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A-Star Algorithm

A-star is a graph-based, path search algorithm. It is used in many fields of computer science as a search algorithm. It is often used due to its completeness, optimality, and optimal efficiency.

Salient Features of the Algorithm

  1. Resolution complete and Resolution optimal : The algorithm finds the optimal solution to the given problem at a chosen discretization, if one exits.
  2. A-Star uses a hueristic to estimate the total cost of a solution constrained to pass through a state. Thus, it searches in order of decreasing solution quality and is optimally efficient.
  3. Any other optimal algorithm using the same hueristic will expand at least as many vertices as A-Star.

Idea of Hueristics Functions

  • Hueristic functions are used to map every node in the graph to a non-negative value.

  • Criteria for Hueristics Functions

    • Should be a monotonic function
    • Should satisfy \(H(goal) = 0\)
    • For any two adjacent nodes \(x\) and \(y\):
      • \(H(x, y) \leq H(y) + d(x, y)\)
      • \(d(x, y) = EdgeCost(x, y)\)
    • These properties ensure that for all nodes \(n\):
      • \(H(n) \leq length of Shortest Path(n, GOAL)\)

  • For path Planning on a grid:

    • Euclidean Distance: \(H(x_n, y_n) = \sqrt{(x_n-x_g)^2 + (y_n-y_g)^2}\)
    • Manhattan Distance: \(H(x_n, y_n) = \lvert(x_n - x_g) + (y_n - y_g)\rvert\)

Where \(x_n\), \(y_n\) and \(x_g\), \(y_g\) are the \(x\), \(y\) coordinates of a the node and the goal respectively.

Psuedo Code for the Algorithm

def Astar(start, goal, graph):
    # Set the g, f values for all nodes in the graph
    for node in graph:
        node.f = Infinity
        node.g = Infinity

    # Create an empty list to store visited nodes
    nodes = []

    # Add Start to nodes list
    nodes.add(start)

    # Loop to traverse the graph
    while nodes is not EMPTY:
        # Obtain bode with the least f-value
        CURRENT = argmin(node, criteria=node.f)

        # Check if current node is the goal Node
        # which means the graph has been completely traversed
        if CURRENT == goal:
            report "SUCCESS"
            break
        # Update parameters for adjacent nodes
        for adjacent_node in CURRENT.adjacent_nodes:
            if adjacent_node.g > CURRENT.g + cost of edge from n to current:
                adjacent_node.g = CURRENT.g + cost of edge from n to current
                adjacent_node.f = adjacent_node.g + H(node)
                adjacent_node.parent = CURRENT

                # Add the adjacent node to nodes list if not there already
                if adjacent_node not in nodes:
                    nodes.add(adjacent_node)

Notations in the Psuedo Code explained:

g-value = distance between a node and the start node
H-function = Hueristic funciton
f-value = g-value + Hueristic value of the node

References