“Classic” Examples of Greedy Algorithms - PowerPoint Presentation

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“Classic” Examples of Greedy Algorithms

Minimum spanning tree (MST)

Single source shortest path (SSSP), e.g., Dijkstra’s algorithm

We will explore the main properties, with focus on theoretical foundations

MST:

Graph G(V,E): undirected, connected, weighted (arbitrary real weights w() on edges)

Goal: find acyclic subset

E of edges (acyclic subgraph of G) that:Connects all vertices in V (spanning V; acyclic property implies it is a tree) is minimizedT acyclic, connects all vertices  T is a spanning tree of G, of minimum weight  minimum spanning treeWill discuss two greedy strategies to find MSTIn general, MST for a graph G is not unique

Two greedy algorithms

We will “abuse” notation so V also denotes the

number

of vertices and E also denotes the number of edges (instead of using |V| and |E|)Kruskal’s algorithm: O(E log(V)) time, usesSortingDisjoint set operations (Union, Find)Prim’s algorithm:O(E log (V)) time with binary heap implementationO(E + V log(V)) time with Fibonacci heap implementationWill not discuss Fibonacci heaps in class – see the textbookFocus on theoretical aspects of the problem

Generic MST algorithm

Grows a MST by adding one edge at a time

Captures the greedy strategy (edge added by greedy choice)

Invariant: The set A of added edges is always a subset of some MSTAt each step, finds edge (u,v) to add to ASuch edge (u,v) is called a safe edgeHas V-1 steps (a tree on V vertices has V-1 edges)We are not concerned with the running time of the generic algorithm

Generic MST algorithm

Generic_MST

(

G,w)A  ⏀while A is not a spanning tree of G doFind edge (u,v) safe for AA  A {(u,v)}return ADifficulty: find a safe edgeone must exist sinceA T, for some MST TThus there exists (u,v) T, (u,v) A  (

u,v

) safe for A (by definition)

Need rule to

recognize

safe edges

Definitions:

Cut

(S,V-S) of G(V,E): partition of V into two disjoint subsets

Edge (u,v) crosses the cut if u S, v V-S (or vice versa)Cut (S,V-S) respects A if no edge in A crosses the cut Edge (u,v) is called a light edge for a cut (S,V-S) if:(u,v) crosses the cutw(u,v) is the smallest among all edges crossing the cut: w(u,v) = min{w(x,y) | (x,y) edge crossing the cut}In general, given a cut, a light edge is not unique 

a

c

g

e

d

b

f

8

5

9

5

8

6

4

9

S

V-S

(

b,d

)

and

(

c,e

)

are light

G

Theorem

Given graph G(V,E), let A be a subset of E included in some MST T of G

Let (S, V-S) be a cut of V that respects A (any such cut)

Let (u,v) be a light edge crossing the cut (S,V-S)Then (u,v) is safe for AProof:We know A T and (u,v) A. Then either (a) (u,v) T and we are done (by definition, (u,v) is safe for A) or (b) (u,v) T. Thus, assume (u,v) T. 

Proof (cont.):

(

u,v

) TWe will construct another MST, T’, such that (A (u,v)) T’  (by definition) (u,v) is safe for A.Consider path P(u,v) in T, from u to v:(u,v) forms a cycle if added to P(u,v)u S, v V-S (x,y) P(u,v) that crosses cut (S,V-S) (x S, y

V-S)

Edge (

x,y

)

A, since cut (S,V-S) respects A

Removing (

x,y

) breaks T in two subtrees, adding (

u,v

) reconnects them into another (spanning) tree T’

w(

u,v

)

w(x,y) (both cross cut (S,V-S)), thus w(T’)

w(T) so T’ is a MST of G Remains to prove (u,v) is safe for A (w.r.t. T’)A T, (x,y)

A  A T’ (A (u,v))

T’ , T’ is a MST  (by definition) (u,v) is safe for A 

Observations:

A is always acyclic

At any step of the generic MST algorithm, subgraph G

A(V,A) is a forest each connected component of GA is a treeAt first (A=⏀), GA has V components, each a 1-node treeA safe edge (u,v) connects two distinct components of GAThe loop of Generic_MST:executes V-1 times terminates when GA has one component (GA is a tree)

Next: MST algorithms

MST algorithms use the following

Corollary

: Given G(V,E) let A be a subset of E included in some MST T of G.Let C be a connected component (tree) of the graph GA(V,A)Let (u,v) be a smallest weight edge connecting C to another component (tree) of GAThen, (u,v) is safe for AProof: (C,V-C) is a cut of G that respects A  (u,v) is a light edge for cut (C,V-C)  we have the setup of the Theorem  (u,v) is safe for A