diff --git a/AlgorithmPROJECT.html b/AlgorithmPROJECT.html deleted file mode 100644 index 39d125e..0000000 --- a/AlgorithmPROJECT.html +++ /dev/null @@ -1,3910 +0,0 @@ - - - - - -Algorithm Visualiser - - - - - -
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Algorithm Visualiser
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- Interactive Visual Learning
- for Advanced Algorithms -

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- This tool teaches three core algorithms from algorithm design and analysis through - step-by-step interactive visualisations. Each tab walks you through how the algorithm - works, why it works, and what its theoretical guarantees are — with live animation, - pseudocode highlighting, and educational explanations at every step. -

- Topics covered: LP Duality & Network Flow, - FFT Polynomial Multiplication, - and Pollard’s Rho Factorization. -

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Algorithms
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Part 1 — CLO 1
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LP Duality & Network Flow
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- Visualise the Edmonds-Karp algorithm finding augmenting paths, - watch the residual graph update at each step, and see the min-cut - highlighted when max-flow is reached. Strong duality is verified live. -

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- Ford-Fulkerson - BFS - O(VE²) - LP Duality -
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- Open visualiser → -
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Part 2 — CLO 2
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FFT: Polynomial Multiplication
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- Step through the Cooley-Tukey butterfly network level by level. - Watch roots of unity animate on the unit circle, see pointwise - multiplication, and follow the inverse FFT back to coefficients. -

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- Butterfly Network - O(n log n) - Roots of Unity -
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ρ
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Part 3 — CLO 3
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Pollard’s Rho Factorization
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- Animate the pseudo-random sequence forming a rho-shaped graph. - Follow the tortoise and hare with Floyd’s cycle detection, - watching GCD values at each step until a factor is revealed. -

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- Floyd Detection - O(n^¼) - Birthday Paradox -
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- Open visualiser → -
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- Academic References -
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- [1] - - Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. - Introduction to Algorithms, 4th edition. - MIT Press, 2022. - — Chapters 26 (Maximum Flow), 29 (Linear Programming), 30 (FFT), 31 (Number Theory). - -
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- [2] - - Edmonds, J. and Karp, R. M. - Theoretical improvements in algorithmic efficiency for network flow problems. - Journal of the ACM, 19(2):248–264, 1972. - — Original proof that BFS augmentation gives O(VE²) complexity. - -
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- [3] - - Cooley, J. W. and Tukey, J. W. - An algorithm for the machine calculation of complex Fourier series. - Mathematics of Computation, 19(90):297–301, 1965. - — The foundational paper establishing the O(n log n) FFT algorithm. - -
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- [4] - - Pollard, J. M. - A Monte Carlo method for factorization. - BIT Numerical Mathematics, 15:331–334, 1975. - — Introduces the rho algorithm achieving O(n^¼) expected factorization. - -
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Suggested Google Scholar search terms
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- Edmonds Karp 1972  ·  - Cooley Tukey 1965 FFT  ·  - Pollard 1975 factorization -
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Network Input
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- - - Use s = source  |  t = sink -
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Playback Controls
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- SPEED - - 1x -
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Live Statistics
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0Max Flow
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0Augmentations
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—Min-Cut Cap.
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LP Formulation — Strong Duality
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PRIMAL (Max-Flow)
- maximize  ∑ f(s,v)
- s.t.   f(u,v) ≤ c(u,v)
-    ∑in = ∑out
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DUAL (Min-Cut)
- minimize  ∑ c(u,v)·d
- s.t.   d(u,v) ≥ p(v)−p(u)
-    p(s)=0, p(t)=1
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- Strong Duality: Max-Flow = Min-Cut
- Optimal values will match when done. -
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Pseudocode — Edmonds-Karp (BFS)
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- 1EDMONDS-KARP(G, s, t): -
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- 2 for each (u,v) in E: f[u,v] = 0 -
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- 3 while BFS(Gf, s to t) finds path p: -
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- 4 cf = min{ cf(u,v) : (u,v) in p } -
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- 5 for each (u,v) in p: -
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- 6 f[u,v] = f[u,v] + cf -
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- 7 f[v,u] = f[v,u] - cf -
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- 8 return sum_v f[s,v] // max flow -
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- 9 // Min-Cut S = {reachable from s in Gf} -
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Step Explanation
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- Build a network, then press Play or Step to animate. -
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Step History
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#TypePathTotal Flow
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What is Max-Flow?

