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An in-depth analysis of various graph implementations, including adjacency matrix, adjacency list, and adjacency set/map. It covers their representations, space requirements, and time requirements. Additionally, it discusses the concept of graph density and the choice between neighbor-based and connection-based algorithms. Lastly, it introduces dijkstra's algorithm for finding the shortest path in a graph.
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Adjacency matrix 2D array of neighbors Adjacency list List of neighbors Adjacency set / map Set / map of neighbors
Affects efficiency / storage
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Adjacency Matrix
2D array Position j, k ⇒ edge between nodes nj , nk
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Adjacency Matrix
Single array for entire graph Undirected graph Only upper / lower triangle matrix needed Since nj, nk implies nk , nj Unweighted graph Matrix elements ⇒ boolean Weighted graph Matrix elements ⇒ weight
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Adjacency Set / Map
For each node, store Set or map of neighbors / successors For unweighted graph Use set of neighbors For weighted graph Use map of neighbors, w/ value = weight of edge Undirected graph For every edge (a, b) Nodes a & b need to store each other as neighbor For directed graph May also need to store map of predecessors
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Graph Space Requirements
½ N^2 entries (for graph with N nodes, E edges) Many empty entries for large, sparse graphs
2×E entries
2×E entries Space overhead per entry Higher than for adjacency list
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Graph Time Requirements
Can find individual edge (a,b) quickly Examine entry in array Edge[a,b] Constant time operation
Can find all edges for node (a) quickly Iterate through collection of edges for a On average E / N edges per node
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Graph Time Requirements
For graph with N nodes, E edges
Adj Set/Map
Find edge O(1) O(E/N)
Enumerate O(N) O(E/N)
edges for node
Delete edge O(1) O(E/N)
Insert edge O(1) O(E/N)
Operation Adj Matrix Adj List
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Shortest Path – Dijkstra’s Algorithm
Nodes with known shortest path from start ⇒ S Cost of shortest path to node K from start ⇒ C[K] Only for paths through nodes in S Predecessor to K on shortest path ⇒ P[K] Updated whenever new (lower) C[K] discovered Remembers actual path with lowest cost
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Shortest Path – Intuition for Dijkstra’s
Shortest paths are known for nodes in S Store in C[K] length of shortest path to node K (for all paths through nodes in { S } ) Add to { S } next closest node
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Shortest Path – Intuition for Djikstra’s
Previous shortest path to K already in C[ K ] Possibly shorter path to J by going through node K Compare C[ J ] with C[ K ] + weight of (K,J), update C[ J ] if needed
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Shortest Path – Dijkstra’s Algorithm
P[ ] = none for all nodes C[start] = 0, C[ ] = ∞ for all other nodes
find node K not in S with smallest C[K] add K to S for each node J not in S adjacent to K if ( C[K] + cost of (K,J) < C[J] ) C[J] = C[K] + cost of (K,J) P[J] = K
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Djikstra’s Shortest Path Example
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Djikstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
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Dijkstra’s Shortest Path Example
Start at K Trace back predecessors in P[ ]
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