.. index:: single: Dynamic Programming; Shortest Paths
Contents
The shortest path problem is a classic problem in mathematics and computer science with applications in
- Economics (sequential decision making, analysis of social networks, etc.)
- Operations research and transportation
- Robotics and artificial intelligence
- Telecommunication network design and routing
- etc., etc.
Variations of the methods we discuss in this lecture are used millions of times every day, in applications such as
- Google Maps
- routing packets on the internet
For us, the shortest path problem also provides a nice introduction to the logic of dynamic programming.
Dynamic programming is an extremely powerful optimization technique that we apply in many lectures on this site.
The shortest path problem is one of finding how to traverse a graph from one specified node to another at minimum cost.
Consider the following graph
We wish to travel from node (vertex) A to node G at minimum cost.
- Arrows (edges) indicate the movements we can take.
- Numbers on edges indicate the cost of traveling that edge.
Possible interpretations of the graph include
- Minimum cost for supplier to reach a destination.
- Routing of packets on the internet (minimize time).
- Etc., etc.
For this simple graph, a quick scan of the edges shows that the optimal paths are
- A, C, F, G at cost 8
- A, D, F, G at cost 8
For large graphs we need a systematic solution.
Let J(v) denote the minimum cost-to-go from node v, understood as the total cost from v if we take the best route.
Suppose that we know J(v) for each node v, as shown below for the graph from the preceding example
Note that J(G) = 0.
The best path can now be found as follows
- Start at A.
- From node v, move to any node that solves
\min_{w \in F_v} \{ c(v, w) + J(w) \}
where
- F_v is the set of nodes that can be reached from v in one step
- c(v, w) is the cost of traveling from v to w
Hence, if we know the function J, then finding the best path is almost trivial.
But how to find J?
Some thought will convince you that, for every node v, the function J satisfies
J(v) = \min_{w \in F_v} \{ c(v, w) + J(w) \}
This is known as the Bellman equation, after the mathematician Richard Bellman.
The standard algorithm for finding J is to start with
J_0(v) = M \text{ if } v \not= \text{ destination, else } J_0(v) = 0
where M is some large number.
Now we use the following algorithm
- Set n = 0.
- Set J_{n+1} (v) = \min_{w \in F_v} \{ c(v, w) + J_n(w) \} for all v.
- If J_{n+1} and J_n are not equal then increment n, go to 2.
In general, this sequence converges to J---the proof is omitted.
Use the algorithm given above to find the optimal path (and its cost) for the following graph.
.. literalinclude:: /_static/includes/deps_generic.jl :class: hide-output
using LinearAlgebra, Statistics
using Test
graph = Dict(zip(0:99, [[(14, 72.21), (8, 11.11), (1, 0.04)],[(13, 64.94), (6, 20.59), (46, 1247.25)],[(45, 1561.45), (31, 166.8), (66, 54.18)],[(11, 42.43), (6, 2.06), (20, 133.65)],[(7, 1.02), (5, 0.73), (75, 3706.67)],[(11, 34.54),(7, 3.33),(45, 1382.97)],[(10, 13.1), (9, 0.72), (31, 63.17)],[(10, 5.85), (9, 3.15), (50, 478.14)], [(12, 3.18), (11, 7.45), (69, 577.91)],[(20, 16.53), (13, 4.42), (70, 2454.28)],[(16, 25.16), (12, 1.87), (89, 5352.79)],[(20, 65.08), (18, 37.55), (94, 4961.32)],[(28, 170.04), (24, 34.32), (84, 3914.62)],[(40, 475.33), (38, 236.33), (60, 2135.95)],[(24, 38.65), (16, 2.7),(67, 1878.96)],[(18, 2.57),(17, 1.01),(91, 3597.11)],[(38, 278.71),(19, 3.49),(36, 392.92)],[(23, 26.45), (22, 24.78), (76, 783.29)],[(28, 55.84), (23, 16.23), (91, 3363.17)],[(28, 70.54), (20, 0.24), (26, 20.09)],[(33, 145.8), (24, 9.81),(98, 3523.33)],[(31, 27.06),(28, 36.65),(56, 626.04)], [(40, 124.22), (39, 136.32), (72, 1447.22)],[(33, 22.37), (26, 2.66), (52, 336.73)],[(28, 14.25), (26, 1.8), (66, 875.19)],[(35, 45.55),(32, 36.58),(70, 1343.63)],[(42, 122.0),(27, 0.01), (47, 135.78)],[(43, 246.24), (35, 48.1),(65, 480.55)],[(36, 15.52), (34, 21.79), (82, 2538.18)],[(33, 12.61), (32, 4.22),(64, 635.52)], [(35, 13.95), (33, 