Floyd-Warshall Algorithm
Chandler Burfield
February 20, 2013
Chandler Burfield
Floyd-Warshall
February 20, 2013
1 / 15
All-Pairs Shortest Paths
Problem
To find the shortest path
between all vertices v ∈ V
for a weighted graph
G = (V , E ).
Chandler Burfield
Floyd-Warshall
February 20, 2013
2 / 15
Dynamic-Programming
How to Develop a Dynamic-Programming Algorithm
1. Characterize the structure of an optimal solution.
2. Recursively define the value of an optimal solution.
3. Compute the value of an optimal solution in a bottom-up manner.
Chandler Burfield
Floyd-Warshall
February 20, 2013
3 / 15
!
Intermediate Vertices
An intermediate vertex of a simple path p = {v1 , v2 , ..., vl } is any vertex
other than v1 or vl i.e. a vertex from the set {v2 , ..., vl−1 }.
Intermediate!Vertices!of!Path!
V1 #
!
Chandler Burfield
!
!
Floyd-Warshall
Vl #
February 20, 2013
4 / 15
Weight Matrix
wij
The weight of an edge between vertex i and vertex j in graph G = (V , E ),
where

if i = j
 0
the weight of a directed edge (i, j) if i 6= j and (i, j) ∈ E
wij =

∞
if i 6= j and (i, j) ∈
/E
W
An n x n matrix representing the edge weights of an n-vertex graph,
where W = (wij ).
Chandler Burfield
Floyd-Warshall
February 20, 2013
5 / 15
Path Matrix
(k)
dij
The weight of the shortest path from vertex i to vertex j for which all
intermediate vertices are in the set {1, 2, ..., k}.
D (k)
An n x n matrix representing the path distances between vertices in a
(k)
directed n-vertex graph, where D (k) = (dij ).
Chandler Burfield
Floyd-Warshall
February 20, 2013
6 / 15
Observations
A shortest path does not contain the same vertex more than once.
For a shortest path from i to j such that any intermediate vertices on
the path are chosen from the set {1, 2, ..., k}, there are two
possibilities:
1. k is not a vertex on the path,
so the shortest such path has length dijk−1
2. k is a vertex on the path,
so the shortest such path is dikk−1 +dkjk−1
(k)
So we see that we can recursively define dij as
(
wij
if k = 0
(k)
dij =
(k−1)
(k−1)
(k−1)
min(dij
, dik
+ dkj
) if k ≥ 1
Chandler Burfield
Floyd-Warshall
February 20, 2013
7 / 15
Predecessor Matrix
(k)
πij
The predecessor of vertex j on a shortest path from vertex i with all
intermediate vertices in the set {1, 2, ..., k}. Where
NIL
if i = j or wij = ∞
(0)
πij =
i
if i 6= j and wij < ∞
And
(
(k)
πij
=
(k−1)
if dij
(k−1)
if dij
πij
πkj
(k−1)
≤ dik
(k−1)
+ dkj
(k−1)
> dik
(k−1)
(k−1)
+ dkj
(k−1)
Π(k)
(k)
The predecessor matrix, where Π(k) = (πij ).
Chandler Burfield
Floyd-Warshall
February 20, 2013
8 / 15
Algorithm
Floyd-Warshall(W)
n = W .rows
D (0) = W
for k = 1 to n
(k)
let D (k) = (dij ) be a new matrix
for i = 1 to n
for j = 1 to n
(k)
(k−1)
(k−1)
(k−1)
dij = min(dij
, dik
+ dkj
)
(n)
return D
Chandler Burfield
Floyd-Warshall
February 20, 2013
9 / 15
Analysis of Algorithm
Runtime
Floyd-Warshall(W)
n = W .rows
D (0) = W
for k = 1 to n
(k)
let D (k) = (dij ) be a new matrix
for i = 1 to n
for j = 1 to n
(k)
(k−1)
(k−1)
(k−1)
dij = min(dij
, dik
+ dkj
)
(n)
return D
Chandler Burfield
Floyd-Warshall
Θ(n3 )
Space
Θ(n3 ) here,
but if we reuse space
this can be done in
Θ(n2 ).
February 20, 2013
10 / 15
Analysis of Improved Algorithm
Floyd-Warshall(W)
n = W .rows
D=W
Π initialization
for k = 1 to n
for i = 1 to n
for j = 1 to n
if dij > dik + dkj
then dij = dik + dkj
πij = πkj
return D
Chandler Burfield
Floyd-Warshall
Analysis
The shortest path
can be constructed,
not just the lengths
of the paths.
Runtime: Θ(n3 ).
Space: Θ(n2 ).
February 20, 2013
11 / 15
Example
Floyd-Warshall(W)
n = W .rows
D=W
Π initialization
for k = 1 to n
for i = 1 to n
for j = 1 to n
if dij > dik + dkj
then dij = dik + dkj
πij = πkj
return D
Chandler Burfield
Floyd-Warshall
February 20, 2013
12 / 15
Proof of Correctness
Inductive Hypothesis
Suppose that prior to the kth iteration it holds that for i, j ∈ V , dij
contains the length of the shortest path Q from i to j in G containing only
vertices in the set {1, 2, ..., k − 1}, and πij contains the immediate
predecessor of j on path Q.
Chandler Burfield
Floyd-Warshall
February 20, 2013
13 / 15
Applications
Detecting the Presence of a Negative Cycle
Transitive Closure of a Directed Graph
Chandler Burfield
Floyd-Warshall
February 20, 2013
14 / 15
Other All-Pairs Shortest Paths Algorithms
Dynamic Programming Approach Based on Matrix Multiplication
Johnson’s Algorithm for Sparse Graphs
Chandler Burfield
Floyd-Warshall
February 20, 2013
15 / 15