1. science
2. mathematics
3. geometry

Analysis of Algorithms, I
CSOR W4231.002
Eleni Drinea
Computer Science Department
Columbia University
Tuesday, March 10, 2015
Outline
1 Recap
2 Graphs with negative edge weights: why Dijkstra fails
3 Single-source shortest paths (negative edges): Bellman-Ford
A DP solution
An alternative formulation of the algorithm
4 All-pairs shortest paths (negative edges): Floyd-Warshall
Today
1 Recap
2 Graphs with negative edge weights: why Dijkstra fails
3 Single-source shortest paths (negative edges): Bellman-Ford
A DP solution
An alternative formulation of the algorithm
4 All-pairs shortest paths (negative edges): Floyd-Warshall
Review of the last lecture
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Connectivity in graphs
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I
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Undirected: find connected components via BFS or DFS
Directed: find strongly connected components via DFS
Single-source shortest paths in directed weighted graphs
with non-negative edge weights
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Dijkstra’s algorithm computes shortest path distances
from a vertex s to every other vertex in the graph
Best implementation depends on the density of the graph
Today
1 Recap
2 Graphs with negative edge weights: why Dijkstra fails
3 Single-source shortest paths (negative edges): Bellman-Ford
A DP solution
An alternative formulation of the algorithm
4 All-pairs shortest paths (negative edges): Floyd-Warshall
Example graph with negative edge weights
2
a
1
c
s
5
b
-4
Dijkstra’s output and correct output for example graph
(2)
2
a
s
5
b
(5)
1
(3)
c
2
(2)
a
(1)
c
s
5
b
-4
(5)
Dijkstra’s algorithm will first include a to S and then c, thus missing
the shorter path from s to b to c.
Negative edge weights: why Dijkstra’s algorithm fails
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Intuitively, a path may start on expensive edges but then
compensate along the way with cheap edges.
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Formally, in the proof of correctness of the algorithm, the
last statement about P does not hold anymore: even if the
cost (weight) of path Pv is smaller than the cost of the
subpath s → x → y, negative edges on the subpath y → v
may now result in P being cheaper than Pv .
Bigger problems in graphs with negative edges?
2
a
1
s
5
b
dist(a) =?
c
1
-4
Bigger problems in graphs with negative edges?
2
a
1
s
c
1
5
b
-4
1. dist(v) goes to −∞ for every v on the cycle
2. no solution to shortest paths when negative cycles
⇒ need to detect negative cycles
Today
1 Recap
2 Graphs with negative edge weights: why Dijkstra fails
3 Single-source shortest paths (negative edges): Bellman-Ford
A DP solution
An alternative formulation of the algorithm
4 All-pairs shortest paths (negative edges): Floyd-Warshall
Single-source shortest paths, negative edge weights
Input: weighted directed graph G = (V, E, w), w : E → R;
a source (origin) vertex s ∈ V
Output:
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Decide if G has a negative cycle
If no negative cycles
1. find the costs (weights) of shortest s-v paths for all v ∈ V
2. reconstruct shortest s-v paths
Properties of shortest paths
Suppose the problem has a solution for an input graph.
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Can there be negative cycles in the graph?
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Can there be positive cycles in the graph?
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Can the shortest paths contain positive cycles?
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Consider a shortest s-t path; are its subpaths shortest?
Towards a DP solution
Key observation: if there are no negative cycles, a path
cannot become cheaper by traversing a cycle.
Fact 1.
If G has no negative cycles, then there is a shortest s-v path
that is simple, thus has at most n − 1 edges.
Fact 2.
The shortest paths problem exhibits optimal substructure.
Facts 1 and 2 suggest a DP solution.
Subproblems
w
s
v
Let
OP T (i, v) = cost of a shortest s-v path with at most i edges
Consider a shortest s-v path using at most i edges.
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If the path uses at most i − 1 edges, then
OP T (i, v) = OP T (i − 1, v).
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If the path uses i edges, then
OP T (i, v) =
min
x:(x,v)∈E
{OP T (i − 1, x) + w(x, v)} .
Subproblems and recurrence
Let
OP T (i, v) = cost of a shortest s-v path using at most i edges
Then

