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Graphs & Graph Algorithms 3
CMSC 132
Department of Computer Science
University of Maryland, College Park
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Graphs and Graph Algorithms: A Comprehensive Study - Prof. Nelson Padua-Perez, Study notes of Computer Science
An overview of graphs and graph algorithms, including graph implementation using adjacency matrix, list, and set, spanning trees, minimum spanning trees using prim's and kruskal's algorithms, and dijkstra's algorithm for shortest paths. It covers the basics of graph theory, data structures, and algorithms.
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Pre 2010
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Graphs & Graph Algorithms 3
CMSC 132
Department of Computer ScienceUniversity of Maryland, College Park
Overview
Graph implementation^ Adjacency list / matrix / set Spanning trees Minimum spanning tree^ Prim’s algorithm^ Kruskal’s algorithm
Adjacency Matrix
Representation^ 2D array^ Position j, k
⇒⇒⇒⇒^ edge between nodes n
, njk
Unweighted graph^ Matrix elements
⇒⇒⇒⇒^ boolean Weighted graph^ Matrix elements
⇒⇒⇒⇒^ weight
Adjacency Matrix
Example
Adjacency List
Representation^ Linked or array list for each node ofneighbors/successors^ for directed graph, may need predecessors aswell^ Unweighted graph^ store neighbor^ Weighted graph^ store neighbor, weight
Adjacency List
Example^ Unweighted graph^ Weighted graph
Graph Space Requirements Adjacency matrix^2 ½ Nentries (for graph with N nodes, E edges) Many empty entries for large graphs Adjacency list E entries Adjacency Set/Map E entries Space overhead per entry higher than for adjacencylist
Graph Time Requirements
Average Complexity of operations^ For graph with N nodes, E edges
Adj Set/Map O(1)O(1)O(1) O(E/N) O(E/N) O(1) Find edge
O(E/N)
O(N)
Enumerateedges
O(E/N)
O(1)
Delete edge
O(E/N)
O(1)
Insert edge
Adj List Adj Matrix Operation
Recursive Spanning Tree
Construction
Known = { start }explore ( start );void explore (Node X) {^ for each successor Y of X
if (Y is not in Known)^ Parent[Y] = XAdd Y to Knownexplore(Y)
Spanning Tree Construction
Known = { start }Discovered = { start }while ( Discovered
≠≠≠≠ ∅∅∅∅^ )
take node X out of Discoveredfor each successor Y of X^ if (Y is not in Known)
Parent[Y] = XAdd Y to DiscoveredAdd Y to Known
Depth-First Spanning Tree Example
Breadth-First Spanning Tree Example
Minimum Spanning Tree (MST) Spanning tree with minimum total edge weight Multiple MSTs possible (with same weight)
Algorithms for MST