Iterative Deepening A Star in Python

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def iterative_deepening_a_star(tree, heuristic, start, goal):
    Performs the iterative deepening A Star (A*) algorithm to find the shortest path from a start to a target node.
    Can be modified to handle graphs by keeping track of already visited nodes.
    :param tree:      An adjacency-matrix-representation of the tree where (x,y) is the weight of the edge or 0 if there is no edge.
    :param heuristic: An estimation of distance from node x to y that is guaranteed to be lower than the actual distance. E.g. straight-line distance.
    :param start:      The node to start from.
    :param goal:      The node we're searching for.
    :return: number shortest distance to the goal node. Can be easily modified to return the path.
    threshold = heuristic[start][goal]
    while True:
        print("Iteration with threshold: " + str(threshold))
        distance = iterative_deepening_a_star_rec(tree, heuristic, start, goal, 0, threshold)
        if distance == float("inf"):
            # Node not found and no more nodes to visit
            return -1
        elif distance < 0:
            # if we found the node, the function returns the negative distance
            print("Found the node we're looking for!")
            return -distance
            # if it hasn't found the node, it returns the (positive) next-bigger threshold
            threshold = distance

def iterative_deepening_a_star_rec(tree, heuristic, node, goal, distance, threshold):
    Performs DFS up to a depth where a threshold is reached (as opposed to interative-deepening DFS which stops at a fixed depth).
    Can be modified to handle graphs by keeping track of already visited nodes.
    :param tree:      An adjacency-matrix-representation of the tree where (x,y) is the weight of the edge or 0 if there is no edge.
    :param heuristic: An estimation of distance from node x to y that is guaranteed to be lower than the actual distance. E.g. straight-line distance.
    :param node:      The node to continue from.
    :param goal:      The node we're searching for.
    :param distance:  Distance from start node to current node.
    :param threshold: Until which distance to search in this iteration.
    :return: number shortest distance to the goal node. Can be easily modified to return the path.
    print("Visiting Node " + str(node))

    if node == goal:
        # We have found the goal node we we're searching for
        return -distance

    estimate = distance + heuristic[node][goal]
    if estimate > threshold:
        print("Breached threshold with heuristic: " + str(estimate))
        return estimate

    # ...then, for all neighboring nodes....
    min = float("inf")
    for i in range(len(tree[node])):
        if tree[node][i] != 0:
            t = iterative_deepening_a_star_rec(tree, heuristic, i, goal, distance + tree[node][i], threshold)
            if t < 0:
                # Node found
                return t
            elif t < min:
                min = t

    return min

About the algorithm and language used in this code snippet:

Iterative Deepening A Star Algorithm

The Iterative Deepening A Star (IDA*) algorithm is an algorithm used to solve the shortest path problem in a tree, but can be modified to handle graphs (i.e. cycles). It builds on Iterative Deepening Depth-First Search (ID-DFS) by adding an heuristic to explore only relevant nodes.

Description of the Algorithm

Whereas Iterative Deepening DFS uses simple depth to decide when to abort the current iteration and continue with a higher depth, Iterative Deepening A Star uses a heuristic to determine which nodes to explore and at which depth to stop. This is similar to how Dijkstra always explores the node with the currently shortest difference and A Star adds an heuristic to this to only explore nodes that are actually closer to the goal.

In more detail, this leads to the following Steps:

  1. For each child of the current node
  2. If it is the target node, return
  3. If the distance plus the heuristic exceeds the current threshold, return this exceeding threshold
  4. Set the current node to this node and go back to 1.
  5. After having gone through all children, go to the next child of the parent (the next sibling)
  6. After having gone through all children of the start node, increase the threshold to the smallest of the exceeding thresholds.
  7. If we have reached all leaf (bottom) nodes, the goal node doesn’t exist.

Example of the Algorithm

Consider the following graph: Graph for the Iterative Deepening A Star (IDA*) shortest path algorithm

The steps the algorithm performs on this graph if given node 0 as a starting point and node 6 as the goal, in order, are:

  1. Iteration with threshold: 6.32
  2. Visiting Node 0
  3. Visiting Node 1
  4. Breached threshold with heuristic: 8.66
  5. Visiting Node 2
  6. Breached threshold with heuristic: 7.00
  7. Iteration with threshold: 7.00
  8. Visiting Node 0
  9. Visiting Node 1
  10. Breached threshold with heuristic: 8.66
  11. Visiting Node 2
  12. Visiting Node 5
  13. Breached threshold with heuristic: 8.83
  14. Iteration with threshold: 8.66
  15. Visiting Node 0
  16. Visiting Node 1
  17. Visiting Node 3
  18. Breached threshold with heuristic: 12.32
  19. Visiting Node 4
  20. Breached threshold with heuristic: 8.83
  21. Visiting Node 2
  22. Visiting Node 5
  23. Breached threshold with heuristic: 8.83
  24. Iteration with threshold: 8.83
  25. Visiting Node 0
  26. Visiting Node 1
  27. Visiting Node 3
  28. Breached threshold with heuristic: 12.32
  29. Visiting Node 4
  30. Visiting Node 2
  31. Visiting Node 5
  32. Visiting Node 6
  33. Found the node we’re looking for!

