Shortest Paths & Matrix Powers

Agenda

1. The Shortest Path Problem: Formulation & Cost Semantics
2. Weighted vs. Unweighted Graphs: Real-World Applications
3. The Shortest Paths Landscape: Classical Complexities
4. The Matrix Powering Perspective: Walks & Distances
5. How to Compute A^k: Repeated Squaring in O(\log k) Steps
6. Reachability & Path Existence via A, A^2, \dots, A^n
7. The Resolvent Trick: Summing Powers via (I - A)^{-1}

What is the Shortest Path Problem?

Given a graph G = (V, E) (directed or undirected) with n = |V| vertices and m = |E| edges:

• Weight Function: Each edge e = (u,v) \in E has a weight w(e) \in \mathbb{R}.
• Path Weight: Total cost w(p) = \sum_{i=1}^k w(v_{i-1}, v_i).
• Shortest Distance: \text{dist}(u, v) = \min_{p: u \rightsquigarrow v} w(p).

Graph Models: Weighted vs. Unweighted

Real-World Applications: Key Domains

Shortest paths are among the most pervasive algorithmic primitives in modern computing:

Real-World Applications: Problem Mapping

How practical domains map to our graph-theoretic models:

The Shortest Paths Landscape

The Matrix Powering Perspective: Walks & Paths

Can linear algebra compute shortest paths?

• Let A be the unweighted adjacency matrix of graph G.

• Matrix Multiplication Property: (A^k)_{u,v} = \sum_{i_1, i_2, \dots, i_{k-1}} A_{u, i_1} A_{i_1, i_2} \cdots A_{i_{k-1}, v}

• (A^k)_{u,v} counts the exact number of walks of length k from u to v.

• Unweighted Shortest Path Distance: d(u,v) = \min \{ k \ge 0 \mid (A^k)_{u,v} > 0 \}

• In ordinary matrix algebra over ring (\mathbb{R}, +, \times), computing A^2 takes O(n^\omega) time (\omega < 2.372).

How to Compute A^k

Can we compute high powers A^k without performing k sequential multiplications?

• Recursive Formulation: A^k = \begin{cases} (A^{k/2})^2 & \text{if } k \text{ is even} \\ A \cdot (A^{(k-1)/2})^2 & \text{if } k \text{ is odd} \end{cases}

• Operation Count: Requires only \lceil \log_2 k \rceil matrix multiplications!

• Total Runtime: O(n^\omega \log k)

• When k = n, we can compute A^n in O(n^\omega \log n) time.

Reachability & Path Existence

Does there exist a path between u and v in G?

• The Path Length Bound: In any graph with n vertices, any simple path has length at most n - 1.

• The Reachability Criterion: \exists \text{ path } u \rightsquigarrow v \iff (A + A^2 + A^3 + \dots + A^{n-1})_{u,v} > 0

• If (A^k)_{u,v} > 0 for any 1 \le k \le n - 1, there exists a walk (and hence a simple path) of length \le n - 1 between u and v.

• Transitive Closure Matrix: T = \sum_{k=0}^{n-1} A^k A pair (u, v) is connected if and only if T_{u,v} > 0.

The Resolvent Trick: (I - A)^{-1}

Can we sum all powers of A in a single matrix inversion?

• Algebraic Verification: (I - A) \big(I + A + A^2 + \dots + A^k\big) = I - A^{k+1}

• If A^{k+1} \to 0 (or working over formal power series \mathbb{R}[[A]]): (I - A) \sum_{k=0}^\infty A^k = I \implies \sum_{k=0}^\infty A^k = (I - A)^{-1}

• Algorithmic Takeaway: The entries of (I - \epsilon A)^{-1} encode all-paths reachability and walk counts across all lengths simultaneously!

Next Lecture

Seidel’s Algorithm & The Isolation Lemma — Solving unweighted APSP in \tilde{O}(n^\omega) via fast matrix multiplication over \mathbb{Z}, and parallel matchings via the MVV Isolation Lemma.