Greedy 2

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Greedy 2

• Jose Rolim
• University of Geneva

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Examples Greedy

• Minimum Spanning Trees
• Shortest Paths Dijkstra

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Definition MST

• Given a connected graph G (V, E), with weight
  function w E --gt 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

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Example
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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

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Kruskals 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

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Pseudo-code
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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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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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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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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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Prims 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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Prims 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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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims Algorithm in Action
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Prims 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
• Dijkstras 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.
• ssource
• vdestination.

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Shortest path

• The weight of path pltv0,v1,,vk gt is the sum of
  the weights of its constituent edges

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 v?V , 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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Dijkstras Algorithm

• Dijkstras algorithm assumes that w(e)?0 for each
  e in the graph.
• maintain a set S of vertices such that
• Every vertex v ?S, dv?(s, v), i.e., the
  shortest-path from s to v has been found. (Intial
  values Sempty, ds0 and dv?)
• (a) select the vertex u?V-S such that
• dumin dxx ?V-S. Set
  SS?u
• (b) for each node v adjacent to u do
  RELAX(u, v, w).
• Repeat step (a) and (b) until SV.

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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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u
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(s,x) is the shortest path using one edge. It is
also the shortest path from s to x.
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(e)
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Time complexity of Dijkstras 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