Chapter 4
Shortest Paths: Label-Setting
Algorithms
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4.1 Introduction
 Directed network G = (N, A) with an arc length (cost) cij associated with each
arc (i, j)A. C = max{ cij: (i, j)A}
Given source node s, find shortest path from s to every other node iN\{s}.
(finding s-t shortest path gives shortest paths to every other iN\{s}as
byproducts.)
 May interpret as sending one unit of flow from s to each of the nodes iN\{s}
in an uncapacitated network.
 Minimize
subject to
(i, j)A cijxij
{j: (i, j)A} xij - {j: (j, i)A} xji = n-1,
= -1,
xij  0 for all (i, j)A
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for i = s
for all iN\{s}
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 Assumption 4.1: Arc lengths are integers
Some algorithms use this assumption in complexity analysis
 Assumption 4.2: G contains a directed path from s to every other node.
May add arc (s, i) with large cost for every node i not connected to s by a directed
path.
 Assumption 4.3: G does not contain a negative cycle. (negative arc length may
be allowed, but no negative cycle)
LP formulation unbounded if there exists a negative cycle in G.
Combinatorial algorithms try to find shortest length directed walks, hence
unbounded if there exists a negative cycle in G.
 Assumption 4.4: The network is directed.
If arc lengths nonnegative, use two arcs with opposite direction
If negative length arc exists, above transformation creates negative cycle. (need
other transformation to handle this (matching).
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 Various Types of Shortest Path Problems:
Finding shortest path from one node to all other nodes when arc lengths are
nonnegative
Finding shortest path from one node to all other nodes with arbitrary arc lengths.
Finding shortest paths from every node to every other node (all pairs shortest path
problem)
Various generalizations (e.g., k-shortest path problem)
 Analog solution of the shortest path problem.
See the 3 observations in text.
 Label-Setting and Label-Correcting Algorithms
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4.2 Applications
 Application 4.3 Knapsack problem
maximize i=1p uixi
subject to i=1p wixi  W
xi  {0, 1} for all i.
 Can be modeled as a longest path problem on an acyclic network.
 Compare the dynamic programming algorithm and the algorithm on the
network.
Ex in text)
j
1
2
3
4
uj
40
15
20
10
wj
4
2
3
1
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 Longest path problem → shortest path problem: take negative of arc lengths
If negative cycle exists, cannot be solved easily. But, the graph is acyclic.
 Recall that for dynamic programming, d(i, j) is maximum utility (longest path
length in the transformed graph) using items 1 through i and weight restriction
j.
 Exists algorithm for shortest path problem in acyclic networks: O(m)
Does it imply that knapsack problem can be solved in polynomial time? No
Transformed network has O(pW) arcs, hence running time is still O(pW),
which is pseudopolynomial time.
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 Application 4.4 Tramp Steamer Problem (minimum cost-to-time ratio cycle
problem)
A voyage of tramp steamer from port i to port j earns pij units of profit and
requires ij units of time.
Find tour W that achieves the largest possible mean daily profit
(W) = ( (i, j)W pij) / ((i, j)W ij)
 We assume ij  0 for (i, j)A, and (i, j)W ij > 0 for every directed cycle W.
 For given 0, consider if there exists W such that
( (i, j)W pij) / ((i, j)W ij) > 0 ,
→
(i, j)W (0ij - pij) < 0
Identify a negative cycle in G with arc cost (0ij - pij). Then, use binary
search.
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 Application 4.5 System of Difference Constraints:
Consider LP with constraints: having one +1 and one -1 in each constraint
m difference constraints in n variables x = {x(1), x(2), … , x(n)}:
x(jk) – x(ik)  b(k)
for each k = 1, … , m.
Want to determine whether it has a feasible solution, and if so, identify a
feasible solution.
 Use constraint graph:
Use arc (ik, jk) of length b(k) for the constraint x(jk) – x(ik)  b(k)
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x(3) – x(4)  5
x(4) – x(1)  -10
x(1) – x(3)  8
x(2) – x(1)  -11
x(3) – x(2)  2.
 Ex)
4
5
0
3
4
5
3
0
8
-10
2
s
1
0
2
-11
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0
8
-10
1
2
2
-11
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 Constraints are identical to the optimality conditions for the shortest path
problem in G (later in Chapter 5). Conditions satisfied if and only if there
does not exist a negative cycle.
 Interpret x(i) as the shortest distance from an additional node s to i. Connect
node s to node i with arc cost 0 so that the existence of a path from s to i is
guaranteed for all i.
