Greedy algorithms

1
Greedy algorithms

• Optimization problems solved through a sequence
  of choices that are
• feasible
• locally optimal
• irrevocable
• Not all optimization problems can be approached
  in this manner!

2
Applications of the Greedy Strategy

• Optimal solutions
• Change making
• Minimum Spanning Tree (MST)
• Single-source shortest paths
• Simple scheduling problems
• Huffman codes
• Approximations
• Traveling Salesman Problem (TSP)
• Knapsack problem
• Other combinatorial optimization problems

3
Minimum Spanning Tree (MST)

• Spanning tree of a connected graph G a connected
  acyclic subgraph of G that includes all of Gs
  vertices.
• Minimum Spanning Tree of a weighted, connected
  graph G a spanning tree of G of minimum total
  weight.
• Example

4
Prims MST algorithm

• Start with tree consisting of one vertex
• Grow tree one vertex/edge at a time to produce
  MST
• Construct a series of expanding subtrees T1, T2,
  
• At each stage construct Ti1 from Ti add minimum
  weight edge connecting a vertex in tree (Ti) to
  one not yet in tree
• choose from fringe edges (this is the greedy
  step)
• Algorithm stops when all vertices are included

5
Examples

6
Prims algorithm formally

• Algorithm Prim(G)
• VT ? v0
• ET ?
• for i ? 1 to V-1 do
• find a minimum weight edge e(v,u) among
• all the edges (v,u) such that v ? VT and u ?
  V-VT
• VT ? VT ? u
• ET ? ET ? e
• return ET

7
Notes about Prims algorithm

• Need to prove that this construction actually
  yields MST
• Need priority queue for locating lowest cost
  fringe edge use min-heap
• Efficiency For graph with n vertices and m
  edges (n1m) log n
• T(m log n)

insertion/deletion from min-heap
number of stages (min-heap deletions)
number of edges considered (min-heap insertions)
8
Another Greedy algorithm for MST Kruskal

• Start with empty forest of trees
• Grow MST one edge at a time
• intermediate stages usually have forest of trees
  (not connected)
• at each stage add minimum weight edge among those
  not yet used that does not create a cycle
• edges are initially sorted by increasing weight
• at each stage the edge may
• expand an existing tree
• combine two existing trees into a single tree
• create a new tree
• need efficient way of detecting/avoiding cycles
• algorithm stops when all vertices are included

9
Examples

10
Kruskals algorithm

• Algorithm Kruskal(G)
• Sort E in nondecreasing order of the weights
• ET ? k ? 0 ecounter ? 0
• while ecounter lt V-1
• k ? k1
• if ET ? eik is acyclic
• ET ? ET ? eik
• ecounter ? ecounter 1
• return ET

11
Notes about Kruskals algorithm

• Algorithm looks easier than Prims but is
• harder to implement (checking for cycles!)
• less efficient T(m log m)
• Cycle checking a cycle exists iff edge connects
  vertices in the same component.
• Union-find algorithms see section 9.2

12
Shortest paths-Dijkstras algorithm

• Single Source Shortest Paths Problem Given a
  weighted graph G, find the shortest paths from a
  source vertex s to each of the other vertices.
• Doesnt work with negative weights
• Applicable to both undirected and directed graphs

13
Shortest paths-Dijkstras algorithm

• Dijkstras algorithm Similar to Prims MST
  algorithm, with the following difference
• Start with tree consisting of one vertex
• grow tree one vertex/edge at a time to produce
  MST
• Construct a series of expanding subtrees T1, T2,
  
• Keep track of shortest path from source to each
  of the vertices in Ti
• at each stage construct Ti1 from Ti add minimum
  weight edge connecting a vertex in tree (Ti) to
  one not yet in tree
• choose from fringe edges
• (this is the greedy step!)
• algorithm stops when all vertices are included

edge (v,w) with lowest d(s,v) d(v,w)
14
Example

15
Dijkstras algorithm

• Algorithm Dijkstra(G,s)
• Initialize(Q)
• for every vertex v ? V do
• dv?8 pv?null Insert(Q,v, dv)
• ds?0 Decrease(Q,s,ds) VT?0
• for i?0 to V-1 do
• u?DeleteMin(Q) VT ? VT ? u
• for every vertex u in V-VT that is adjacent to
  u do
• if duw(u,u)ltdu
• du?duw(u,u) pu?u Decrease(Q,u,du)
• Efficiency