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1. Working with
Dijkstra’s algorithm.
by Aditya Y. Bhargava
Save 42% off Grokking Algorithms
with code algrokking at
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2. Here’s a little info about Dijkstra’s algorithm:
You can use Dijkstra’s algorithm to find out how to go from start to
finish in the most expedient manner.
There are four steps to Dijkstra’s algorithm:
1. Find the “cheapest” node. This is the node you can get to in the least
amount of time.
2. Update the costs of the neighbors of this node. I’ll explain what I
mean by this shortly.
3. Repeat until you’ve done this for every node in the graph.
4. Calculate the final path.

3. Here’s a basic example:
Each segment has a travel time in minutes. We’ll use
Dijkstra’s algorithm to go from start to finish in the shortest
possible time.

4. Let’s go through
the 4 steps.
You’re standing at the start, wondering if you should go to
node A or node B. How long does it take to get to each node?
(6 mins to A and 2 mins to B – B is the cheapest node!)
Step 1 - find the
cheapest node:

5. Hey, wait a second… we just
found a shorter path to node
A! It used to take 6 minutes
to get to node A, but we can
get there in 5 mins by going
through B.
Step 2 – calculate how
long it takes to get to all of
node B’s neighbors by
following an edge from B:

6. Let’s recap what we just learned:
Since we found a shorter path for a neighbor of B, we’re going
to update its cost. In this case, we found:
• A shorter path to A (down from 6 minutes to 5 minutes)
• A shorter path to the finish (down from infinity to 7 minutes)
Now we go on to step 3: repeating steps 1 & 2 for all other
nodes, in this case that just leaves node A.

7. We’ve done node B already, so node A has the next smallest
time estimate.
Step 1 again - find
the cheapest
node:

8. Woo, it takes only 6 minutes to get to the finish now!
Step 2 again –
update the costs
for node A’s
neighbors:

9. We’ve run Dijkstra’s algorithm for every
node (you don’t need to run it for the
finish node). At this point, you know:
• It takes 2 minutes to get to node B.
• It takes 5 minutes to get to node A.
• It takes 6 minutes to get to the finish.

10. We just used Dijkstra’s
algorithm to calculate the
shortest path in a “weighted”
graph!
Step 4 – Calculate
the final path
(start to fin = 6):
*Hint* when you work with Dijkstra’s
algorithm, each edge in the graph has a
number associated with it. These are called
weights.

11. Let’s look at another example:
This is Rama.
Rama is trying to trade a music
book for a piano.

12. Luckily, Rama has friends who are willing to
trade some of their possessions.
“I’ll give you this poster for your book,” says Alex. “It’s a poster of
my favorite band, Destroyer. Or I’ll give you this rare LP of Rick
Astley for your book and $5 more.”
“Ooh, I’ve heard that LP has a really great song,” says Amy. “I’ll trade
you my guitar or drum set for the poster or the LP.”
“I’ve been meaning to get into guitar!” exclaims Beethoven. “Hey,
I’ll trade you my piano for either of Amy’s things.”

13. Perfect! With a little bit of
money, Rama can trade his
way from a piano book to a
real piano. Now he just needs
to figure out how to spend
the least amount of money to
make those trades. Let’s
graph out what he’s been
offered. The weights on the edges are the
amount of money he would have
to pay to make the trade.

14. Make a table of the cost for each node. The cost of a node is
how expensive it is to get to. You’ll keep updating this table as
the algorithm goes on.
Before we start, we
need this chart
(feel free to grab some
scratch paper and make
your own version).

15. To calculate the final path, you also need a parent column on
this table. We’ll see how this column works soon. For now,
let’s start the algorithm.
We also need a chart
like this one.

16. Let’s get started with
step one – finding the
cheapest node.
Trading for the poster
is the cheapest at $0.
Finding the cheapest node is
important, so make certain to take
your time on this step.

17. Next, were going to set the poster as the
new parent and figure out how long it takes
to get to its neighbors (the cost).
We now have two neighbors that we can reach from
the poster: the drums for $35; and the bass guitar
for $30.
Newly
unlocked
prices with
poster as
parent.

18. Now we have prices for the bass guitar and the drum set in the
table. Their value was set when we went through the poster, so
the poster gets set as their parent. That means, to get to the bass
guitar, we’re going to follow the edge from the poster, and the
same for the drums.

19. We’re going to repeat steps 1&2, finding
the next cheapest node and updating the
values of it’s neighbors.
Hey, now it’s cheaper to get to the drums and guitar
by following the edge from the LP. Let’s set the LP as
our new parent.

20. Repeat steps 1&2, again!
The bass guitar is our new cheapest node. And when
we update it’s neighbors, we get a price for the
piano - $40!

21. We finally have a price for the piano!
But it is the cheapest price? Let’s set the drums as
parent and see if we can’t get that piano for less.

22. Rama can get the piano more cheaply by trading
the drum set for the piano instead. So the cheapest
set of trades will cost Rama $35 total.
So far, we know that the shortest
path costs $35, but how do we
figure out the path? To start with,
look at the parent for piano.

23. The piano has drums as its parent. That means
Rama will trade the drums for the piano. So we
follow that edge back toward the book.

24. The drums have the LP as parent, so Rama
will trade the LP to get the drums.

25. Of course, this means that he’ll trade the book
for the LP. By following the parents backward,
we now have the complete path.

26. Here is the final path
Rama will need to take to
get the piano via the
shortest path (i.e. the
cheapest path). Thanks
Dijkstra!

27. Hopefully you’ve seen just how useful
Dijkstra’s algorithm can be.
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