 Note that for negative cycle 1-2-3 in G, we can add the lhs and rhs of the
corresponding constraints respectively and get 0 < -1, which is inconsistent.
 Value of a shortest path from s to i gives a feasible solution x(i) for all i. If
negative cycle exists, label correcting algorithm can detect it, and it indicates
inconsistent set of constraints.
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 Application 4.6 Telephone Operator Scheduling
Telephone company need to schedule operators around the clock.
Let b(i), i = 0, 1, 2, … , 23 denote the minimum number of operators needed
for the i-th hour of the day. (b(0): number needed for 0-1 a.m.)
Each operator works 8 consecutive hours and a shift can begin at any hour.
Want cyclic schedule that repeats daily.(using minimum number of operators )

minimize i=023 yi
subject to
yi-7 + yi-6 + … + yi  b(i)
y17+i + … + y23 + y0 + … + yi  b(i)
yi  0
for all i = 8 to 23,
for all i = 0 to 7,
for all i =0 to 23.
 Note that the rows have 1’s consecutively (in wraparound fashion)
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 Let x(i) = k=0i yk, i = 0, … , 23
and x(23) = p
yi-7 + yi-6 + … + yi  b(i)
for all i = 8 to 23,
y17+i + … + y23 + y0 + … + yi  b(i)
for all i = 0 to 7,
yi  0
for all i =0 to 23.
Rewrite the constraints as
x(i) – x(i-8)  b(i)
for all i = 8 to 23,
x(23) – x(16+i) + x(i)
= p – x(16+i) + x(i)  b(i) for all i = 0 to 7,
x(i) – x(i-1)  0
 System of difference constraints. Use binary search.
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4.3 Tree of Shortest Paths
 Shortest paths from s to all other nodes constitutes a directed out-tree (Shortest
path tree)
 Property 4.1 Let s = i1 – i2 - … - ih = k be a shortest path from s to k.
Then subpath s = i1 – i2 - … - iq is a shortest s-iq path for q = 2, 3,…, h-1
P1
s
P3
p
k
P2
 P1-P3 is shortest s-k path. If P2 is shorter s-p path, P2-P3 is s-k walk with
shorter length. Can obtain s-k path after eliminating cycles (if no negative
cycle)
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 Let d() denote the shortest path distances.
P is a shortest s-k path.  d(j) = d(i) + cij for every arc (i, j)P.
Converse?
Suppose d(j) = d(i) + cij for every arc in an s-k path P.
Let s = i1 – i2 - … - ih = k be the node sequence in P. Then
d(k) = d(ih) = (d(ih) – d(ih-1)) + (d(ih-1) – d(ih-2)) + … + (d(i2) – d(i1))
= ci(h-1),i(h) + ci(h-2),i(h-1) + … + ci2,i1 = (i, j)P cij
Hence P is a shortest s-k path.
 Property 4.2 Let the vector d represent the shortest path distances. Then
s-t path P is a shortest path.  d(j) = d(i) + cij for every arc (i, j)P.
 Use breadth-first search to identify shortest path tree satisfying d(j) = d(i) + cij.
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4.4 Shortest Path Problems in Acyclic Networks
 For acyclic network  topological ordering with i < j for (i, j)A.
Then for node k, all incoming arcs must come from nodes 1, 2, … , k-1.
By property 4.1, d(k) = minimumi<k {d(i) + cik}.
 Algorithm
Initially, d(s) = 0, d(i) = M for large M
scan nodes in topological order
for all arcs (i, j) in adjacency list A(i), if d(j) > d(i) + cij, set d(j) = d(i) + cij.
at termination, d(i)’s are shortest distances (use pointers to identify the paths)
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 Proof of correctness)
Use induction. Suppose scanned nodes 1, 2, … , k and the distance labels are
optimal. For node k+1, suppose s = i1 – i2 – ih – (k+1) is shortest path to k+1.
By property 4.1, the path i1 – i2 – ih is a shortest s - ih path.
By topological ordering, (ih, k+1)A implies ih {1, 2, … , k}.
By inductive hypothesis, the label of ih is equal to the length of i1 – i2 – ih .
Hence, when examining node ih, algorithm scans arc (ih, k+1) and set the label
of node k+1 as the length of the path i1 – i2 – ih – (k+1).
So, when the algorithm examines node k+1, its distance label is optimal.
 Thm 4.3. The reaching algorithm solves the shortest path problem on acyclic
networks in O(m) time.
 If the network contains cycles, we need more elaborate schemes.
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