About Searching in Tree and Graph Data Structures

Introduction

In this article, we will provide a detailed explanation of Tree and Graph data structures, which play extremely important roles in computer science, from their basic concepts to search algorithms.

In the field of data structures and algorithms, these are unavoidable themes. In particular, Depth-First Search (DFS), Breadth-First Search (BFS), and Dijkstra’s Algorithm for solving the shortest path problem frequently appear in programming contests and practical business.

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

1. Basics of Tree Data Structure

The tree structure is a data structure suitable for representing data with hierarchical relationships. It is used in various situations such as file systems, organization charts, and HTML DOM trees.

A tree structure consists of the following elements:

• Node : The element that holds data
• Edge : The line connecting nodes
• Root Node : The topmost node in the tree. It is a node that does not have a parent.
• Leaf Node : A node that does not have children.

  graph TD
  "Root" --> "NodeA"
  "Root" --> "NodeB"
  "NodeA" --> "Leaf1"
  "NodeA" --> "Leaf2"
  "NodeB" --> "Leaf3"

As the basics of searching in tree structures, there are Depth-First Search (DFS) and Breadth-First Search (BFS).

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

2. Depth-First Search (DFS)

Depth-First Search is an algorithm that starts from a certain node, goes as deep as possible, and when it reaches a dead end, returns to the previous node to continue the search. It can be implemented very simply by using recursive functions. It also sometimes uses a data structure called a Stack.

Python Implementation Example of DFS in a Tree Structure

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class TreeNode:
    def __init__(self, value):
        self.value = value
        self.children = []

def dfs_tree(node):
    if node is None:
        return
    print(f"Visiting {node.value}")
    for child in node.children:
        dfs_tree(child)

# Tree construction
root = TreeNode("Root")
node_a = TreeNode("A")
node_b = TreeNode("B")
root.children.extend([node_a, node_b])
node_a.children.extend([TreeNode("C"), TreeNode("D")])

print("DFS Traversal:")
dfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

3. Breadth-First Search (BFS)

Breadth-First Search is an algorithm that starts from the root node, searches all nodes at the same depth, and then proceeds to the nodes at the next depth. It uses a data structure called a Queue. It is often used when finding the shortest path.

Python Implementation Example of BFS in a Tree Structure

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from collections import deque

def bfs_tree(root):
    if root is None:
        return
    queue = deque([root])
    while queue:
        current = queue.popleft()
        print(f"Visiting {current.value}")
        for child in current.children:
            queue.append(child)

print("BFS Traversal:")
bfs_tree(root)

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

4. Basics of Graph Data Structure

A graph structure consists of a set of nodes (vertices) and edges. A tree structure is also a type of graph (an undirected or directed graph without cycles), but a general graph can have cycles and multiple parents.

Graphs have the following types:

• Undirected Graph : A graph where edges have no direction
• Directed Graph : A graph where edges have a direction
• Weighted Graph : A graph where edges have weights (costs)

  graph LR
  "A" -- "5" --> "B"
  "A" -- "2" --> "C"
  "B" -- "1" --> "D"
  "C" -- "8" --> "D"
  "C" -- "4" --> "E"
  "D" -- "3" --> "E"

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")

5. Dijkstra’s Algorithm

Dijkstra’s Algorithm is an algorithm for finding the shortest path from a starting point to all other vertices in a weighted graph. However, the edge weights must be non-negative (0 or greater).

By using a Priority Queue, searches can be performed efficiently. As a mathematical expression, letting $ d(v) $ be the shortest distance from the starting point to vertex $ v $, for an edge $ (u, v) $ with weight $ w(u, v) $, we update it as $ d(v) = \min(d(v), d(u) + w(u, v)) $. Mathematically, it satisfies the property $ d(v) \le d(u) + w(u, v) $. Here, we choose the path where $ \text{cost} $ is minimized.

Python Implementation Example of Dijkstra’s Algorithm

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import heapq

def dijkstra(graph, start):
    # Initialize shortest distances to infinity
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)

        if current_distance > distances[current_node]:
            continue

        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

# Definition of the graph (adjacency list format)
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 1},
    'C': {'D': 8, 'E': 4},
    'D': {'E': 3},
    'E': {}
}

start_node = 'A'
shortest_paths = dijkstra(graph, start_node)
print(f"Shortest paths from {start_node}: {shortest_paths}")