5.61), (98, 2616.03)],[(44, 125.88),(36, 20.44), (98, 3350.98)],[(35, 1.46), (34, 3.33), (97, 2613.92)], [(47, 111.54), (41, 3.23), (81, 1854.73)],[(48, 129.45), (42, 51.52), (73, 1075.38)],[(50, 78.81), (41, 2.09), (52, 17.57)], [(57, 260.46), (54, 101.08), (71, 1171.6)],[(46, 80.49),(38, 0.36), (75, 269.97)],[(42, 8.78), (40, 1.79), (93, 2767.85)],[(41, 1.34), (40, 0.95), (50, 39.88)],[(54, 53.46), (47, 28.57), (75, 548.68)], [(54, 162.24), (46, 0.28), (53, 18.23)],[(72, 437.49), (47, 10.08), (59, 141.86)],[(60, 116.23), (54, 95.06), (98, 2984.83)], [(47, 2.14), (46, 1.56), (91, 807.39)],[(49, 15.51), (47, 3.68), (58, 79.93)],[(67, 65.48), (57, 27.5), (52, 22.68)],[(61, 172.64), (56, 49.31), (50, 2.82)],[(60, 66.44), (59, 34.52), (99, 2564.12)], [(56, 10.89), (50, 0.51), (78, 53.79)],[(55, 20.1), (53, 1.38), (85, 251.76)],[(60, 73.79),(59, 23.67),(98, 2110.67)], [(66, 123.03), (64, 102.41), (94, 1471.8)],[(67, 88.35),(56, 4.33), (72, 22.85)],[(73, 238.61), (59, 24.3), (88, 967.59)],[(64, 60.8), (57, 2.13), (84, 86.09)],[(61, 11.06), (57, 0.02), (76, 197.03)], [(60, 7.01), (58, 0.46), (86, 701.09)],[(65, 34.32), (64, 29.85), (83, 556.7)],[(71, 0.67), (60, 0.72), (90, 820.66)],[(67, 1.63), (65, 4.76), (76, 48.03)],[(64, 4.88), (63, 0.95), (98, 1057.59)], [(76, 38.43), (64, 2.94), (91, 132.23)],[(75, 56.34), (72, 70.08), (66, 4.43)],[(76, 11.98), (65, 0.3), (80, 47.73)],[(73, 33.23), (66, 0.64), (94, 594.93)],[(73, 37.53), (68, 2.66), (98, 395.63)], [(70, 0.98), (68, 0.09), (82, 153.53)],[(71, 1.66), (70, 3.35), (94, 232.1)],[(73, 8.99), (70, 0.06), (99, 247.8)],[(73, 8.37), (72, 1.5), (76, 27.18)],[(91, 284.64), (74, 8.86), (89, 104.5)], [(92, 133.06), (84, 102.77), (76, 15.32)],[(90, 243.0), (76, 1.4), (83, 52.22)],[(78, 8.08), (76, 0.52), (81, 1.07)],[(77, 1.19), (76, 0.81), (92, 68.53)],[(78, 2.36), (77, 0.45), (85, 13.18)], [(86, 64.32), (78, 0.98), (80, 8.94)],[(81, 2.59), (98, 355.9)],[(91, 22.35), (85, 1.45), (81, 0.09)],[(98, 264.34), (88, 28.78), (92, 121.87)],[(92, 99.89), (89, 39.52), (94, 99.78)],[(93, 11.99), (88, 28.05), (91, 47.44)],[(88, 5.78), (86, 8.75), (94, 114.95)], [(98, 121.05), (94, 30.41), (89, 19.14)],[(89, 4.9), (87, 2.66), (97, 94.51)],[(97, 85.09)],[(92, 21.23), (91, 11.14), (88, 0.21)], [(98, 6.12), (91, 6.83), (93, 1.31)],[(99, 82.12), (97, 36.97)], [(99, 50.99), (94, 10.47), (96, 23.53)],[(97, 22.17)],[(99, 34.68), (97, 11.24), (96, 10.83)],[(99, 32.77), (97, 6.71), (94, 0.19)], [(96, 2.03), (98, 5.91)],[(99, 0.27), (98, 6.17)],[(99, 5.87), (97, 0.43), (98, 3.32)],[(98, 0.3)],[(99, 0.33)],[(99, 0.0)]]))
The cost from node 68 to node 71 is 1.66 and so on.
function update_J!(J, graph)
next_J = Dict()
for node in keys(graph)
if node == 99
next_J[node] = 0
else
next_J[node] = minimum(cost + J[dest] for (dest, cost) in graph[node])
end
end
return next_J
end
function print_best_path(J, graph)
sum_costs = 0.0
current_location, destination = extrema(keys(graph))
while current_location != destination
println("node $current_location")
running_min = 1e10
minimizer_dest = Inf
minimizer_cost = 1e10
for (dest, cost) in graph[current_location]
cost_of_path = cost + J[dest]
if cost_of_path < running_min
running_min = cost_of_path
minimizer_cost = cost
minimizer_dest = dest
end
end
current_location = minimizer_dest
sum_costs += minimizer_cost
end
sum_costs = round(sum_costs, digits = 2)
println("node $destination\nCost: $sum_costs")
end
J = Dict((node => Inf) for node in keys(graph))
while true
next_J = update_J!(J, graph)
if next_J == J
break
else
J = next_J
end
end
print_best_path(J, graph)
sum_costs = 0.0
current_location, destination = extrema(keys(graph))
while current_location != destination
println("node $current_location")
running_min = 1e10
minimizer_dest = Inf
minimizer_cost = 1e10
for (dest, cost) in graph[current_location]
cost_of_path = cost + J[dest]
if cost_of_path < running_min
running_min = cost_of_path
minimizer_cost = cost
minimizer_dest = dest
end
end
current_location = minimizer_dest
sum_costs += minimizer_cost
end
sum_costs = round(sum_costs, digits = 2)
@test sum_costs ≈ 160.55