- Find the maximum flow from source s to sink t in a directed network - where each edge has a capacity constraint. -
- Ford-Fulkerson Method: Repeatedly augment flow along paths - in the residual graph. Edmonds-Karp uses BFS (shortest path) - to guarantee O(VE²) runtime regardless of capacity values. -
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Residual Graph

- For edge (u,v) with capacity c and flow f: the residual graph has a forward - edge with capacity c−f and a backward edge with capacity f. The backward - edge allows "undoing" previous flow decisions. -
- Max-Flow Min-Cut Theorem (LP Duality): - Maximum s-t flow = Minimum s-t cut capacity. - This follows from LP Strong Duality: max-flow is the primal LP; min-cut is its dual. -
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Worked Example

- Load "Classic CLRS". BFS finds s→a→c→t (bottleneck 4), - then s→a→d→t (bottleneck 6), then s→b→d→t (bottleneck 4). - Total max flow = 14 = min-cut capacity. ✓ -
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- Edmonds-Karp — Flow Network & Residual Graph -
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Flow Edge
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Aug. Path
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Min-Cut
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Residual
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Polynomial Input
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- - - "1,2,3" represents 1 + 2x + 3x² - - - - -
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Live Statistics
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0Multiplications
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—Naive n² ops
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Complexity Comparison
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NAIVE O(n²)
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FFT O(n log n)
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— ops
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- The n-th roots of unity w_n^k = e^(2pi*i*k/n) satisfy - w_n^(n/2) = -1 (cancellation), halving the work at each recursion level. -
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Pseudocode — Cooley-Tukey FFT
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- 1FFT(a[0..n-1], invert): -
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- 2 if n == 1: return -
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- 3 a_even = a[0,2,4,...]; a_odd = a[1,3,5,...] -
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- 4 FFT(a_even, invert); FFT(a_odd, invert) -
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- 5 w = e^(+/-2*pi*i/n) [- sign for IFFT] -
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- 6 for k = 0 to n/2 - 1: -
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- 7 t = w^k * a_odd[k] -
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- 8 a[k] = a_even[k] + t -
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- 9 a[k+n/2] = a_even[k] - t -
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- 10 if invert: a[k] = a[k] / n -
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Step Explanation
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- Enter two polynomials and press Build to begin. -
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Step History
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#PhaseDetailTotal Ops
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Why FFT for Polynomials?

- Naive multiplication of degree-n polynomials costs O(n²). FFT evaluates each polynomial - at n cleverly chosen points in O(n log n), multiplies pointwise in O(n), then interpolates back. -
- 3-Step Process:
- (1) FFT(A) and FFT(B) → value representation at n roots of unity
- (2) Pointwise: C[k] = A[k] · B[k] — O(n)
- (3) IFFT(C) → coefficient form of the product -
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Butterfly Network

- At each recursion level, n/2 butterfly operations combine adjacent pairs. - With log⊂2 n levels total, the overall work is O(n log n). -
- Divide & Conquer: - A(x) = A_even(x²) + x·A_odd(x²). - Recurrence: T(n) = 2T(n/2) + O(n) → O(n log n). -
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Worked Example

- A=[1,2,3,4], B=[1,1,1,1]. FFT pads to n=8, evaluates at 8 roots of unity, - multiplies pointwise, inverse-FFTs to get C=[1,3,6,10,9,7,4]. -
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- FFT Butterfly Network — 3-Step Polynomial Multiplication -
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Poly A
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Poly B
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Result C
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Active Level
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Pending
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- Enter two polynomials on the left
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Factorization Input
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Live Statistics
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0Steps
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—Factor Found
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—Expected O(n^0.25)
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Pointer State
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TORTOISE
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xi (slow)
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HARE
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GCD
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Pseudocode — Pollard's Rho + Floyd Detection
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- 1POLLARD-RHO(n, c=1, x0=2): -
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- 2 x = x0; y = x0; d = 1 -
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- 3 f(x) = (x*x + c) mod n -
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- 4 while d == 1: -
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- 5 x = f(x) // tortoise: 1 step -
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- 6 y = f(f(y)) // hare: 2 steps -
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- 7 d = gcd(|x - y|, n) -
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- 8 if d == n: FAILURE (retry with new c) -
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- 9 return d // non-trivial factor of n -
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Step Explanation
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- Enter a composite number and press Build to begin. -
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Step History
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#Tortoise xiHare x2igcd
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What is Pollard's Rho?

- A probabilistic factorization algorithm (John Pollard, 1975) with expected runtime O(n^(1/4)) — - far faster than trial division at O(n^(1/2)). -
- Birthday Paradox: Among about sqrt(p) random values mod p - (p = smallest prime factor of n), two will collide with high probability. - This collision mod p reveals p via GCD. -
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Floyd's Cycle Detection

- The sequence x0, f(x0), f(f(x0)), ... is eventually periodic modulo p — - it forms a rho shape: a tail leading into a cycle. - The tortoise moves 1 step per iteration; the hare moves 2. -
- Key insight: When tortoise x equals hare y mod p, - then p divides gcd(|x−y|, n). Since p divides n but p does not equal n, - we get 1 < gcd < n — a real non-trivial factor. -
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Worked Example

- n=77, c=1, x0=2. Sequence: 2, 5, 26, 677 mod 77 = 677-8*77=61, ... - Eventually gcd(|xi - x2i|, 77) = 7. Factor found: 7 × 11 = 77. ✓ -
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- Pollard's Rho — Pseudo-Random Sequence & Floyd's Cycle Detection -
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Tortoise (slow)
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Hare (fast)
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Factor Found
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