Solved and Unsolved in Elementary Algorithms

ion of shortest-paths problem: Definition 19. A commutative semi-ring is a set K and two operations ⊗,⊕ satisfying: 1. ⊕ is commutative and associative; there is an identity “0” s.t. k + 0 = k for all k ∈ K. 2. ⊗ is commutative and associative; there is a multiplicative identity “1” s.t. k ⊗ 1 = k for all k ∈ K. 3. The distributive law holds: a⊗ (b⊕ c) = (a⊗ b)⊕ (a⊗ c). Definition 20. The min-sum semi-ring over the reals is: K = R ∪∞, ⊗ = +, 1 = 0, ⊕ =min, 0 =∞. Check the distributive law: a+ (bmin c) = (a+ b) min(a+ c) for a, b, c ∈ R ∪∞. This semi-ring plays an important role in computation. Consider a (possibly directed) weighted graph G = (V,E,w). Distance adds along any path, is minimized across the choice of paths. One of the (many) places that this comes up: max-likelihood decoding. Let’s pose the computational problem algebraically. Form the n× n matrix of edge weights AG =  0 2 ∞ ∞ 0 −3 2 ∞ 0  Squaring AG (w.r.t. min-sum) gives shortest paths consisting of ≤ 2 edges: (AG)i,k = min j {Aij +Ajk} For any t ≥ n− 1, AG = matrix of shortest-path distances =  0 2 −1 −1 0 −3 2 4 0  Let TM (n) = runtime for min-sum multiplication of matrices of size n× n. All-pairs shortest-paths runtime ≤ O(TM (n) · log n) ≤ O(n log n). (Repeated squaring.) Let TC(n) = runtime for min-sum convolution of vectors of length n. Convolution with respect to this semi-ring was first studied (to my knowledge) by Bellman and Karush in a series of papers in the early 1960’s [5, 6, 7, 9, 8, 10].