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1
Greedy 2
Jose RolimUniversity of Geneva
Algorithmique Greedy 2
Jose Rolim 2
Examples Greedy
Minimum Spanning Trees Shortest Paths Dijkstra
Algorithmique Greedy 2
Jose Rolim 3
Definition MST
Given a connected graph G = (V, E), with weight function w : E --> R
Min-weight connected subgraph Spanning tree T:
A tree that includes all nodes from V T = (V, E’), where E’ E Weight of T: W( T ) = w(e)
Minimum spanning tree (MST): A tree with minimum weight among all
spanning trees
Algorithmique Greedy 2
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Example:
Algorithmique Greedy 2
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MST:
MST for given G may not be unique
Since MST is a spanning tree: # edges : |V| - 1
If the graph is unweighted: All spanning trees have same weight
Algorithmique Greedy 2
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Kruskal’s algorithm (Greedy)
Start with A empty, and each vertex being its own connected component
Repeatedly merge two components by connecting them with a light edge crossing them
Two issues: Maintain sets of components
Choose light edges
Algorithmique Greedy 2
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Pseudo-code
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The Greedy Algorithm in Action
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Algorithmique Greedy 2
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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Algorithmique Greedy 2
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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The Greedy Algorithm in Action
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Algorithmique Greedy 2
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Complexity
Time complexity: #make-set, find-set and union
operations: O(|V| + |E|) • O( (|V| + |E|) (|V| + |E|))
Sorting:• O(|E| log |E|) = O(|E| log |V|)
Total: • O(|E| log |V|)
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Prim’s algorithm (also Greedy)
Start with an arbitrary node from V
Instead of maintaining a forest, grow a MST At any time, maintain a MST for V’ V
At any moment, find a light edge connecting V’ with (V-V’) I.e., the edge with smallest weight connecting
some vertex in V’ with some vertex in V-V’ !
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Issues:
Again two issues: Maintain the tree already build at any
moment• Easy: simply a tree rooted at r : the starting
node
Find the next light edge efficiently• For v V - V’, define key(v) = the min
distance between v and some node from V’• At any moment, find the node with min key.
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Pseudo-code
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Prim’s Algorithm in Action
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The minimum cost arc from yellow nodes to green nodes can be found by placing arc values in a priority queue.
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Prim’s Algorithm in Action
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Complexity
Time complexity # insert:
• O(|V|)
# Decrease-Key: • O( |E|)
# Extract-Min• O( |V| )
Using heap for priority queue: Each operation is O ( log |V| )
Total time complexity: O (|E| log |V|)
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Single-Source Shortest Paths
Problem Definition Shortest paths Dijkstra’s algorithm (can be viewed
as a greedy algorithm)
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Problem Definition: Real problem: A motorist wishes to find the
shortest possible route from Chicago to Boston.Given a road map of the United States on which the distance between each pair of adjacent intersections is marked, how can we determine this shortest route?
Formal definition: Given a graph G=(V, E, W), where each edge has a weight, find a shortest path from s to v for some interesting vertices s and v.
s—source v—destination.
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Shortest path: The weight of path p=<v0,v1,…,vk > is
the sum of the weights of its constituent edges:
k
i
ii vvwpw1
1 ),()(
The cost of the shortest path from s to v is denoted as (s, v).
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Representing shortest paths:
we maintain for each vertex vV , a predecessor
[ v] that is the vertex in the shortest path
right before v. With the values of , a backtracking
process can give the shortest path. (We will discuss that after the algorithm is given)
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Dijkstra’s Algorithm:
Dijkstra’s algorithm assumes that w(e)0 for each e in the graph.
maintain a set S of vertices such that Every vertex v S, d[v]=(s, v), i.e., the shortest-path
from s to v has been found. (Intial values: S=empty, d[s]=0 and d[v]=)
(a) select the vertex uV-S such that d[u]=min {d[x]|x V-S}. Set S=S{u} (b) for each node v adjacent to u do RELAX(u,
v, w). Repeat step (a) and (b) until S=V.
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Implementation:
a priority queue Q stores vertices in V-S, keyed by their d[] values.
the graph G is represented by adjacency lists.
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0
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(b)(s,x) is the shortest path using one edge. It is also the shortest path from s to x.
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Time complexity of Dijkstra’s Algorithm:
Time complexity: #Extract-min: O(|V|) #Decrease-key = #Relax: O(|E|) Use binary heap for priority heap:
• Time complexity: O((|V| + |E|) log |V|) Use Fibonacci heap for priority heap:
• Amortized time complexity: O(|V| log |V| + |E|)
Greedy algorithm: Any moment, select the lightest
vertex in V-S to add to set S