Naive Implementations of Priority Queues

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From the course by University of California San Diego

Data Structures

1610 ratings

University of California San Diego

Data Structures

1610 ratings

Course 2 of 6 in the Specialization Data Structures and Algorithms

A good algorithm usually comes together with a set of good data structures that allow the algorithm to manipulate the data efficiently. In this course, we consider the common data structures that are used in various computational problems. You will learn how these data structures are implemented in different programming languages and will practice implementing them in our programming assignments. This will help you to understand what is going on inside a particular built-in implementation of a data structure and what to expect from it. You will also learn typical use cases for these data structures. A few examples of questions that we are going to cover in this class are the following: 1. What is a good strategy of resizing a dynamic array? 2. How priority queues are implemented in C++, Java, and Python? 3. How to implement a hash table so that the amortized running time of all operations is O(1) on average? 4. What are good strategies to keep a binary tree balanced? You will also learn how services like Dropbox manage to upload some large files instantly and to save a lot of storage space!

From the lesson

Priority Queues and Disjoint Sets

We start this module by considering priority queues which are used to efficiently schedule jobs, either in the context of a computer operating system or in real life, to sort huge files, which is the most important building block for any Big Data processing algorithm, and to efficiently compute shortest paths in graphs, which is a topic we will cover in our next course. For this reason, priority queues have built-in implementations in many programming languages, including C++, Java, and Python. We will see that these implementations are based on a beautiful idea of storing a complete binary tree in an array that allows to implement all priority queue methods in just few lines of code. We will then switch to disjoint sets data structure that is used, for example, in dynamic graph connectivity and image processing. We will see again how simple and natural ideas lead to an implementation that is both easy to code and very efficient. By completing this module, you will be able to implement both these data structures efficiently from scratch.

Introduction6:07

Naive Implementations of Priority Queues5:57

Meet the Instructors

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering

As usual before going into the details of efficient implementation

let's check what is wrong with naive implementations?

For example, what if we store the contents of a priority

queue just in an unsorted array or in an unsorted list?

In this example on the slide, we use a doubly linked list.

Well, in this case inserting a new element is very easy.

We just append the new element to the end of our array or list.

For example, as follows,

if our new element is seven we can just put it to the next available cell in our

array where we can just append it to the end of the list.

So we put 7 to the end.

We say that the previous element of 7 is 2 and that there is no next element, right?

So it is easy and it takes constant time, okay?

Now, what about extracting the maximum element in this case?

Well, unfortunately we need to scan the whole array to find the maximum element.

And we need to scan the whole list to find the maximum element

which gives us a linear running time.

That is we O(n), right?

In our previous naive implementation using an unsorted array or

list, the running time of the extract max operation is linear.

Well, a reasonable approach to try to improve this,

is to keep the contents of our array, for example array, sorted.

Well, what are the advantages of this approach?

Well, of course, in this case, extract max is very easy.

So, the maximum element, is just the last element of our array.

Right?

Which means that the running time of ExtractMax in this case is just constant.

However, the disadvantage is that now the insertion operation takes linear time,

and this is why.

Well, to find the right position for the new element we can use the binary search.

This is actually good, well it can be done in logarithmic time.

For example, if we need to insert 7 in our priority queue,

then in logarithmic time we will find out that it should be

inserted between 3 and 9 in this for example.

However unfortunately after finding this right position,

we need to shift everything to the right of this position by one.

Right just to create a vacant position for 7.

For this we need to first shift 16 to this cell.

Then we move 10 then to this cell, then we move 9 to this cell, and

finally we put 7 in to this cell, and we get it sorted already.

So in the worst case, we need to shift a linear number of cells, a linear number of

elements, which gives us a linear running time for the insertion operation.

As we've just seen, inserting an element into a sorted array is expensive

because to insert an element into the middle we need to shift

all elements to the right of this position by one.

So, this makes the running time of the insertion procedure linear.

However if we use a doubly linked list,

then inserting into the middle of this list is actually constant time operation.

So let's try to use a sorted list. Well, the first advantage is

that the extract max operation still takes constant time.

Well this is just because, well,

the maximum element in our list is just the last element, right?

So for this reason, we have a constant time for extract max.

Also, another advantage is that inserting in the middle of this list

actually takes a constant amount of work, not linear, and this is why.

Again let's try to insert 7 into our list.

Well, this can be done as follows.

We know that inserting 7 should be done between 3 and 9.

So we do just the following.

We will remove this.

We remove this pointer and this pointer.

We'll say that now the next element after 3 is 7 and

the previous element before 7 is 3.

And also the next element after 3, after 7 is 9, and previous element before 9 is 7.

So inserting an element just involves changing four pointers.

Right? So it is a constant time operation.

However everything is not so easy, unfortunately.

And this is because just finding the right position for

inserting this new element takes a linear amount of work.

And this in particular because we cannot use binary search for lists.

Given the first element of this list and the last element of this list, we cannot

find the position of the middle element of this list because this is not an arry.

We cannot just compute the middle index in this array.

So for this reason, just finding the right position for the new element

I mean to keep the list sorted takes already a linear amount of work.

And for this reason,

inserting into a sorted list still takes a linear amount of work.

Well to conclude if you implement a priority queue using a list or

an array sorted or not then one of the operations insert and

extract max takes a linear amount of work.

In the next video we will show a data structure called binary heap

which allows to implement a priority queue so

that both of these operations can be performed in logarithmic amount of work.

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