Algorithmsesslab.hanyang.ac.kr/uploads/algorithm_2018_2/lecture... · 2018-11-27 · The...

Algorithms

All-Pairs Shortest Paths

Dong Kyue Kim

Hanyang University

[email protected]

2

Contents

• Using single source shortest path algorithms

• Presents O(V4)-time algorithm, O(V3 log V)-time algorithm, O(V3)-time algorithm

– O(V4)-time algorithm

– O(V3 log V)-time algorithm

– Floyd-Warshall algorithm

– Transitive closure of a directed graph

2

3

Using single source shortest path algorithms

• Positive edges

• Negative edges

4


• Using single source shortest path algorithms– We can solve an all-pairs shortest-paths problem by

running a single-source shortest-paths algorithm |V| times, once for each vertex as the source.

5


• Positive edges– Using dijkstra algorithm:

• The linear-array implementation– O(V3 + V E) = O(V3).

• The binary min-heap implementation O(V E lg V),

– Fibonacci heap• O(V2 lg V + V E).

6


• Negative edges– Using Bellman-Ford algorithm:

• O(V2E)

• Dense graph– O(V4)

7


• Use adjacency matrix

• Predecessor matrix

8


• Use adjacency matrix• We assume that the vertices are numbered 1, 2,..., |V|

• The input is an n×n matrix W representing the edge weights of an n-vertex directed graph G = (V, E)

• Negative-weight edges are allowed, but we assume for the time being that the input graph contains no negative-weight cycles.

E j) (i, and j i if

E, j) (i, and j i if j) (i, edge directed of weight the

j, i if 0

ijw

9


• Predecessor matrix

– Say Π = (πij)

– πij = NIL if either i = j or there is not path from i to j.

– πij is the predecessor of j on some shortest path from i to j.

10


• For each vertex i ∈ V, we define predecessor subgraph of G for i as Gπ,i = (Vπ,i, Eπ,i), where– Vπ,i = {j ∈ V : πij = NIL} ∪ {i}

– Eπ,i = {(πij, j)|j ∈ Vπ,i − {i}}

• If Gπ,i is a shortest-paths tree, then PRINT-ALL-PAIRS-SHORTEST-PATH will print a shortest path from i to j.

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PRINT-ALL-PAIRS-SHORTEST-PATH(Π, i, j)

1 if i = j

2 then print i

3 else if πij = NIL

4 then print “no path from” i “to” j “exists”5 else PRINT-ALL-PAIRS-SHORTEST-PATH(Π,i, πij)

6 print j

12

Shortest paths and matrix multiplication

• O(V4)-time algorithm– Use dynamic programming

– Use matrix multiplication

– Associative (short explanation)

– Computing predecessor matrix

13


• Use dynamic programming

– Characterize the structure of an optimal solution.

– Recursively define the value of an optimal solution.

– Compute the value of an optimal solution in a bottom-up fashion.

14


• The structure of a shortest path

– On a graph G = (V, E), all subpaths of a shortest path are shortest paths. (Lemma 24.1)

– p’ is a shortest path from i to k, and so δ(i, j) = δ(i, k) + wkj.

i

k

j

P1P2

P` : {1,2,…,k-1}

15


• A recursive solution to the all-pairs shortest-paths problem

– Let be the minimum weight of any path from vertex ito vertex j that contains at most m edges

• When m = 0, then

• When m ≥ 1, then

)(m

ijl

j i if

ji if 0)0(

ijl

}{min

}){min ,min(

)1(

nk1

)1(

nk1

)1()(

kj

m

ij

kj

m

ij

m

ij

m

ij

wl

wlll

16


– If the graph contains no negative-weight cycles,

– For every pair of vertices i and j for which δ(i, j) < ∞

• There is a shortest path from i to j that is simple and thus contains at most n - 1 edges.

– A path from vertex i to vertex j with more than n - 1 edges cannot have lower weight than a shortest path from i to j.

– Therefore,

...),( )1()()1( n

ij

n

ij

n

ij lllji

17


• Computing the shortest-path weights bottom up

– Input the matrix W = (wij), we now compute a series of matrices L(1), L(2), ... , L(n-1),

• Where for m = 1, 2,..., n - 1, we have .

– The final matrix L(n-1) contains the actual shortest-path weights.

– Observe that for all vertices i, j ∈ V , and so L(1) = Wij.

)( )()( m

ij

m ll

18


– Let L(m−1) = L, L(m) = L. We have

– Costs Θ(n3) time.