0,



 ∞,
(
OP T (i, v) =



 min
if i = 0, v = s
if i = 0, v 6= s
OP T (i − 1, v)
min {OP T (i − 1, x) + w(x, v)},
x:(x,v)∈E
if i > 0
Example of Bellman-Ford
Compute shortest s-v paths in the graph below, for all v ∈ V .
a
4
s
-1
1
c
2
5
b
-1
Pseudocode
n × n dynamic programming table M storing OP T (i, v)
Bellman-Ford(G = (V, E, w), s ∈ V )
for v ∈ V do
M [0, v] = ∞
end for
M [0, s] = 0
for i = 1, . . . , n − 1 do
for v ∈ V (in any order) do

 M [i −n1, v]
o
M [i, v] = min
min
M
[i
−
1,
x]
+
w(x,
v)

x:(x,v)∈E
end for
end for
Running time & Space
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Running time: O(nm)
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Space: Θ(n2 ) —can be improved (coming up)
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To reconstruct actual shortest paths: keep array prev of
size n such that
prev[v] = predecessor of v in current shortest s-v path.
Improving space requirements
Only need two columns of M at all times.
4 Actually, only need one! (see Remark 1) Thus drop the index
i from M [i, v] and only use it as a counter for #repetitions.
n
o
M [v] = min M [v], min M [x] + w(x, v)
x:(x,v)∈E
Remark 1.
Throughout the algorithm, M [v] is the cost of some s-v path.
After i repetitions, M [v] is no larger than the cost of the
current shortest s-v path with at most i edges.
Early termination condition: if at some iteration i no value
in M changed, then stop (why?) –also allows to detect negative
cycles.
An alternative way to view Bellman-Ford
s
v1
v2
…
vk-1
v
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Let P = (s = v0 , v1 , v2 , . . . , vk = v) be a shortest s-v path.
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Then P can contain at most n − 1 edges.
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How can we correctly compute dist(v) on this path?
Key observations about subroutine Update(u, v)
Recall subroutine Update from Dijkstra’s algorithm:
Update(u, v) : dist(v) = min{dist(v), dist(u) + w(u, v)}
Fact 3.
Suppose u is the last node on the shortest s-v path, and suppose
dist(u) has been correctly set. The call Update(u, v) returns the
correct value for dist(v).
Fact 4.
No matter how many times Update(u, v) is performed, it will
never make dist(v) too small. That is, Update is a safe
operation: performing few extra updates can’t hurt.
Main idea
Suppose we update the edges on the shortest path P in the
order they appear on the path (though not necessarily
consecutively). Hence we update
(s, v1 ), (v1 , v2 ), (v2 , v3 ), . . . , (vk−1 , v)
This sequence of updates correctly computes dist(v1 ), dist(v2 ),
. . ., dist(v) (by induction and Fact 3).
How can we guarantee that updates will happen in this order?
Bellman-Ford algorithm
Update all m edges in the graph, n − 1 times in a row.
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By Fact 4, it is ok to update an edge several times in
between.
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All we care for is that the edges on the path are updated in
this particular order. This is guaranteed if we update all
edges n − 1 times in a row.
Pseudocode
We will use Initialize and Update from Dijkstra’s algorithm.
Initialize(G, s)
for v ∈ V do
dist[v] = ∞
prev[v] = N IL
end for
dist[s] = 0
Update(u, v)
if dist[v] > dist[u] + w(u, v) then
dist[v] = dist[u] + w(u, v)
prev[v] = u
end if
Bellman-Ford
Bellman-Ford(G = (V, E, w), s)
Initialize(G, s)
for i = 1, . . . , n − 1 do
for (u, v) ∈ E do
Update(u, v)
end for
end for
Running time? Space?
Detecting negative cycles
2
a
1
s
c
1
5
b
-4
1. dist(v) goes to −∞ for every v on the cycle.
2. Any shortest s-v path can have at most n − 1 edges.
3. Update every edge n times (instead of n − 1): if dist(v)
changes for any v ∈ V , there is a negative cycle.
Today
1 Recap
2 Graphs with negative edge weights: why Dijkstra fails
3 Single-source shortest paths (negative edges): Bellman-Ford
A DP solution
An alternative formulation of the algorithm
4 All-pairs shortest paths (negative edges): Floyd-Warshall
All pairs shortest-paths
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Input: a directed graph G with real edge weights
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Output: an n × n matrix D such that
D[i, j] = weight of shortest path from i to j
Solving all pairs shortest-paths
1. Straightforward solution: run Bellman–Ford once for every
vertex (O(n2 m) time).
2. Improved solution: a dynamic programming algorithm for
generating shortest paths (Floyd–Warshall, O(n3 ) time).
Towards a DP formulation
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Consider a shortest path P from s to t.
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This path uses some intermediate vertices: that is, if
P = (s, v1 , v2 , . . . , vk , t), then v1 , . . . , vk are intermediate
vertices.
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For simplicity, let V = {1, 2, 3, . . . , n} and consider a
shortest path from i to j when intermediate vertices may
only be from {1, 2, . . . , k}.
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Goal: find the shortest i-j path using {1, 2, . . . , n} as
intermediate vertices.
Example
a
4
s
-1
1
c
2
5
b
-1
Rename {s, a, b, c} as {1, 2, 3, 4}
2
4
1
-1
1
2
5
3
4
-1
Examples of shortest paths
Shortest (1, 2)-path may use:
(i) no intermediate nodes (P); or
(ii) node 3 (P1)
P
1
1
P1
5
4
2
P1
Shortest (1, 3)-path may use:
(i) no intermediate nodes (P); or
(ii) node 2 (P1)
2
4
3
2
4
-1
1
P1 P1
1 5
-1
P
3
4
A shortest i-j path using nodes from {1, . . . , k}
Consider a shortest path P from i to j where intermediate
nodes may only be from the set of nodes {1, 2, . . . , k}.
i
Fact: any subpath of P must be shortest itself.
j
A useful observation
Focus on the last node k from this set. Either
1. P completely avoids k: then a shortest i-j path with
intermediate nodes from {1, . . . , k} is the same as a shortest
i-j path with intermediate nodes from {1, . . . , k − 1}.
2. Or, k is an intermediate node of P .
k
i
j
Decompose P into an i-k subpath P1 and a k-j subpath P2 .
i. P1 , P2 are shortest subpaths themselves.
ii. All intermediate nodes of P1 , P2 are from {1, . . . , k − 1}.
Subproblems
Let
OP Tk (i, j) = cost of shortest i − j path P using
{1, . . . , k} as intermediate vertices
1. Either k does not appear in P , hence
OP Tk (i, j) = OP Tk−1 (i, j)
2. Or, k appears in P , hence
OP Tk (i, j) = OP Tk−1 (i, k) + OP Tk−1 (k, j)
Recurrence
Hence

