The Tower of Epiphany | Think Like A Coder, Ep 7

Ethic and Hedge are on the ground floor
of a massive tower.

Barriers of energy separate them
from their quest’s second goal:

the Node of Creation.

To reach it, Ethic must use
three energy streams to climb the tower.

As soon as she steps forward a timer
will begin counting down from 60 seconds.

At the back of the room
there’s a basin made of invisible towers

that can hold energy between them.

After one minute, a torrent of energy
will pour down from above,

filling one unit at a time,

with a force field preventing it
from spilling out the front or back.

During the 60 calm seconds,

Ethic and Hedge must decide exactly
how many units of energy will fall.

For each of the three challenges,

they must choose the amount
that will fill the basin exactly.

If they do so, the energy will propel them
further upwards.

But if they get the amount at all wrong,
the energy lift will fail,

dropping them.

Diagrams on the walls
illustrate some examples.

This configuration
will capture exactly 2 units of energy.

This configuration will capture 4—
3 here, and 1 here.

And this one will also capture 4,

because any energy on the right
would spill out.

The energy will rain down in such a way

that it’ll only overflow
if there’s no space that could hold it.

Hedge can make one tower of blocks visible
at a time and count how tall it is,

but he can’t look at
the whole structure all at once.

How does Ethic program Hedge
to figure out

exactly how much energy
each basin can hold?

Pause now to figure it out for yourself.

Here’s one way of thinking about
what’s happening:

each unoccupied cell will hold energy

if and only if there is a wall
eventually to its left,

and a wall eventually to its right.

But it would take a long time for Hedge
to check this for each individual cell.

So what if he were to consider
a whole column of blocks at a time?

How many units of energy
can this hold, for instance?

Pause now to figure it out for yourself.

Let’s analyze the problem
by looking at our example.

There are 5 columns of blocks here.

The leftmost one can’t hold any energy,
because there’s nothing higher than it.

The 2nd stack can have 3 units above it,

as they would be trapped
between these two 4 block stacks.

We get 3 units by taking the height
where the energy would level off— 4,

and subtracting the height of the stack—
so that’s 4 minus 1.

The 3rd stack is similar— 4 to the left,
4 to the right, and it’s 3 high,

so it’ll hold 4 minus 3 equals 1 unit.

The 4th stack and 5th stacks have
nothing higher than them to the right,

so they can’t hold any energy.

We can adapt this idea into an algorithm.

Considering one column at a time
as the point of reference,

Hedge can look to the left stack by stack
to find the height of the tallest one,

look to the right to find the height
of the tallest one,

and take the smaller of the two
as the height the energy can fill up to.

If the result is higher than the column
in question,

subtract the height
of the original column,

and the result will be the number of units
that column can hold.

If it's equal to or below the level
of the column in question,

the energy would spill off.

Hedge can apply that
to an entire basin with a loop

that starts on the left-most column
and moves right, one column at a time.

For each column, he’ll run the same steps—
look all the way left for the tallest,

do the same to the right,
take the lower height of the two,

subtract the original column height,

and increase the grand total
if that number is positive.

His loop will repeat
as many times as there are columns.

That will work, but it’ll take a long time
for a large basin.

At every step Hedge repeats the action
of looking left and looking right.

If there are N stacks,
he’ll look at all N stacks N times.

Is there a faster way?

Here’s one time saver:
before doing anything else,

Hedge can start on the left,

and keep a running tally
of what the highest stack is.

Here that would be 2, 2 again,
since the first was higher,

then 4, 4, 4.

He can then find
the highest right-most stacks

by doing the same going right-to-left:
1, 3, 4, 4, 4.

In the end he’ll have a table
like this in his memory.

Now, Hedge can take one more pass
to calculate how much energy there will be

above every stack
with the same equation from before:

take the smaller of the stored left
and right values,

and subtract the height
of the current tower.

Instead of looking at N stacks N times,
he’ll look at N stacks just 3 times—

which is what’s called linear time.

There are ways to optimize
the solution even further,

but this is good enough for our heroes.

Ethic and Hedge work as one.

The first cascade is a breeze,
and they rise up the tower.

The second is a little tougher.

The third is huge,
with dozens of stacks of blocks.

The timer ticks down towards zero,
but Ethic’s program is fast.

She gets the wheel in position
just in time,

and the energy lifts them
to the Node of Creation.

Like the first, it reveals a vision:
memories of years gone by.

The world machine changed everything,

and Ethic, in her position
as chief robotics engineer,

grew troubled by what she saw.

When the Bradbarrier went up
to keep the people in,

she knew something was seriously wrong.

So she created three artifacts

with the ability to restore people’s
power, creativity, and memory,

and smuggled them to three communities.

Before she could tell people
how to use them,

the government discovered her efforts
and sent bots to arrest her

and the other programmers.

The last thing Ethic
used the world machine to create

was a robot that would protect
the ancient device

from the forces of ignorance
by enclosing it in a giant maze.

She named her creation Hedge.

Without warning, the energy lift flickers,
then fizzles out.