DeepSeek vs Gemini vs Mistral: Graph Algorithms

Prompt

Graph algorithms - shortest paths, spanning trees, network flows

DeepSeekDeepSeek

Graph algorithms - shortest paths, spanning trees, network flows

Of course. This is a fantastic and fundamental area of computer science. Let's break down these core graph algorithm categories: Shortest Paths, Spanning Trees, and Network Flows.

I'll explain the key problems, the most important algorithms, their intuition, and their common applications.

1. Shortest Paths

The problem is simple: find the shortest path from a starting node to a target node (or to all other nodes). "Shortest" can mean minimum number of edges, minimum distance, minimum time, or minimum cost.

Key Algorithms:

a) Dijkstra's Algorithm

Use Case: Finds shortest paths from a single source node to all other nodes in a graph with non-negative edge weights.

Intuition: A greedy algorithm. It maintains a set of nodes whose shortest distance is known. It repeatedly picks the unvisited node with the smallest known distance, updates the distances of its neighbors, and marks it as visited.

Complexity: O((V + E) log V) with a priority queue (where V is vertices, E is edges).

Analogy: It's like a "controlled wavefront" expanding from the source, always extending the shortest known path first.

Applications:

GPS navigation (finding the fastest route).

Network routing protocols (e.g., OSPF).

Social networks (finding the degree of separation).

b) Bellman-Ford Algorithm

Use Case: Finds shortest paths from a single source in graphs that can have negative edge weights. It can also detect negative weight cycles.

Intuition: Relax all edges repeatedly. It does this |V| - 1 times. If a shorter path can still be found on the |V|-th iteration, a negative cycle exists.

Complexity: O(V * E), which is slower than Dijkstra.

Applications:

Routing in networks where costs can be negative.

Arbitrage detection in currency exchange markets (by transforming the problem and looking for negative cycles).

c) Floyd-Warshall Algorithm

Use Case: Finds the shortest paths between every pair of nodes.

Intuition: A dynamic programming algorithm. It considers each node as a potential intermediate point in the shortest path between two other nodes.

Complexity: O(V³), so it's only practical for graphs that aren't too large.

Applications:

Precomputing all-pairs shortest paths for fast query response (e.g., "is there a connection between any two users?").

Computing the transitive closure of a graph.

2. Spanning Trees

A spanning tree of a connected, undirected graph is a subgraph that is a tree and includes all the vertices. A Minimum Spanning Tree (MST) is a spanning tree with the minimum possible total edge weight.

Key Algorithms:

a) Kruskal's Algorithm

Intuition: A greedy algorithm. It starts with all vertices and no edges. It then sorts all edges by weight and adds them one by one to the growing forest, but only if adding the edge does not form a cycle. Union-Find (Disjoint Set) data structure is used for efficient cycle detection.

Complexity: O(E log V) due to sorting.

Analogy: Building a network by always choosing the cheapest possible connection that links two previously unconnected components.

b) Prim's Algorithm

Intuition: Also a greedy algorithm. It starts from an arbitrary vertex and grows the MST one vertex at a time. At each step, it adds the cheapest edge that connects a vertex in the MST to a vertex outside the MST.

Complexity: O(E log V) with a priority queue.

Analogy: Growing a "crystal" from a seed. You always add the cheapest piece attached to the existing crystal.

Applications of MST:

Designing network infrastructure (least-cost fiber optic cable layout connecting all cities).

Circuit design (minimizing wire length).

Cluster analysis in machine learning.

3. Network Flows

The problem is modeled as a flow network: a directed graph where each edge has a capacity. There is a source node (S) that produces flow and a sink node (T) that consumes it. The goal is to find the maximum flow from S to T that does not exceed any edge capacity.

Key Algorithm:

Ford-Fulkerson Method (with Edmonds-Karp implementation)

Intuition: The core idea is to start with a flow of zero and repeatedly find an "augmenting path" from the source to the sink in the residual graph and push as much flow as possible along it. The residual graph is a key concept—it shows the remaining capacity on edges and allows for "undoing" flow.

Residual Graph: For an edge with capacity c and flow f, you create:

A forward edge with remaining capacity c - f.

A backward edge with capacity f (representing the ability to reduce flow).

Edmonds-Karp: An implementation of Ford-Fulkerson that uses BFS to find the augmenting path. This guarantees a polynomial time complexity.

Complexity: O(V * E²) for Edmonds-Karp.

Max-Flow Min-Cut Theorem: The value of the maximum flow is equal to the capacity of the minimum cut (a partition of nodes separating S and T with the smallest total capacity). This is a fundamental duality theorem.

