# Shortest Paths¶

## Overview¶

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.

## Outline of the Problem¶

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

## Finding Least-Cost Paths¶

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

$$ \min_{w \in F_v} \{ c(v, w) + J(w) \} \tag{1} $$

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

$$ J(v) = \min_{w \in F_v} \{ c(v, w) + J(w) \} \tag{2} $$

This is known as the *Bellman equation*, after the mathematician Richard Bellman.

## Solving for $ J $¶

The standard algorithm for finding $ J $ is to start with

$$ J_0(v) = M \text{ if } v \not= \text{ destination, else } J_0(v) = 0 \tag{3} $$

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.

## Exercises¶

### Exercise 1¶

Use the algorithm given above to find the optimal path (and its cost) for the following graph.

### Setup¶

```
using InstantiateFromURL
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.5.0")
# github_project("QuantEcon/quantecon-notebooks-julia", version = "0.5.0", instantiate = true) # uncomment to force package installation
```

```
using LinearAlgebra, Statistics
```

```
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.

## Solutions¶

### Exercise 1¶

```
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)
```