Now, look at an interesting application of priority queues that is actually

representative of whole family of a critically important applications in

applications of computing. It's called event driven simulation. And the idea is

we want to study some property of the natural world by simulating it. And that's

something that's very, very common in, in scientific inquiry nowadays. And this is a

very natural idea. And actually, the idea goes back to Einstein. So, we want to

simulate the motion of N moving particles that might collide with the priority.

This, this kind of stimulation is enabled by priority queues. And without something

like priority queues, you couldn't do this for a large number of particles because it

would require quadratic time and simply can't be afforded for a huge number of

particles. So, and let's take a look at how we can possibly make this happen. So

we use a simple scientific model called the hard disc model. And then, this is

just for simplicity to get this done and just part of a lecture. Clearly, these

things can be extended in many ways. So, we're going to have moving particles that

either collide with each other and with the walls. And each particle is a disc

that's got known position, velocity, mass, and radius. And there's no other forces

involved. It gets more complicated if there's more forces, like gravity

involved. And this point by itself is very significant. As I mentioned, it goes back

to the study of physics with [cough] the trying to understand the pressure and

temperature in Einstein's famous experiment on a pollen grain showing that

their motion was brownian and random. So whether it's individual atoms and

molecules or some bigger kinds of particles. It's a complex dynamic

situation that is better understood through computer simulation. And nowadays

that means priority queues. [cough] So, as a wa rm-up, here's code to implement

bouncing balls without the collisions. And this is an elementary programing exercise

that is the, the code at the left has the effects shown at the right. So, we have a

data type called ball that represents just one of the particles and has instance

variables that has the position and the velocity. So, that's why we make a bunch

of them and then we have a, a while loop which is just every 50 milliseconds clear

the, the whole drawing and then move the balls a little bit and then draw them in

their current position. And then the only to move [cough] operation does is to

update the position of the ball by the velocity, which is just another number and

then it does the bouncing off the walls. If it happens to hit the left of the wall

then you reflect the x-coordinate in the right wall, you reflect the x-coordinate

bottom to top, you do the same for the y-coordinate. So, this the is an easy

programming exercise given the right display primitives. And it's a good

exercise in object-oriented programming showing how just one implementation then

we can use that same implementation to simulate a number of instances. So, that's

our starting point in terms of the code. So this is the implementation of the ball

class. So, it's got position and velocity as I mentioned, and every ball has a, a

radius. And then there is a constructor and maybe we have a constructor that takes

arguments that would initialize the position and the velocity or maybe

initialize them to a random position if there's no arguments. And then, here's the

move method. And the move method again, most of the times, just takes the x and y

coordinates and adds the current velocity times the speed constant. The dt speed,

speed variable that's given as argument dt. And then these tests are for whether

it hits the walls in which case, you have to flip the x or y velocity. And then

draw, it's just using standard draw. Just draw the ball. So, that's all the code for

doing the bouncing ball simulation. Now, what's missing in this is what happens

when the balls collide with each other. And to cope with that we need to do both.

A little bit of high school Physics and a little bit of basic Computer Science. The

Physics problem is exactly what happens when two balls hit and they bounce off

each other according to some well-understood physical process, and

that's the high school Physics. And the CS problem is how and when to we exactly do

these computations for each of the balls. And how can we do it efficiently that is

in, in log N time versus quadratic time. Because if we have a computational process

that takes quadratic time, then it's not going to scale, we're not going to be able

to do large number of particles. Simulations in the real world, usually, we

wind up doing huge amounts of data and we cannot have a quadratic algorithm. This is

just first indication of that of why if you want to do this simulation, you better

know about some data structure like priority queues. If you try to do it

without it, you're not going to be successful. Alright, so, let's take a look

at what happens. So there's a number of things that you might consider trying. So,

one idea is the so-called time driven simulation. And we just say, we're going

to update everything every dt seconds. Then we go ahead and then we could check

if there's a collision, if the two balls, pieces of the two balls are occupying the

same space. And if there is, then we could roll back time just a little bit and I'll

try to figure out exactly, the moment of which they collided and then figure out

how the position and velocity should change accordingly and then continue the

simulation. But this has a huge problem. The first one is that you have t o check

all pairs of balls for overlap so that's quadratic, so it's going to be really,

really lot of overall texture you're not going to be able to do it for a huge, huge

value of N. But the other thing is even if N is small if you do a very small dt, then

you're just doing this calculation over and over again and there's just too much

computation moving the balls little bit at a time. On the other hand, if you try to

improve things by making dt too large you might completely miss a collision as shown

in the example at right. So figuring out the value of dt that would really work is

a huge problem for the time driven simulation. Instead, what we want to do is

called an event driven simulation. And this is a very general concept that's

useful in all kinds of context. And we are going to change things when something

happens. So, since the only thing that matters is collisions, we are going to

figure the particles move in a straight line, between collisions. And what we are

going to do is focus only on the times when the collisions are going to occur.

And the way we are going to that, is to maintain a priority queue and that

priority queue is going to have all the possible collisions that could happen in

the future and they're going to be prioritized by time. And when we remove

the minimum element from the priority queue, that's the next collision that we

have to deal with. And so we have two phases, we have prediction and resolution.