Applications of Max Flow:

Bipartite Matching: Finding the maximum matching in a bipartite graph (e.g., assigning jobs to applicants).

Airline Scheduling: Maximizing the utilization of airplanes and crews.

Image Segmentation: Separating an image into foreground and background.

Baseball Elimination: Determining if a team is mathematically eliminated from winning their division.

Summary Table

Category	Problem	Key Algorithms	Key Idea	Best For
Shortest Path	Single-Source	Dijkstra's	Greedy expansion from closest node	Non-negative weights
Single-Source	Bellman-Ford	Relax all edges repeatedly	Graphs with negative weights
All-Pairs	Floyd-Warshall	Dynamic Programming with intermediates	Dense graphs, small V
Spanning Tree	Minimum Spanning Tree	Kruskal's	Add cheapest edges, avoid cycles	Sparse graphs
Minimum Spanning Tree	Prim's	Grow a tree from a single node	Dense graphs
Network Flow	Maximum Flow	Ford-Fulkerson/Edmonds-Karp	Find augmenting paths in residual graph	Modeling flow/capacity systems

These algorithms form the backbone of graph theory applications and are essential knowledge for fields like computer networking, operations research, logistics, and social network analysis.

11:29

GeminiGoogle

Graph algorithms - shortest paths, spanning trees, network flows

Okay, let's break down the core concepts and applications of graph algorithms related to shortest paths, spanning trees, and network flows. These are fundamental and widely used in computer science and various fields.

1. Shortest Paths

Goal: Find the path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized.

Types of Shortest Path Problems:

Single-Source Shortest Path (SSSP): Find the shortest path from a designated source vertex to all other vertices in the graph. This is the most common variant.

Single-Destination Shortest Path: Find the shortest path from all vertices in the graph to a designated destination vertex. This is equivalent to the SSSP problem on the graph with the direction of every edge reversed.

All-Pairs Shortest Path (APSP): Find the shortest path between every pair of vertices in the graph.

Algorithms:

Dijkstra's Algorithm:

Purpose: Finds the SSSP in a weighted graph with non-negative edge weights.

Mechanism: Uses a greedy approach. It maintains a set of visited vertices and a priority queue (usually a min-heap) of unvisited vertices, prioritized by their current shortest distance estimate from the source. It iteratively selects the vertex with the smallest distance estimate, marks it as visited, and updates the distance estimates of its neighbors.

Complexity: O(V²) with an adjacency matrix, or O((V + E)log V) with an adjacency list and a binary heap, or O(E + V log V) with a Fibonacci heap (though Fibonacci heaps are often impractical due to constant factor overhead). V is the number of vertices and E is the number of edges.

Limitations: Doesn't work with negative edge weights. If negative cycles are present, it can lead to incorrect results (infinite loops).

Bellman-Ford Algorithm:

Purpose: Finds the SSSP in a weighted graph, including graphs with negative edge weights. It can also detect the presence of negative cycles.

Mechanism: Works by iteratively relaxing edges. It repeatedly goes through all edges in the graph and updates the distance estimates of vertices. After V-1 iterations, if any edge can still be relaxed, it means a negative cycle exists.

Complexity: O(V * E)

Advantages: Handles negative edge weights and detects negative cycles.

Disadvantages: Slower than Dijkstra's for graphs with only non-negative edge weights.

Floyd-Warshall Algorithm:

Purpose: Finds the APSP in a weighted graph. It can handle negative edge weights and detect negative cycles.

Mechanism: Uses dynamic programming. It iteratively considers each vertex as an intermediate vertex in the shortest path between all pairs of vertices.

Complexity: O(V³)

Advantages: Simple to implement. Good for dense graphs where the number of edges is close to V².

Disadvantages: Less efficient than using Dijkstra's algorithm V times for sparse graphs.

**A* Search (A-Star):**

Purpose: Find the shortest path from a starting node to a goal node. It's particularly useful when you know (or can estimate) the "distance" to the goal.

Mechanism: An informed search algorithm that uses a heuristic function, h(n), to estimate the cost of the cheapest path from node n to the goal. It combines this with the actual cost from the start node to node n, denoted by g(n). The algorithm evaluates nodes by combining g(n) and h(n), i.e. f(n) = g(n) + h(n).

Heuristic Function: The choice of the heuristic function affects the algorithm's performance. If h(n) is admissible (never overestimates the actual cost to reach the goal), A* is guaranteed to find the optimal solution.

Complexity: Depends on the heuristic function. In the worst case, it can be exponential. With a good heuristic, it can be significantly faster than Dijkstra's.

Applications: Pathfinding in games, robotics, and route planning.

