ACCRA INSTITUTE OF TECHNOLOGY
ADVANCE SCHOOL OF SYSTEMS AND DATA STUDIES
DATA STRUCTURES AND ALGORITHMS II – CS204
SPANNING TREE
PRESENTED BY GAMELI ANYAM
Contents
1. Foundational Concepts
2. The Universe as a Graph
3. What is Spanning Tree?
4. Number of Possible Spanning Trees
5. Properties of Spanning Tree
6. Minimum Spanning Trees (MST) Algorithms
7. Applications of Spanning Tree
Gameli Anyam
Foundational Concepts
Graphs are structures that consists of vertices and edges.
The edges can either connect all vertices in one direction (directed) or
not (undirected).
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Foundational Concepts
1. Complete graph: every vertex has an edge to all vertices.
2. Closed graph: number vertices equals number of edges.
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Foundational Concepts
The Universe as a Graph
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Gameli Anyam
What is Spanning Tree?
A spanning tree of a graph is a subgraph that contains the least number of edges
needed to create a path from one vertex to another until all vertices are connected.
An edge that has a numerical value is called a weighted edge.
Minimum Spanning Tree (MSP) is a spanning tree with the least sum edge
weights over all possible paths or spanning tree.
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Number of Possible Spanning Trees
According to graph theory, in order to determine the number of
feasible spanning trees, you must first determine the graph's type.
For a cycle graph with equal number of vertices as edges,
⇒ 𝑛 𝑆𝑇 𝑐𝑦𝑙𝑒 𝑔𝑟𝑎𝑝ℎ = 𝑛 𝑉 = 𝑛 𝐸 .
For a completed graph, the possible number of spanning tree is given
as by;
⇒ 𝑛 𝑆𝑇 = 𝑛 𝑉
𝑉−2
. This is known as Cayley’s Formula.
Gameli Anyam
Properties of Spanning Tree
Here are few properties of spanning tree:
❑ A spanning tree whose overall resultant weight value is minimal is
considered to be a Minimal Spanning Tree.
❑ A connected graph can have more than one spanning tree.
❑ All Spanning trees must contain the same number of vertices as of
graph, and the number of edges must be equal to |V| - 1.
❑ The spanning tree must not contain any cycle.
Gameli Anyam
MST Algorithms
Most commonly used are Prim’s and Kruskal’s.
Others: Reverse-deleted and Boruvka’s algorithm.
Prim's Algorithm
Prim's algorithm finds the subset of edges that includes every vertex of the graph such
that the sum of the weights of the edges can be minimized. It starts with the single node
and explore all the adjacent nodes with all the connecting edges at every step. The edges
with the minimal weights causing no cycles in the graph got selected.
Kruskal’s Algorithm
Kruskal's Algorithm is used to find the minimum spanning tree for a connected weighted
graph. Kruskal's algorithm follows greedy approach which finds an optimum solution at
every stage instead of focusing on a global optimum.
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Applications of MST
Telecommunication Network
Building: If we want to develop a
telecommunication network for
the entire city, a basic naive
approach will be more expensive.
We can create a communications
system at a lower cost by using the
Minimum Spanning Tree
technique. The image given below
explains the difference between
Naive and MST routing.
Gameli Anyam
Applications of MST
Constructing Highways or
Railroads: For constructing
highways or railroads the
Minimum Spanning Tree
approach is utilized everywhere.
Given the possible routes
between two cities, MST
technique provides you with the
optimal route. Basically, the
algorithm treats the cities as the
vertices and paths joining them
as edges to create a subtree that
will make the route fully
connected and cost efficient.
Gameli Anyam