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Advanced Algorithms

All-pairs SPs DP algorithm

Floyd-Warshall alg.

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APSP Problem

Given a directed graph G = (V, E), weight function w : E → R, |V| = n determine the length of the shortest path between all pairs of vertices in G.

In other words, we want to create an n × n matrix of shortest-path distances δ(u, v).

Here we assume that there are no cycles with zero or negative cost.

Could run BELLMAN-FORD once from each vertex:

• O(V2E) - which is O(V4) if the graph is dense (E = Ө(V2)).

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APSP (cont.)

If no negative-weight edges, could run Dijkstra.s algorithm once from each vertex:

O(V E lg V) with binary heap - O(V3lg V) if dense,

O(V2lg V + V E) with Fibonacci heap - O(V3) if dense.

We will see how to do in O(V3) in all cases, with no fancy data structure.

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Terminologies

G = (V, E) where V are numbered 1, 2, .. n=|V|

Input : , n n matrix Output: , n n matrix

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Dynamic Programming Approach

Optimal substructure property holds Define subproblem:

: minimum weight of any path from vertex i to j that contains at most m edges.

Recursive solution Base case : m = 0

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Recursive Solution cont.

Goal:

Example:

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Build DP-table

Compute in order

= W

Go from

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Pseudo-code cont.

Time complexity:

Extend:

Slow-Apsp:

Can we do better ?

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Matrix Multiplication

Extend procedure same as matrix multiplication L’ = L W !

=L(m)

W(m)

Goal: compute W(n-1)

Only need to compute , , , …, W2

W1

W(n-1)

W4

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Faster Algorithm

Time complexity:

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Floyd-Warshall Algorithm

Structure of a shortest path

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Structure of a shortest path (cont.)

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Recursive solution

Recursive solution: : the weight of shortest path from vertex i to j

where all intermediate vertices are from set {1, 2, … , k}.

Goal: To compute = D

(n)

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Pseudo-code

Time complexity:

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Constructing a shortest Path

Compute a sequence of matrices Π(0), Π(1), Π(2),…, Π(n), where Π= Π(n).

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Construct a shortest path

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Thank you.