1+T4 , C2,2=T5-T1+T3+T6 T(n) = 7·T(n/2) +18·(n/2)² = O(nlog27) World record:O(n2.37) [F.Le Gall'14]

  20. 1. Computational problems and algorithmic solutions pedagogy This course focuses on understanding, less about knowledge! Bloom's Taxonomy

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