So, that's sometime t, we can take two particles. We know their position and

velocities shown at the bottom here and we can predict exactly the moment, which

they'll collide assuming that something else doesn't happen to them in between and

then so they will put that predicted collision time on the priority queue and

later on, when that time comes to pass we will be right at moment when they collide

and we can figure out what to do. Now, there is a possibly that something else

happened to t hem in between and we'll talk about that change, too. So, we have

to do collision prediction, which is given position, velocity, and radius when's it

going to hit with another particle or, or the wall. And then there's resolution

which is to figure out how to change the velocities of the particles according to

physical laws. Now this part I'm not going to talk about in that much detail right

now because it's high school Physics. And so, I think most students have had high

school Physics and will be able to do, do this Math or at least be convinced that

the code that does this Math is correct. So, if you know that you have a particle

that's at a certain position or x or y and has got a certain velocity, the x in the

x-direction and y in the y-direction, then you can from the distance to the pro,

vertical wall you can figure out how many seconds this is going to take until it

hits it. It's basically that distance divided by the by the velocity. And so

that's the prediction. And then, the resolution. When it hits the wall is, is

just going to change the velocity. So that's in, you know what the position is.

So that's just an example of collision, of collision prediction, when's it going to

hit the wall and resolution what do you do when it gets to the wall. When you have

two particles there's definitely more math. And again, this is high school

Physics. And we're not going to test on it or even go through the details. But it's

just a little bit of arithmetic with the velocities and positions to deal with what

happens when, when how to predict when a given particle is going to collide with

another given particle knowing their velocity and position. So, you have to

take both velocities and divide their distance by those and, and so forth. So

there's simple formulas to tell us what to do and we can also figure out the formulas

for what we do o nce they do collide. And again nobody's claiming that this is easy

but this is the Physics part and it's worked out and it comes from Newton's

Second Law. And, and, anybody taking high school Physics will, be able to deal with

these formulas and the rest of this may have to go to a reference book to get up

to speed on them. [cough] but once it's reduced to code we can be, it might have

some trouble debugging at first but at least we can be convinced that it works.

But now, let's look at the computer science code. So, this is just extending

our ball data type that we use for the bouncing balls that didn't collide to take

in, into account these extra things. So, ours will have mass, so there will be some

big heavy ones that make things more interesting. And there's also a variable

called count, which is the number of collisions of particles have been involved

in. And that's useful for a couple of purposes. So, we're going to need a bunch

of procedures which do the prediction and the collision resolution. I want, what's

the, given a particle what's the time till we hit that particle? What's the time till

we hit vertical horizontal wall? And the same thing is if we're at the point where

we're hitting a particle, what would we do, the, the same way with the vertical

and horizontal wall. So, that's the skeleton. We need those procedures that

implement those Physics rules for every particle. And, and this is what they look

like and again this is high school Physics so we're not going to do it in detail

other than to point out it's really not a huge amount of code. Lots of the xs and

the ys and the vs but really not a huge amount of code. And the other point is

we're going to return infinity if there's no collision at all so that it's going to

keep, keep that on the priority queue, that ran on the priority queue forever.

Okay, so that's the procedures that we need and then they're similar ones for the

horizontal and vertical walls. So now, let's look at the main loop for the event

driven simulation. So, the first thing is we're going to for every particle we're

going to compute the next time that it might hit every horizontal and vertical

wall. Well, actually if it's going away from a wall, it's not going to hit it so

that would be infinity. But if it's going towards a wall, then we'll compute the

time. And then that's a time in the future and we'll put that event on the priority

queue with that time as the key. And then, we'll do the same thing for all pairs of

particles. So, we do have a quadratic initialization phase that we perform just

once to get the priority queue filled up. Now, all collisions are, might not happen

so we might have two particles that are on a collision course that and we're going to

predict that point for both of those particles, you know, even right at the

beginning. But it might be the case that there's a third particle that knocks one

of those out before that thing happens and that event would be invalidated. So, the

simulation has to be careful to take that into account. But that's not difficult to

do. So, here's what the main loop is. So, we're going to take the next event from

the priority queue. That's the next collision that's going to happen from all

our calculations. There's one collision that's going to happen next. Then, we test

whether that event has been invalidated. And we do that by using that count field

in the particle. So, then that tells us what time it's going to be next. So, then

we have to go through all the particles and change their positions on a straight

line trajectory, where would they'll be after that much time? Then we have to take

the two particles that collide and change their velocity. They bounce off one

another. Now those two particles' velocities have changed , essentially that

invalidates the future collisions involving those. And then we, what we have

to do is for those two particles is go through and predict the future collisions

with any walls and collisions with any other particles. And put all those new

events on to the priority queue. But that's it. You got two particles, change

your velocities figure out the future collision of those particles with the wall

and update the priority queue and then the main loop is take the next thing off the

priority queue and keep going. That's the code that we'll look at next. So we have

a, a, a bunch of conventions just to reduce the code. And if we this the, the

thing called event which involves it says between two particles, something is going

to happen at a certain time and we're going to adopt the conventions that, if,