Skip to main content

Dijkstra's algorithm - Part 2 - Tutorial


For Part 2 of the series, we'll be developing a Priority Queue implemented using a Heap . Continuing from Part 1, we'll be using Python for the implementation.

Fortunately, Python has a PriorityQueue class in the `queue` module. It's more than enough for our purposes, but the Queue module was designed thread safe, so there is a small overhead that comes with it. The Priority Queue is a Minimum Priority Queue by default, it takes in tuples which are compared together to determine their position in the queue. For those unfamiliar with tuple comparison in Python, it works something like this.

>>> (1,2) < (3,4)
True
>>> (1,2) < (1,3)
True
>>> (1,2) < (1,1)
False
>>> (1,2) < (1,2,3)
True
>>> (1,2,3,4) < (1,2,3)
False

The first non equal element  of the both tuples determine the output of the comparison operation. So say you have a bunch of Tasks Ti with Priorities Pi, then we push (Pi, Ti) into the queue, the Queue will automatically order it such that the first element popped will be the one with the lowest priority value.

Here's an example showing this in Action.
Now, what if we wanted the priority queue to work in the reverse order (ie, Largest Value has the higher Priority)? Well, there isn't any feature provided for this by the module, but we can do a clever little hack, observe when we negate the priority the output of the comparison operation flips. Therefore, For a Maximum Priority Queue, simply insert (- Pi, Ti)....and presto, now the elements are ordered by decreasing order of priority.

For the sake of learning, we'll implement our own Priority Queue. Using a heap for a backend implementation. Python comes with the heapq module. The heapq module lets us use a list as a heap, it has the heappush and heappop functions push an element onto a heap. The comparison of the elements work as  before with the Priority Queue.

Here's the class I wrote, it has a priority-tie-breaking mechanism. I simply add a counter as the second element to the tuple, this is essential for two things. One, when two elements have the same priority, the one inserted first will be popped first. Second, if we don't add this, then we'll have to make the data-elements comparable otherwise python will attempt to compare 2 elements (possibly non-comparable) with 2 equal priorities. In order to avoid this, we introduce the counter. So the Tuple Comparison will be resolved at the counter itself. Also, I've abstracted the min/max priority by having the pqueue class take care of negating the priorities. Far more convenient in my opinion.

Here's the complete source code.

We now have the essential tools for implementing our Dijkstra's Shortest Path algorithm. Until next time...
Labels:

Comments

Post a Comment

Popular posts from this blog

Find Increasing Triplet Subsequence - Medium

Problem - Given an integer array A[1..n], find an instance of i,j,k where 0 < i < j < k <= n and A[i] < A[j] < A[k]. Let's start with the obvious solution, bruteforce every three element combination until we find a solution. Let's try to reduce this by searching left and right for each element, we search left for an element smaller than the current one, towards the right an element greater than the current one. When we find an element that has both such elements, we found the solution. The complexity of this solution is O(n^2). To reduce this even further, One can simply apply the longest increasing subsequence algorithm and break when we get an LIS of length 3. But the best algorithm that can find an LIS is O(nlogn) with O( n ) space . An O(nlogn) algorithm seems like an overkill! Can this be done in linear time? The Algorithm: We iterate over the array and keep track of two things. The minimum value iterated over (min) The minimum increa...
2 comments
Read more

Merge k-sorted lists - Medium

Problem - Given k-sorted lists, merge them into a single sorted list. A daft way of doing this would be to copy all the list into a new array and sorting the new array. ie O(n log(n)) The naive method would be to simply perform k-way merge similar to the auxiliary method in Merge Sort. But that is reduces the problem to a minimum selection from a list of k-elements. The Complexity of this algorithm is an abysmal O(nk). Here's how it looks in Python. We maintain an additional array called Index[1..k] to maintain the head of each list. We improve upon this by optimizing the minimum selection process by using a familiar data structure, the Heap! Using a MinHeap, we extract the minimum element from a list and then push the next element from the same list into the heap, until all the list get exhausted. This reduces the Time complexity to O(nlogk) since for each element we perform O(logk) operations on the heap. An important implementation detail is we need to keep track ...
Post a Comment
Read more

3SUM - Hard

Problem - Given an Array of integers, A. Find out if there exists a triple (i,j,k) such that A[i] + A[j] + A[k] == 0. The 3SUM  problem is very similar to the 2SUM  problem in many aspects. The solutions I'll be discussing are also very similar. I highly recommend you read the previous post first, since I'll explain only the differences in the algorithm from the previous post. Let's begin, We start with the naive algorithm. An O(n^3) solution with 3 nested loops each checking if the sum of the triple is 0. Since O(n^3) is the higher order term, we can sort the array in O(nlogn) and add a guard at the nested loops to prune of parts of the arrays. But the complexity still remains O(n^3). The code is pretty simple and similar to the naive algorithm of 2SUM. Moving on, we'll do the same thing we did in 2SUM, replace the inner-most linear search with a binary search. The Complexity now drops to O(n^2 logn) Now, the hash table method, this is strictly not ...
Post a Comment
Read more