Shortest Paths

classical shortest paths.

• dijkstra's algorithm
• floyd's algorithm. similarity to matrix multiplication

Matrices

• length 2 paths by squaring
• matrix multiplication. strassen.
• shortest paths by ``funny multiplication.''
• huge integer implementation
• base-(n + 1) integers

Boolean matrix multiplication

• easy.
• gives objects at distance 2.
• gives nMM(n) algorithm for problem
• what about recursive?
• well can get to within 2: let T_k be boolean ``distance less than or equal to 2^k. Squaring gives T_{k + 1}.
• O(log n) squares for unit length
• what about exact?

Seidel's distance algorithm for unit lengths.

• log-size integers:
• parities suffice:
• square G to get adjacency A', distance D'
• if D_ij even then D_ij = 2D'_ij
• if D_ij odd then D_ij = 2D'_ij - 1
• For neighbors i, k,
• D_ij - 1$\le$D_kj$\le$D_ij + 1
• exists k, D_kj = D_ij - 1
• Parities
• If D_ij even, then D'_kj$\ge$D'_ij for every neighbor k
• If D_ij odd, then D'_kj$\le$D'_ij for every neighbor k, and strict for at least one
• Add
• D_ij even iff S_ij = $\sum_{k}^{}$D'_kj$\ge$D_ijd (i)
• D_ij odd iff $\sum_{k}^{}$D'_kj < D_ijd (i)
• How determine? find S = AD'

To find paths: Witness product.

• easy case: unique witness
• multiply column c by c.
• read off witness identity
• reduction to easy case:
• Suppose r columns have witness
• Suppose choose each with prob. p
• Prob. exactly 1 witness: rp(1 - p)^{r - 1} $\approx$1/e
• Try all values of r
• Wait, too many.
• Approx
• Suppose p = 2/r
• Then prob. exactly 1 is $\approx$2/e²
• So anything in range 1/r...1/2r will do.
• So try p all powers of 2.
• suppose 2^k$\le$r$\le$2^{k + 1}
• choose each column with probability 2^-k.
• prob. exactly one witness: r ^. 2^-k(1 - 2^-k)^{r - 1}$\ge$(1/2)(1/e²)
• so try log n distinct powers of 2, each O(log n) times

¬…       Mod 3:

o      Recall some neighbor distance down by one

o      so compute distances mod 3.

o      suppose D_ij = 1 mod 3

o      then look for k neighbor of i such that D_kj = 0 mod 3

o      let D_ij^(s) = 1 iff D_ij = s mod 3

o      than AD^(s) has ij = 1 iff a neighbor k of i has D_kj^(s)

o      so, witness matrix mul!

Minimum Cut

• deterministic algorithms

¬…       Min-cut implementation

¬…       data structure for contractions

¬…       alternative view--permutations.

¬…       deterministic leaf algo

¬…       recursion:

$${\textstyle\frac{1}{4}}$$

¬…      

¬…       cut counting

¬…       Reliability

¬…       Sampling