Applications of Shortest Path Algorithms:

Navigation: Finding the shortest route between two locations (GPS, mapping applications).

Network Routing: Determining the optimal path for data packets to travel across a network.

Transportation Planning: Optimizing delivery routes and transportation schedules.

Resource Allocation: Finding the most efficient way to allocate resources in a network.

Social Network Analysis: Determining the "shortest" social connection between two people.

Compiler Optimization: Instruction scheduling.

2. Spanning Trees

Goal: Find a subset of the edges of a connected, undirected graph that forms a tree that connects all the vertices.

Minimum Spanning Tree (MST): A spanning tree where the sum of the weights of the edges is minimized.

Algorithms:

Kruskal's Algorithm:

Purpose: Finds the MST of a weighted, undirected graph.

Mechanism: A greedy algorithm. It sorts the edges of the graph in ascending order of weight. Then, it iteratively adds edges to the MST, as long as adding the edge does not create a cycle. A disjoint-set data structure (Union-Find) is typically used to efficiently detect cycles.

Complexity: O(E log E) or O(E log V) (since E can be at most V², log E is O(log V))

Advantages: Simple to implement. Often efficient for sparse graphs.

Prim's Algorithm:

Purpose: Finds the MST of a weighted, undirected graph.

Mechanism: A greedy algorithm. It starts with an arbitrary vertex and iteratively adds the minimum-weight edge that connects a vertex in the MST to a vertex not yet in the MST. A priority queue (min-heap) is typically used to efficiently find the minimum-weight edge.

Complexity: O(V²) with an adjacency matrix, or O(E log V) with an adjacency list and a binary heap, or O(E + V log V) with a Fibonacci heap.

Advantages: Can be more efficient than Kruskal's for dense graphs.

Applications of Spanning Trees:

Network Design: Connecting all nodes in a network with the minimum possible cost.

Clustering: Grouping similar data points together.

Image Segmentation: Dividing an image into different regions.

Computer Vision: Feature extraction.

Bioinformatics: Phylogenetic tree construction.

Infrastructure Planning: Designing efficient road networks, power grids, or communication networks.

3. Network Flows

Goal: Determine the maximum amount of "flow" that can be sent from a source vertex to a sink vertex in a directed graph, subject to capacity constraints on the edges.

Key Concepts:

Network: A directed graph where each edge has a capacity (a non-negative integer or real number) representing the maximum amount of flow that can pass through that edge.

Source (s): The vertex where the flow originates.

Sink (t): The vertex where the flow terminates.

Flow: An assignment of a flow value to each edge, subject to the following constraints:

The flow on an edge cannot exceed its capacity.

For every vertex (except the source and sink), the total flow entering the vertex must equal the total flow leaving the vertex (flow conservation).

Residual Graph: A graph that represents the remaining capacity on each edge after a certain amount of flow has been assigned. It contains both forward edges (with remaining capacity) and backward edges (with the amount of flow currently on the original edge).

Augmenting Path: A path from the source to the sink in the residual graph. Sending flow along an augmenting path increases the overall flow in the network.

Algorithms:

Ford-Fulkerson Algorithm:

Purpose: Finds the maximum flow in a network.

Mechanism: Repeatedly finds augmenting paths in the residual graph and increases the flow along those paths until no more augmenting paths can be found.

Complexity: O(E * f), where f is the maximum flow value. This is pseudo-polynomial time. The algorithm can be inefficient if the capacities are large integers.

Issues: Can be very slow with poorly chosen augmenting paths or irrational capacities.

Edmonds-Karp Algorithm:

Purpose: Finds the maximum flow in a network. It's a specialization of Ford-Fulkerson.

Mechanism: Uses a Breadth-First Search (BFS) to find the shortest augmenting path in the residual graph.

Complexity: O(V * E²).

Advantages: Guaranteed polynomial time complexity, regardless of the capacities.

Dinic's Algorithm:

Purpose: Finds the maximum flow in a network.

Mechanism: A more efficient implementation of the augmenting path method. It uses a layered network (constructed with BFS) to find blocking flows (flows that saturate at least one edge on every path).

Complexity: O(V²E)

Applications of Network Flows:

Transportation Planning: Optimizing the flow of goods through a transportation network.

Network Routing: Determining the maximum data throughput in a communication network.

Matching Problems: Finding the maximum matching in a bipartite graph (e.g., matching students to internships).

Image Segmentation: Dividing an image into foreground and background regions.

Data Mining: Finding patterns in data.

Project Scheduling: Determining the minimum time required to complete a project.

