Iterative Deepening A Star in Javascript

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``````/**
* Performs 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.
*/
const iterativeDeepeningAStar = function (tree, heuristic, start, goal) {
var threshold = heuristic[start][goal];
while (true) {
console.log("Iteration with threshold: " + threshold);
var distance = iterativeDeepeningAStarRec(tree, heuristic, start, goal, 0, threshold);
if (distance === Number.MAX_VALUE) {
return -1;
} else if (distance < 0) {
// if we found the node, the function returns the negative distance
console.log("Found the node we're looking for!");
return -distance;
} else {
// if it hasn't found the node, it returns the (positive) next-bigger threshold
threshold = distance;
}
}
};

/**
* 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.
*/
const iterativeDeepeningAStarRec = function (tree, heuristic, node, goal, distance, threshold) {
console.log("Visiting Node " + node);

if (node === goal) {
// We have found the goal node we we're searching for
return -distance;
}

var estimate = distance + heuristic[node][goal];
if (estimate > threshold) {
console.log("Breached threshold with heuristic: " + estimate);
return estimate;
}

//...then, for all neighboring nodes....
var min = Number.MAX_VALUE;
for (var i = 0; i < tree[node].length; i++) {
if (tree[node][i] !== 0) {
var t = iterativeDeepeningAStarRec(tree, heuristic, i, goal, distance + tree[node][i], threshold);
if (t < 0) {
// Node found
return t;
} else if (t < min) {
min = t;
}
}
}
return min;
};

module.exports = {iterativeDeepeningAStar};``````

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:

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.

Iterative Deepening A Star Algorithm in other languages:

JavaScript

JavaScript is an interpreted scripting language previously primarily used in web pages (executed in browsers) that has since found popularity for back-end and other tasks as well through node.js

While it borrows a lot of its syntax from Java, it is a very different language and should not be confused.

Getting to “Hello World” in JavaScript

The most important things first - here’s how you can run your first line of code in JavaScript. If you want to use JavaScript for backend, follow the chapter on how to print Hello World using Node.js. If you want to use JavaScript in the frontend (i.e. in web pages), follow the chapter on how print Hello World in the browser.

Getting to “Hello World” in JavaScript using the browser

1. Create a file named hello_world.html
2. Open it using a text editor (e.g. Sublime Text, or just the default Windows editor)
3. Paste the following code snippet:

``````<html>
<script type="application/javascript">
// This prints to the browsers console
console.log("Hello World")
// This opens a popup
</script>
<body>
(Website content)
</body>
</html>``````
4. Open this file using your browser (by typing the location into the address bar)
5. You should see a pop-up saying “Hello World”
6. If you use your browsers console (e.g. in Chrome: right-click -> inspect), you will see it printed there as well.

The reason we’re wrapping the script in HTML is that the browser will otherwise not execute the JavaScript, but just show it’s contents.

Getting to “Hello World” in JavaScript using Node.js

2. Open a terminal, make sure the `node` command is working. 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 each platform:

3. As soon as that’s working, copy the following snippet into a file named hello_world.js:

``console.log("Hello World");``
4. Change directory by typing `cd path/to/hello_world`, then run `node hello_world.js`. This should print “Hello World” to your Terminal.

That’s it! Notice that the entry barrier is similarly low as it is for Python and many other scripting languages.

Fundamentals in JavaScript

To understand algorithms and technologies implemented in JavaScript, one first needs to understand what basic programming concepts look like in this particular language. Each of the following snippets can be executed using Node.js on its own, as no boilerplate is required. In the browser, the code needs to be surrounded by HTML just like the Hello World example for the browser shown above.

Variables and Arithmetic

Variables in JavaScript are dynamically typed, meaning the content of a variable is determined at runtime and does not need to be specified when writing the code.

``````var number = 5;
var decimalNumber = 3.25;
var result = number * decimalNumber;
var callout = "The number is ";
// In this instance, the values are concatenated rather than added because one of them is a String.
console.log(callout + result);``````

This will print ‘The number is 16.25’.

Arrays

Arrays in JavaScript are implemented as objects, with the index just being the name of the property. This makes them dynamically sized. The whole concepts of objects and arrays are merged in JavaScript, as demonstrated by the following snippet.

``````var integers = {}; // initialized as object
integers[3] = 42; // assigned using array index
console.log(integers["3"]); // Accessed using property name, prints "42"

var strings = ["Hello"]; // strings[0] is now Hi
strings[2] = "World"; // index 1 skipped
strings.beautiful = "Beautiful" // Assigned using property name

console.log(strings[0] + " " + strings["beautiful"] + " " + strings[2]); // Prints "Hello World"``````

Conditions

Just like most programming languages, JavaScript can do `if-else` statements. Additionally, JavaScript can also do `switch-case` statements.

``````var value = 5;
if(value === 5){
console.log("Value is 5");
} else if(value < 5){
console.log("Value is less than 5");
} else {
console.log("Value is something else");
}

switch (value){
case 1:
console.log("Value is 1");
break; // Don't go further down the cases
case 2:
console.log("Value is 2");
break; // Don't go further down the cases
case 3:
console.log("Value is 3");
break; // Don't go further down the cases
case 4:
console.log("Value is 4");
break; // Don't go further down the cases
case 5:
console.log("Value is 5");
break; // Don't go further down the cases
default:
console.log("Value is something else");
}``````

The above JavaScript code will print “Value is 5” twice.

Loops

JavaScript supports `for`, `while` as well as `do while` loops. `break` and `continue` statements are also supported. The below example illustrates the differences:

``````var value = 2;
for (var i = 0; i < value; i++) {
console.log(i);
}
while (value > 0) {
console.log(value);
value--;
}
do {
console.log(value);
value--;
} while (value > 0);``````

This will print the following to the terminal:

``````0
1
2
1
0``````

Note the last `0`: it is printed because in the `do-while`-loop, compared to the `while`-loop. the code block is executed at least once before the condition is checked.

Functions

Functions in JavaScript can be declared using many different syntaxes, for example as object properties, as variables, or, in more recent JavaScript versions, as part of a class.

Here’s an example of a JavaScript function as a variable:

``````var my_function = function(){
console.log("Hello World")
}

my_function()``````

Here’s an example of a JavaScript function as an object property:

``````var function_object = {}
function_object.my_function = function(){
console.log("Hello World")
}

function_object.my_function()``````

And here’s an example of calling a function of an object of a class:

``````class FunctionClass {
my_function(){
console.log("Hello World")
}
}

new FunctionClass().my_function();``````

(all of these examples print “Hello World”.)

Syntax

As previously mentioned, JavaScript shares much of its Syntax with Java. JavaScript requires the use of curly brackets (`{}`) to surround code blocks in conditions, loops, functions etc.; It doesn’t always require semicolons at the end of statements, but their use is encouraged, as their usage means the use of whitespace for preferred formatting (e.g. indentation of code pieces) does not affect the code.