w(i, j)



OP Tk (i, j) =
OP Tk−1 (i, j)


 min
OP Tk−1 (i, k) + OP Tk−1 (k, j)
We want OP Tn (i, j).
Time/space requirements?
, if k = 0
, if k ≥ 1
Floyd-Warshall on example graph
D0 =
0
∞
∞
∞
4
0
2
∞
5
-1
0
∞
∞
1
-1
0
D2 =
0
∞
∞
∞
4
0
2
∞
3
-1
0
∞
5
1
-1
0
D1 =
0
∞
∞
∞
4
0
2
∞
5
-1
0
∞
∞
1
-1
0
D3 =
0
∞
∞
∞
4
0
2
∞
3
-1
0
∞
2
-2
-1
0
Space requirements
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A single n × n dynamic programming table D, initialized
to w(i, j) (the adjacency matrix of G).
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Let {1, . . . , k} be the set of intermediate nodes that may be
used for the shortest i-j path.
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After the k-th iteration, D[i, j] contains the cost of an i-j
path that is no larger than the cost of the shortest i-j path
using {1, . . . , k} as intermediate nodes.
The Floyd-Warshall algorithm
Floyd-Warshall(G = (V, E, w))
for k = 1 to n do
for i = 1 to n do
for j = 1 to n do
D[i, j] = min(D[i, j], D[i, k] + D[k, j])
end for
end for
end for
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Running time: O(n3 )
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Space: Θ(n2 )