Final lowest distance from node 0 to node 6: 9

Notice how the algorithm did not continue to explore down from node 3 in the iteration it found the goal node in. If node 3 would’ve had children, whereas Iterative Deepening DFS would’ve potentially (and needlessly!) explored them, Iterative Deepening A Star did not.

Runtime of the Algorithm

The runtime complexity of Iterative Deepening A Star is in principle the same as Iterative Deepening DFS. In practice, though, if we choose a good heuristic, many of the paths can be eliminated before they are explored making for a significant time improvement. More information on how the heuristic influences the complexity can be found on the Wikipedia Article.

Space of the Algorithm

The space complexity of Iterative Deepening A Star is the amount of storage needed for the tree or graph. O(|N|), |N| = number of Nodes in the tree or graph, which can be replaced with b^d for trees, where b is the branching factor and d is the depth. Additionally, whatever space the heuristic requires.


The Python Logo

Python™ is an interpreted language used for many purposes ranging from embedded programming to web development, with one of the largest use cases being data science.

Getting to “Hello World” in Python

The most important things first - here’s how you can run your first line of code in Python.

  1. Download and install the latest version of Python from You can also download an earlier version if your use case requires it - many technologies still require it due to the breaking changes introduced with Python 3.
  2. Open a terminal, make sure the python or python3 command is working, and that the command your’re going to be using is referring to the version you just installed by running python3 --version or python --version. If you’re getting a “command not found” error (or similar), try restarting your command line, and, if that doesn’t help, your computer. If the issue persists, here are some helpful StackOverflow questions for Windows, Mac and Linux.
  3. As soon as that’s working, you can run the following snippet: print("Hello World"). You have two options to run this: 3.1 Run python in the command line, just paste the code snippet and press enter (Press CTRL + D or write exit() and press enter to exit). 3.2 Save the snippet to a file, name it something ending with .py, e.g., and run python path/to/ Tip: use the ls command (dir in Windows) to figure out which files are in the folder your command line is currently in.

That’s it! Notice how printing something to the console is just a single line in Python - this low entry barrier and lack of required boilerplate code is a big part of the appeal of Python.

Fundamentals in Python

To understand algorithms and technologies implemented in Python, one first needs to understand what basic programming concepts look like in this particular language.

Variables and Arithmetic

Variables in Python are really simple, no need to declare a datatype or even declare that you’re defining a variable; Python knows this implicitly.

a = 1
b = {'c':2}

print(a + b['c']) # prints 3


Working with arrays is similarly simple in Python:

arr = ["Hello", "World"]

print(arr[0]) # Hello
print(arr[1]) # World
# print(arr[2]) # IndexError


print(arr[2]) # !

As those of you familiar with other programming language like Java might have already noticed, those are not native arrays, but rather lists dressed like arrays. This is evident by the fact that no size needs to be specified, and elements can be appended at will. In fact, print(type(arr)) prints <class 'list'>. This means that arrays in Python are considerably slower than in lower level programming languages. There are, however, packages like numpy which implement real arrays that are considerably faster.


Just like most programming languages, Python can do if-else statements:

value = 1
if value==1:
    print("Value is 1")
elif value==2:
    print("Value is 2")
    print("Value is something else")

Python does however not have case-statements that other languages like Java have. In my opinion, this can be excused by the simplicity of the if-statements which make the “syntactic sugar” of case-statements obsolete.


Python supports both for and while loops as well as break and continue statements. While it does not have do-while loops, it does have a number of built-in functions that make make looping very convenient, like ‘enumerate’ or range. Here are some examples:

value = 10
while value > 0:
    value -= 1

for index, character in enumerate("banana"):
    print("The %d-th letter is a %s" % (index + 1, character))

Note that Python does not share the common iterator-variable syntax of other languages (e.g. for(int i = 0; i < arr.length; i++) in Java) - for this, the enumerate function can be used.


Functions in Python are easily defined and, for better or worse, do not require specifying return or arguments types. Optionally, a default for arguments can be specified:

def print_something(something="Hello World"):
    return "Success"


(This will print “Hello World”, “Banana”, and then “Success”)


As you might have noticed, Python does not use curly brackets ({}) to surround code blocks in conditions, loops, functions etc.; This is because Python depends on indentation (whitespace) as part of its syntax. Whereas you can add and delete any amount of whitespace (spaces, tabs, newlines) in Java without changing the program, this will break the Syntax in Python. This also means that semicolons are not required, which is a common syntax error in other languages.

Advanced Knowledge of Python

Python was first released in 1990 and is multi-paradigm, meaning while it is primarily imperative and functional, it also has object-oriented and reflective elements. It’s dynamically typed, but has started offering syntax for gradual typing since version 3.5. For more information, Python has a great Wikipedia article.