# Prim's Algorithm vs. Kruskal's Algorithm — What's the Difference?

By Fiza Rafique — Published on January 16, 2024
Prim's algorithm builds a minimum spanning tree by adding the cheapest available connection from a node; Kruskal's starts with edges, ignoring cycles.

## Key Differences

Prim’s Algorithm begins with a single node and expands the spanning tree by adding the cheapest edge that connects a node in the tree to a node outside it. In contrast, Kruskal’s Algorithm sorts all edges from low to high cost and adds them one by one, without forming any cycles, until all nodes are connected.
Prim’s Algorithm works well for graphs with lots of edges, as it optimizes on nodes and the growing spanning tree. Kruskal’s Algorithm is often better for sparse graphs because it processes edges without consideration of the growing tree, which can be more efficient when the edge count is low.
In Prim’s Algorithm, each step is focused on the nodes and the tree, considering all the edges of the currently growing tree. Kruskal’s Algorithm treats every node as a separate tree and merges these trees, focusing on edges with the least weight.
Prim’s Algorithm requires a priority queue to select the next cheapest edge, which can lead to a simpler implementation for dense graphs. Kruskal’s Algorithm, on the other hand, requires sorting all edges and a disjoint-set data structure to detect cycles, which can be more efficient for graphs with fewer edges.
Prim’s and Kruskal’s Algorithms both find a minimum spanning tree for a connected weighted graph. However, Prim’s is more efficient for dense graphs where the number of edges is high, and Kruskal’s is better for sparse graphs where the number of edges is much less than the square of the number of nodes.

## Comparison Chart

### Starting Point

Begins with a single node and expands.
Starts with the smallest edge and connects trees.

### Edge Selection

Adds the cheapest edge to a node in the tree.
Adds the next cheapest edge without forming a cycle.

### Data Structure

Uses a priority queue to select the next edge.
Requires sorting edges and uses a disjoint-set to avoid cycles.

### Suitability

More suitable for dense graphs.
More suitable for sparse graphs.

### Graph Type

Efficient for graphs with a large number of edges.
Efficient for graphs with fewer edges.

### Edge Processing

Considers edges connected to the growing spanning tree.
Considers all edges independently, sorted by weight.

### Cycle Detection

Naturally avoids cycles as it grows the tree.
Requires a disjoint-set data structure to detect and avoid cycles.

### Algorithm Type

Greedy algorithm expanding from a node.
Greedy algorithm based on edge weights.

## Compare with Definitions

#### Prim's Algorithm

Always works on a single connected component.
Prim's Algorithm quickly connected the isolated cluster to the main grid.

#### Kruskal's Algorithm

Kruskal's Algorithm builds a minimum spanning tree by considering edges.
Kruskal's Algorithm selected the shortest bridges to span the river.

#### Prim's Algorithm

Selects edges with the least possible weight.
Prim's Algorithm chose the edge with the lowest cost to expand the network.

#### Kruskal's Algorithm

Particularly efficient for graphs with fewer edges.
In our rural network, Kruskal's Algorithm was the best choice due to fewer roads.

#### Prim's Algorithm

Prim's Algorithm is a greedy method that builds a minimum spanning tree.
We used Prim's Algorithm to efficiently connect all the nodes in our network.

#### Kruskal's Algorithm

Begins by sorting all the edges of the graph.
We sorted the pathways by length using Kruskal's Algorithm for the park design.

#### Prim's Algorithm

Prim's expands the tree from a chosen starting node.
Prim's Algorithm starts with node A and gradually includes the nearest nodes.

#### Kruskal's Algorithm

Uses disjoint-set to ensure a cycle is not formed.
Kruskal's Algorithm efficiently managed the sections of the new highway.

#### Prim's Algorithm

It finds the minimum spanning tree for a weighted undirected graph.
To minimize the wiring cost, we applied Prim's Algorithm to the layout plan.

#### Kruskal's Algorithm

The algorithm initiated with individual houses, linking them into a community grid.

## Common Curiosities

#### Do both algorithms always give the same result?

Both algorithms always result in a minimum spanning tree, but the tree may not be unique.

#### What is the primary goal of Prim’s Algorithm?

Prim’s Algorithm aims to find the minimum spanning tree in a connected, undirected graph.

#### Can Prim’s Algorithm start at any node?

Yes, Prim’s Algorithm can start at any node in the graph.

#### Which algorithm is faster?

It depends on the graph; Prim’s is generally faster for dense graphs, while Kruskal’s is faster for sparse graphs.

#### What kind of data structure is used in Prim’s Algorithm?

Prim’s Algorithm typically uses a priority queue to manage edges.

#### Does Kruskal’s Algorithm require a starting node?

No, Kruskal’s does not need a starting node as it builds the tree from edges.

#### How does Kruskal’s Algorithm work?

Kruskal’s Algorithm constructs a minimum spanning tree by adding the next lightest edge that doesn’t produce a cycle.

#### Is Kruskal’s Algorithm greedy?

Yes, Kruskal’s Algorithm is a greedy algorithm that selects the lowest-weight edges.

#### What does sorting the edges accomplish in Kruskal’s Algorithm?

Sorting helps in selecting the next smallest edge efficiently.

#### What is a key advantage of Prim’s Algorithm?

Prim’s can be more efficient with dense graphs where edge selection is localized to the new node.

#### Why does Kruskal’s Algorithm need a disjoint-set?

The disjoint-set helps keep track of connected components to avoid cycles.

#### How does Prim’s Algorithm ensure there are no cycles?

Prim’s Algorithm builds on a single tree, which naturally avoids cycles.

#### Can Prim’s Algorithm handle disconnected graphs?

No, it only works on connected graphs.

#### Is it necessary to sort all edges in Prim’s Algorithm?

No, Prim’s Algorithm does not require sorting all edges, only maintaining a priority queue.

#### Can Kruskal’s Algorithm result in multiple trees?

Initially, it can handle multiple trees but ends up with a single minimum spanning tree.