Important Considerations:

Negative Edge Weights: Be mindful of negative edge weights, as they can cause issues with Dijkstra's algorithm. Bellman-Ford or Floyd-Warshall are required for shortest path problems involving negative edges.

Graph Representation: The choice of graph representation (adjacency matrix vs. adjacency list) can significantly affect the performance of graph algorithms. Adjacency lists are generally preferred for sparse graphs, while adjacency matrices are better for dense graphs.

Data Structures: Priority queues (heaps) are commonly used in Dijkstra's and Prim's algorithms for efficient selection of vertices or edges. Disjoint-set data structures are crucial for Kruskal's algorithm.

Algorithm Selection: The best algorithm for a particular problem depends on the specific characteristics of the graph (e.g., density, edge weights, presence of negative cycles).

This overview should give you a good foundation in shortest paths, spanning trees, and network flows. Remember to study implementations and work through practice problems to solidify your understanding. Good luck!

11:29

MistralMistral AI

Graph algorithms - shortest paths, spanning trees, network flows

Graph algorithms are fundamental in computer science and have wide-ranging applications in various fields such as network design, routing, and optimization. Here are some key graph algorithms related to shortest paths, spanning trees, and network flows:

Shortest Paths

Dijkstra's Algorithm:

Description: Finds the shortest path from a single source vertex to all other vertices in a graph with non-negative weights.

Complexity: O(V^2) for a simple implementation, O(E + V log V) using a priority queue (e.g., Fibonacci heap).

Use Case: Routing algorithms, GPS navigation.

Bellman-Ford Algorithm:

Description: Finds the shortest path from a single source vertex to all other vertices in a graph, allowing for negative weights but no negative cycles.

Complexity: O(V * E).

Use Case: Networks with negative weights, detecting negative cycles.

**A* Algorithm**:

Description: An informed search algorithm that uses heuristics to improve efficiency. It is often used in pathfinding and graph traversal.

Complexity: Depends on the heuristic used.

Use Case: Game development, robotics.

Floyd-Warshall Algorithm:

Description: Finds the shortest paths between all pairs of vertices in a weighted graph.

Complexity: O(V^3).

Use Case: Network analysis, all-pairs shortest paths.

Spanning Trees

Kruskal's Algorithm:

Description: Finds the minimum spanning tree (MST) of a connected, undirected graph using a greedy approach.

Complexity: O(E log E) or O(E log V) using a disjoint-set data structure.

Use Case: Network design, clustering.

Prim's Algorithm:

Description: Finds the MST of a connected, undirected graph starting from an arbitrary vertex.

Complexity: O(E log V) using a priority queue.

Use Case: Network design, clustering.

Borůvka's Algorithm:

Description: One of the earliest algorithms for finding the MST, it iteratively merges components.

Complexity: O(E log V).

Use Case: Historical interest, educational purposes.

Network Flows

Ford-Fulkerson Algorithm:

Description: Computes the maximum flow in a flow network. It uses the concept of augmenting paths.

Complexity: O(E * f), where f is the maximum flow.

Use Case: Network flow problems, resource allocation.

Edmonds-Karp Algorithm:

Description: An implementation of the Ford-Fulkerson algorithm using BFS to find augmenting paths.

Complexity: O(V * E^2).

Use Case: Network flow problems, resource allocation.

Push-Relabel Algorithm:

Description: A more efficient algorithm for finding the maximum flow in a flow network.

Complexity: O(V^2 * E).

Use Case: Network flow problems, resource allocation.

Capacity Scaling Algorithm:

Description: An algorithm that scales the capacity of edges to find the maximum flow.

Complexity: O(E^2 log U), where U is the maximum capacity.

Use Case: Network flow problems, resource allocation.

Applications

Shortest Paths: Used in routing protocols, GPS systems, and logistics.

Spanning Trees: Used in network design, clustering, and electrical grid planning.

Network Flows: Used in resource allocation, traffic management, and supply chain optimization.

These algorithms are essential tools in the field of graph theory and have numerous practical applications across various domains.

11:29

Analysis

This comparison demonstrates the different approaches each AI model takes when responding to the same prompt. Here are the key differences observed:

Response Characteristics

DeepSeek: Provides a direct response with 73 sentences.

Gemini: Provides a direct response with 118 sentences.

Mistral: Provides a direct response with 53 sentences.

Key Takeaways

Each model brings unique strengths to this type of query
Response styles vary significantly between models
Consider your specific use case when choosing between these models

Try This Comparison Yourself

Want to test these models with your own prompts? Visit SNEOS.com to compare AI responses side-by-side in real-time.

This comparison was generated using the SNEOS AI Comparison ToolPublished: October 01, 2025 | Models: DeepSeek, Gemini, Mistral