k Shortest Paths

1
k Shortest Paths

• Pierre Fransson

2
Outline

• Focus on data structures
• The Problem
• Notation
• Path representation (heaps galore)
• Complexities of Eppstein

3
The problem

• Find k shortest paths S1, S2, S3, ..., Sk from
  vertex s to vertex t
• S1 is shortest path from s to t .

w0
w4
v0
v2
w1
t
4
Notation

• A path p(u, v) uniquely represented by shortest
  path from u to v and sidetracks(p).
• sidetracks(p) are all edges in G T
• If e(u, v) then
• head(e) v
• tail(e) u

5
Notation cont.

• l(u, v) cost of edge (u, v)
• d(s, t) shortest path distance from u to v
• Cost of detour
• d(u, v) l(u, v) d(v, t) - d(u, t)
• Heaps
• Hout(v), HT(v), HG(v), D(G), P(G)

6
Path Representation (Hout(v))
w0
w4
v0
v2
w1
t
v1
w2
w3
7
Calculating Hout(v)
Hout(t)
Hout(v0)
Hout(v1)
Hout(v2)
outroot(v2)
outroot(v1)
outroot(v0)
t-w1
v0-v1
v1-v2
v2-v0
outroot(t)
d5
d2
d2
d2
t-w4
v0-v2
d2
d5
t-w4
v0-w2
v0-w3
d5
d3
Hout(w1)
Hout(w0)
Hout(w3)
Hout(w4)
w0-v2
w3-w2
outroot(w0)
outroot(w1)
outroot(w3)
outroot(w4)
d2
d3
Hout(w2)
w0-w1
v3-w4
d3
d3
outroot(w2)
d3
w0-w4
8
Calculating HG(v)
HG(v1)
HG(t)
HG(v2)
HG(v0)
h(v1)
v1-v2
d2
h(v2)
v2-v0
d2
t-w1
h(t)
h(v0)
v0-v1
d2
d5
t-w4
t-w1
h(t)
v0-v2
d2
d5
t-w1
h(t)
d5
d5
t-w1
h(t)
d5
t-w4
t-w4
t-w4
v0-w2
v0-w3
d3
d5
d5
d5
t-w4
d5
t-w4
d5
t-w4
d5
t-w4
d5
9
Calculating HG(v)
HG(w0)
HG(w3)
h(v1)
v1-v2
d2
h(v0)
v0-v1
d2
w0-v2
h(w0)
d2
w3-w2
h(w3)
d3
t-w1
h(t)
v0-v2
d2
d5
t-w1
h(t)
d5
t-w4
w0-w1
d3
t-w4
v3-w4
d3
v0-w2
v0-w3
d3
d5
d5
d3
t-w4
d5
w0-w4
t-w4
d5
HG(w1)
HG(w4)
HG(w2)
h(w1)
h(w2)
h(w4)
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Calculating HG(v)
HG(w1)
HG(w4)
HG(w2)
h(w4)
h(w1)
h(w2)
t-w1
h(t)
h(v2)
v1-v2
v2-v0
d2
h(v0)
h(v1)
d2
d5
t-w1
v0-v1
d2
t-w1
h(t)
d5
h(t)
d5
t-w4
v0-v2
d2
t-w4
t-w4
d5
d5
d5
t-w4
d5
t-w4
v0-w2
v0-w3
d3
t-w4
d5
d5
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Calculating D(G)
h(w1)
h(w2)
h(w4)
h(v0)
h(v1)
h(v2)
v0-v1
d2
v1-v2
d2
v2-v0
d2
w0-v2
h(w0)
d2
v0-v2
d2
t-w1
h(t)
w3-w2
h(w3)
d3
d5
w0-w1
d3
t-w4
v0-w2
v0-w3
d3
v3-w4
d3
d5
d3
w0-w4
t-w4
d5
12
Calculating P(G)
h(w4)
2
h(w1)
h(w2)
2
5
2
5
5
r(s)
2
h(v0)
h(v2)
d2
d2
h(v1)
v0-v1
d2
v1-v2
v2-v0
0
2
2
2
0
3
3
3
1
0
w0-v2
h(w0)
d2
v0-v2
d2
0
t-w1
h(t)
w3-w2
h(w3)
2
d3
d5
1
1
1
0
0
0
w0-w1
d3
t-w4
v0-w2
v0-w3
d3
v3-w4
0
d5
d3
3
0
0
0
0
d3
w0-w4
t-w4
d5
0
0
13
Building Complexities

• Building Hout(v) O(m)
• Building HG(v) O(log n) (without modification)
• ? O(n log n)
• Total cost O(m n log n)
• Total space O(m n log n) ?

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Search Complexities

• Finding k shortest paths from t to s
• cost O(m n log n k)
• Finding all shortest paths from t to s
• cost O(m n log n kn)

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Improving Building Complexities

• Total cost O(m n k)
• Total space O(m n k) ?