MST

Review Background

• kruskal
• boruvka
• verification

sampling theorem:

• Heavy edges
• pick F with probability p
• get n/p F-heavy edges

Recursive algorithm without boruvka:

T(m, n) = T(m/2, n) + O(m) + T(2n, n) = O(m + n log n)

(sloppy on expectation on T(2n,n))

Recursive algorithm with 3 boruvka steps:
$$\begin{align*} T(m,n) &= T(m/2,n/8) + c_1( m+n) + T(n/4,n/8)\\ &\le c(m/2+n/8) ... ...(n/4+n/8)\\ &=(c/2+c_1)m + (c/8+c_1+c/4+c/8)n\\ &= (c/2+c_1)(m+n) \end{align*}$$
so set c = 2c₁ (not sloppy expectation).

Notes:

• Chazelle m log$\alpha$(m, n) via relaxed heap
• Ramachandran optimal deterministic algorithm (runtime unknown)
• open questions.