Milda Glebauskaitė Graph theory and software engineering 

PART 1: Basics a.k.a stuff I should have learned in the university but I wasn’t listening 

ORIGINS 

ORIGINS 

ORIGINS ● There should not be any uncrossed bridges ● Each bridge must not be crossed more than once. 

ORIGINS ● There should not be any uncrossed bridges ● Each bridge must not be crossed more than once. 

ORIGINS ● 4 lands, 7 bridges ● There are an odd number of bridges connected to each land ● if you enter a land by crossing one bridge, you can always leave the land by crossing its second bridge. If third bridge appears, you won’t be able to leave a land. 

ORIGINS ● If new bridge appears, you can can solve the problem 

ORIGINS ● If new bridge appears, you can can solve the problem 

ORIGINS 

ORIGINS Vertex Edge 

ORIGINS Degree of a vertex, the number of edges incident connected to the vertex. 

ORIGINS An Euler path of a finite undirected graph G(V, E) is a path such that every edge of G appears on it once. If G has an Euler path, then it is called an Euler graph. 

ORIGINS Theorem. A finite undirected connected graph is an Euler graph if and only if exactly two vertices are of odd degree or all vertices are of even degree. In the latter case, every Euler path of the graph is a circuit, and in the former case, none is. 

Graph representation 

Graph representation Twitter problem 

Graph representation 

Graph representation 

Graph representation 

Graph representation Adjacency list - each list describes the set of neighbors of a vertex in the graph 

Graph representation 

Graph algorithms Any processing done with graphs can be called “graph algorithm” 

Graph algorithms Uber/Google Maps problem 

Graph algorithms 

Graph algorithms 

Dijkstra’s algorithm Graph algorithms 

Dijkstra’s algorithm 1) Mark all nodes unvisited. Create a set of all the unvisited nodes called the unvisited set. 2) Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes. Set the initial node as current. Graph algorithms 

Dijkstra’s algorithm 1) Mark all nodes unvisited. Create a set of all the unvisited nodes called the unvisited set. 2) Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes. Set the initial node as current. Graph algorithms 

Dijkstra’s algorithm 3) For the current node, consider all of its unvisited neighbors and calculate their tentative distances through the current node. Compare the newly calculated tentative distance to the current assigned value and assign the smaller one. Graph algorithms 

Dijkstra’s algorithm 4) When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again. Graph algorithms 

Dijkstra’s algorithm 5) If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the unvisited set is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished. 6) Otherwise, select the unvisited node that is marked with the smallest tentative distance, set it as the new “current node”, and go back to step 3. Graph algorithms 

Dijkstra’s algorithm 5) If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the unvisited set is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished. 6) Otherwise, select the unvisited node that is marked with the smallest tentative distance, set it as the new “current node”, and go back to step 3. Graph algorithms 

PART 2: Graphs are fun a.k.a stuff I found interesting but did not know how to structure into proper presentation 

GT in Model based testing Model based testing is a software testing technique where run time behavior of software under test is checked against predictions made by a model. A model is a description of a system's behavior. 

MBT ▪ You create model of a SUT ▪ It’s a simplified model ▪ It’s a state model defining states and relationships between them ▪ From this model you can generate tests and execute them on SUT 

GT in MBT ▪ Chinese postman problem solutions ▪ de Bruijn sequences algorithm ▪ Random Walk algorithm 

Graph coloring problem Given m colors, find a way of coloring the vertices of a graph such that no two adjacent vertices are colored using same color. 

Real life problems ▪ Schedules and timetables ▪ Map coloring ▪ Sudoku 

More fancy problems ▪ Register allocation 

In compiler optimization, register allocation is the process of assigning a large number of target program variables onto a small number of CPU registers. 

Nodes in the graph represent live ranges (variables, temporaries, virtual/symbolic registers) that are candidates for register allocation. Edges connect live ranges that interfere , i.e., live ranges that are simultaneously live at at least one program point. Register allocation then reduces to the graph coloring problem in which colors (registers) are assigned to the nodes such that two nodes connected by an edge do not receive the same color. 

More fancy problems ▪ Register allocation ▪ Network updates 

OO programs as graphs 

GT in OO programs ▪ Identification of “God” classes - HyperLink Induced Topic Search algorithm 

Hyperlink-Induced Topic Search is a link analysis algorithm that rates Web pages. The scheme assigns two scores for each page: its authority, which estimates the value of the content of the page, and its hub value, which estimates the value of its links to other pages. 

GT in OO programs ▪ Identification of “God” classes - HyperLink Induced Topic Search algorithm ▪ Design pattern detection - graph matching algorithms 

Thank You 

P.S. I stole the illustrations from this post ▪ https://www.freecodecamp.org/news/i-dont-understan d-graph-theory-1c96572a1401/