EXTEND-SHORTEST-PATHS(L, W)

1 n ← rows[L]

2 let L′ = (lij ′) be an n × n matrix

3 for i ← 1 to n

4 do for j ← to n

5 do

6 for k ← 1 to n

7 do

8 return L′

19


EXTEND-SHORTEST-PATHS(L, W)1 n ← rows[L]

2 let C be an n × n matrix

3 for i ← 1 to n

4 do for j ← 1 to n

5 do cij ← 0

6 for k ← 1 to n

7 do cij ← cij + aik · bkj

8 return L’

20


• Costs Θ(n4) time.

SLOW-ALL-PAIRS-SHORTEST-PATHS(W)

1 n ← rows[W]

2 L(1) ←W

3 for m ← 2 to n − 1

4 do L(m) ← EXTEND-SHORTEST-PATHS(L(m−1),W)

5 return L(n−1)

21


• Improving the running time

– We are interested only in matrix L(n-1).

– Recall that in the absence of negative-weight cycles, implies L(m) = L(n-1) for all integers m ≥ n - 1.

– Therefore, we can compute L(n-1) with only ⌈lg(n - 1)⌉ matrix products by computing the sequence

22


2

5 4

1 3

6

7 1

8

-5

43

2

-4

0 6 1 5 8

2 0 5 1 2

3 5 0 4 7

1- 1 4 0 3

4- 2 3- 1 0

0 6 1 5 8

2 0 5 1 2

11 5 0 4 7

1- 1 4 0 3

4- 2 8 3 0

0 6 1 8

2 0 5 1 2

11 5 0 4

7 1 4 0 3

4- 2 8 3 0

0 6

0 5 2

0 4

7 1 0

4- 8 3 0

)4()3(

)2()1(

LL

LL

23


1)1lg(21)1lg(2)1lg(2)1lg(2(

448)8(

224)4(

2)2(

)1(

nnnnWWWL

WWWL

WWWL

WWWL

WL

24


1 2 3

4 5 6

-4

1 2

-810

5

7

-12

3

25


• Costs Θ(n3) time.

FASTER-ALL-PAIRS-SHORTEST-PATHS(W)

1 n ← rows[W]

2 L(1) ← W

3 m ← 1

4 while m < n - 1

5 do L(2m) ← EXTEND-SHORTEST-PATHS(L(m), L(m))

6 m ← 2m

7 return L(m)

26

The Floyd-Warshall algorithm

– Intermediate Vertex• An intermediate vertex of a simple path p = <v1, v2, · · · , vl> is

any vertex of p other than v1 and vl.

27


• The structure of a shortest path

– Floyd-Warshall algorithm is based on the observation of the intermediate vertices, which costs Θ(|V|3) time.

– Let V = {1, 2, · · · , n}.

– For any pair of vertices i, j ∈ V , consider all paths from ito j whose intermediate vertices are all drawn from {1, 2, · ·

· , k}, and let p be a minimum weight path from among them.

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– If k is not an intermediate vertex of path p, then all intermediate vertices of p are in {1, 2, · · · , k − 1}.

– If k is an intermediate vertex of path p, then we break pdown into

i

k

j

P1P2

P: all intermediate vertices in {1,2,…,k-1}

all intermediate vertices in {1,2..,k-1} all intermediate vertices in {1,2..,k-1}

29


• A recursive solution to the all-pairs shortest-paths problem

– Let be the weight of a shortest path from vertex i to vertex j for which all intermediate vertices are in the set {1, 2, · · · , k}.

– We have the following recurrence:

– Because for any path, all intermediate vertices are in the set {1, 2, · · · , n}, the matrix gives the final answer:

for all i, j ∈ V .

)(k

ijd

(25.5) 1.k if )dd,min(d

0, k if

1)-(k

kj

1)-(k

ik

1)-(k

ij

)(

ijk

ij

wd

d D (n)

ij

(n)

),()( jid n

ij

30


• costs Θ(n3) time.

FLOYD-WARSHALL(W)

1 n ← rows[W]

2 D(0) ←W

3 for k ← 1 to n

4 do for i ← 1 to n

5 do for j ← 1 to n

6 do

7 return D(n)

)1()1()1()( ,min k

kj

k

ik

k

ij

k

ij dddd

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32


33


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• Constructing A Shortest Path

– Let be the predecessor of vertex j on a shortest path from vertex i with all intermediate vertices in {1, 2, · · · ,

k}.

k

ij

. wand j i if i

, or w j i if NIL

ij

ij)0(

ij

. if

, if

)1()1()1()1(

)1()1()1()1(

)(

k

kj

k

ik

k

ij

k

ik

k

kj

k

ik

k

ij

k

ijk

ijddd

ddd

35


• Transitive Closure of Graph

– Given a directed graph G = (V, E) with vertex set V = {1, 2, · · · , n}.

– The transitive closure of G is defined as the graph G∗ = (V,

E∗), where E∗ = {(i, j) : there is a path from vertex i to vertex j in G}.

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